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Interaction patterns and web-structures of resonant solitons of the kadomtsev-petviashvili equation

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Interaction patterns and web-structures of resonant solitons of the kadomtsev-petviashvili equation
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Tippabhotla, Anupama
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The kp equation
Solitons
Interaction patterns
Spider-web-like structures
Levels of intersection
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ABSTRACT: In this thesis, the interaction pattern for a class of soliton solutions of the Kadomtsev- Petviashvili (KP) equation is analyzed. The complete asymptotic properties of the soliton solutions for are determined. The resonance characteristic of two sub-classes of the soliton solutions, in which incoming line solitons for interact to form outgoing line solitons for , is described. These two specific sub-classes of -soliton solutions are the following: 1) (2, 3), (2, 4), (2, 5), 2) (3, 2), (3, 3), (3, 4). The intermediate solitons and the interaction regions of the above soliton solutions are determined, and their various interaction patterns are explored. Maple and Mathematica are used to get the 3 dimensional plots and contour plots of the soliton solutions to show their interaction patterns. Finally, the spider-web-structures of the discussed solitons of the KP equation are displayed.
Thesis:
Thesis (M.A.)--University of South Florida, 2005.
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by Anupama Tippabhotla.
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Interaction Patterns and Web-structures of Resonant Solitons of the Kadomtsev-Petviashvili Equation by Anupama Tippabhotla A thesis submitted in partial fulfillment of the requirements for the degree of Master of Arts Department of Mathematics College of Arts and Sciences University of South Florida Major Professor: Wen-Xiu Ma, Ph.D. Yuncheng You, Ph.D. Athanassios G. Kartsatos, Ph.D. July 8, 2005 Keywords: The KP equation, Solitons, Interaction Patterns, Spider-web-like structures, levels of Intersection. Copyright 2005, Anupama Tippabhotla

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ACKNOWLEDGEMENTS It is my great pleasure to thank the many people whose help and suggestions were so valuable in completing this thesis. My greatest acknowledgement is extended to my advisor, Dr. Wen-Xiu Ma for his guidance, support and encouragement; without his help, so generously given, the completion of this thesis would just not have been possible. I would like to thank my committee members Dr. Yuncheng You and Dr. Athanassios G. Kartsatos for their direct and indirect contribution throughout this investigation. I am indebted to my family, especially my parents Murty and Anita for their encouragement, without which I would not have come so far. I would also like to appreciate the help and motivation given by my friends Jagadeesan Siva, Dilip Balachnadran, Sridhar Veeravalli and Maanasa Devi.

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DEDICATION To my brother Siva Kumar

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Table of Contents List of Figures ii Abstract iii 1. Introduction 1.1. Discovery of solitary waves 1 1.2. Objectives of this thesis 2 1.3. Summary 3 2. Basic Facts 2.1. The KP equation and its Wronskian solutions 4 2.2. Asymptotic analysis of the solutions 6 3. Interaction Patterns and Web-structures 3.1. Analysis of the (2,3)-soliton solution 19 3.2. Analysis of the (2,4)-soliton solution 22 3.3. Analysis of the (2,5)-soliton solution 26 3.4. Analysis of the (3,2)-soliton solution 30 3.5. Analysis of the (3,3)-soliton solution 34 3.6. Analysis of the (3,4)-soliton solution 39 References 44

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List of Figures Figure 1. Levels of interaction of (1, 2)-soliton 15 Figure 2. 3-dimensional pattern of (1, 2)-soliton 16 Figure 3. Contour plot of the interaction pattern of (1, 2)-soliton with () = (-1, 0, 1) at t=(-20, 0, 20) 17 321,,kkk Figure 4. Contour plot of the interaction pattern of (1, 2)-soliton with = -3 at t=20 17 1k Figure 5. Contour plot of the interaction pattern of (1, 2)-soliton with = 3 at =20 18 3k t Figure 6. Contour plot of the interaction pattern of (1, 2)-soliton with ( 321,,kkk ) = (-1,-2,-3) at t=20 18 Figure 7. Contour plot of the interaction pattern of (1, 2)-soliton with ( 321,,kkk ) = (1, 2, 3) at t=20 19 Figure 8. Levels of interaction of (2, 3)-soliton 20 Figure 9. Contour plot of the interaction pattern of (2, 3)-soliton with () = (-3, -2, -1, 1, 2) at t= (-20, 0) 21 54321,,,,kkkkk Figure 10. The interaction pattern of the intermediate solitons of the (2, 3)-soliton solution 21 Figure 11. Levels of interaction of (2, 4)-soliton 23 Figure 12. The interaction pattern of the intermediate solitons of the (2, 4)-soliton solution 23 Figure 13. Contour plot of the interaction pattern of (2, 4)-soliton with () = (-3,-2,-1, 1, 2, 3) at 654321,,,,,kkkkkk = (-20, 0, 20) 24 t Figure 14. Contour plot of the interaction pattern of (2, 4)-soliton with = -5 at =20 25 1k t

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Figure 15. Contour plot of the interaction pattern of (2, 4)-soliton with = 5 at =20 25 6k t Figure 16. Levels of interaction of (2, 5)-soliton 27 Figure 17. The interaction pattern of the intermediate solitons of the (2, 5)-soliton solution 27 Figure 18. Contour plot of the interaction pattern of (2, 5)-soliton with () = (-3-2,-1, 0 1 2, 3) and 7654321,,,,,,kkkkkkk , t= (-10, 20 0) 28 Figure 19. Contour plot of the interaction pattern of (2, 5)-soliton with = -10 29 1k Figure 20. Contour plot of the interaction pattern of (2, 5)-soliton with = 5 29 7k Figure 21. Levels of interaction of (3, 2)-soliton 31 Figure 22. The interaction pattern of the intermediate solitons of the (3, 2)-soliton solution. 31 Figure 23. Contour plot of the interaction pattern of (3, 2)-soliton ( 54321,,,,kkkkk ) = (-3, -2, -1, 1, 2) at t = (-20, 0, 20) 32 Figure 24. Contour plot of the interaction pattern of (3, 2)-soliton = -5 at t = -20 33 1k Figure 25. Contour plot of the interaction pattern of (3, 2)-soliton = -5 and = -9/2 at = -20 33 1k 2k t Figure 26. Contour plot of the interaction pattern of (3, 2)-soliton = 5 at = -20 34 6k t Figure 27. Levels of interaction of (3, 3)-soliton 36 Figure 28. Interaction pattern of (3, 3)-soliton 36 Figure 29. Contour plot of the interaction pattern of (3, 3)-soliton with () = (-3,-2,-1, 1 2, 3) and 654321,,,,,kkkkkk t= (-10, 0, 10) 37 Figure 30. Contour plot of the interaction pattern of (3, 3)-soliton with = -10 at t= 10 38 1k

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Figure 31. Contour plot of the interaction pattern of (3, 3)-soliton with = -10 at t= 10 38 1k Figure 32. Levels of interaction of (3, 4)-soliton 40 Figure 33. Levels of interaction of (3, 4)-soliton 40 Figure 34. Contour plot of the interaction pattern of (3, 4)-soliton ( 7654321,,,,,,kkkkkkk ) = (-3, -2, -1, 0, 1, 2, 3) at t = (-20, 0, 20) 41 Figure 35. Contour plot of the interaction pattern of (3, 4)-soliton = -6 at t = 20 42 1k Figure 36. Contour plot of the interaction pattern of (3, 4)-soliton = 6 at = 20 42 1k t Note: In all the pictures, the horizontal and vertical axes denote and x y axes, respectively.

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Interaction Patterns and Web-structures of Resonant Solitons of the Kadomtsev-Petviashvili Equation Anupama Tippabhotla ABSTRACT In this thesis, the interaction pattern for a class of soliton solutions of the KadomtsevPetviashvili (KP) equation is analyzed. The complete asymptotic properties of the soliton solutions for 03)64(yyxxxxxtuuuuu y are determined. The resonance characteristic of two sub-classes of the soliton solutions, in which incoming line solitons for interact to form outgoing line solitons for N y N y is described. These two specific sub-classes of -soliton solutions are the following: ),(NN 1) {(2, 3), (2, 4), (2, 5)}, 2) {(3, 2), (3, 3), (3, 4)}. The intermediate solitons and the interaction regions of the above soliton solutions are determined, and their various interaction patterns are explored. Maple and Mathematica are used to get the 3 dimensional plots and contour plots of the soliton solutions to show their interaction patterns. Finally, the spider-web-structures of the discussed solitons of the KP equation are displayed.

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1. Introduction Waves and vibrations are among the most basic forms of motion, and their study goes a very long way back in time. Small amplitude waves are described mathematically by a linear differential equation, and their behaviour can be studied in detail. In contrast, when the amplitude is not restricted to being small, the differential equation becomes nonlinear, and its analysis becomes in general an extremely difficult problem. Localized large amplitude waves called solitons represent one of the most striking aspects of nonlinear phenomena, though there exist other types of phenomena like positons [14] and complexitons [10]. Solitons propagate without spreading and have particle-like properties, which retain their identities in a collision. During the collision of solitons the solution cannot be represented as a linear combination of two soliton solutions, but after collision solitons recover their shapes and the only result of collision is phase shift. 1.1 Discovery of solitary waves The solitary wave[16,20] was first observed by J. Scott Russell on the Edinburgh-Glasgow canal in 1834; he called it the great wave of translation. Russell reported his observations to the British association in his 1844 Report on Waves in the following words: I believe I shall best introduce the phenomenon by describing the circumstances of my own first acquaintance with it. I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still on at a rate of some 8 or 9 miles an hour, preserving its original figure some 30 feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of 1 or 2 miles, I lost it in the windings of the channel. Russell also performed some laboratory experiments, generating solitary waves by dropping a weight at one end of a water channel. Mikio Sato was the first to discover that the Kadomtsev-Petviashvili (KP) equation is the most fundamental among many soliton equations. Sato discovered that polynomial solutions of the bilinear KP equation are equivalent to the characteristic polynomials of the general linear group. Later, he found a Lax pair for a hierarchy of KP-like equations by means of a pseudo-differential operator and came to the conclusion that the KP equation is equivalent to the motion of a point in a Grassmanian manifold and its bilinear equation is nothing but a plucker relation. Also, Junikichi Satsuma had discovered before Sato that the soliton solutions of the KdV equation could be expressed in terms of Wronskian determinants [19,13]. Later, in 1983, Freeman and Nimmo found that the KP bilinear 1

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equation could be rewritten as a determinantal identity if one expresses its soliton solutions in terms of Wronskian. The KdV equation is a 1+1-dimensional equation describing shallow water waves. The KP equation was introduced in order to discuss the stability of shallow water waves to perpendicular horizontal perturbations. Some physicists may object to the idea that the KP equation is the most fundamental of the soliton equations. First of all, the KdV[14] equation is derived through a certain approximation, and then the KP equation is obtained from the KdV equation under the assumption that horizontal perturbations are small. The KP equation is fundamental because of the simple mathematical structure of its solutions and its relation to the other soliton equations arising from this simplicity. The KP equation is the 2+1-dimensional (two-dimensional space, ( y x ) plus onedimensional time, t) nonlinear partial differential equation: 03)64(yyxxxxxtuuuuu Since we obtain the KdV equation (which is the most celebrated equation in the integral equations) by neglecting the y-derivative term, this is also called the two-dimensional KdV equation (shallow water wave equation proposed at the end of ninetieenth century). At present, a huge number of concrete examples of integrable nonlinear differential (and difference) equations are known. Many techniques for finding exact solutions of these equations have also been discovered: inverse scattering theory[6] which solves the initial value problem, the bilinear method initiated by Ryogo Hirota, the theory of quasi periodic solutions based on Riemann surface and theta functions [1,4,12,13,18], etc. The investigation of soliton-like structures in realistic and intricate multidimensional world obviously is of great interest. The solitons in more dimensions have been found experimentally and theoretically in many branches of physics and applied mathematics. They exist in the ocean as waves bombarding oil wells, also exist in much smaller natural and laboratory systems as plasmas, molecular systems, laser pulses propagating in solids, super fluid helium magnetic system, structural phase transitions, liquid crystals, polymers and fluid flows as well in elementary particles [2]. 1.2 Objectives of this thesis The aim of this thesis is to study the interactions, structures, resonance and asymptotic properties for a class of soliton solutions of the KP equation, where and denote the numbers of incoming and outgoing line solitons, respectively. We will observe the changes in the interaction patterns and web-structures by varying the values of parameters in the above soliton solutions of the KP equation. We will find the intermediate solitons based on the work of [9], and describe the interaction between various intermediate solitons. Here we will be considering the folllowing two specific sub-classes of -soliton solutions: N N ),(NN 1) {(2, 3), (2, 4), (2, 5)}, 2) {(3, 2), (3, 3), (3, 4)}. By considering the idea of fundamental resonance [9], we describe the intermediate and 2

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interaction patterns of the resonant solitons in theplane. Then we will analyze the asymptotic properties of the two above mentioned sub-classes of -soliton solutions in this thesis. ),(yx ),(NN 1.3 Summary In the chapter 1, we have seen the introduction about solitons, discovery of solitary waves and some applications of solitons. In the chapter 2, we will study some basic definitions and theorems required to understand this thesis. Most of the results of this chapter are from Ref [9]. Finally in the chapter 3, we will describe the actual interaction patterns, analysis of the asymptotic properties and web-structures of two specific sub-classes of -soliton solutions of the KP equation. ),(NN 3

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2. Basic Facts 2.1 The KP equation and its Wronskian solutions The aim of this chapter is to study a family of soliton solutions of the Kadomtsev-Petviashvili (KP) equation 03)64(yyxxxxxtuuuuu which can be transformed into the bilinear form 03424yxtxDDDD, Here, and are the Hirota derivatives, for example, the Hirota derivative is defined as mxD myD mtD mxD (2.2) (2.1) ''|),,(),,()('xxmxxmxtyxgtyxfgfD and uis obtained from the tau-function ),,(tyx through ),,(log2),,(22tyxxtyxu It is well known that some solutions of the KP equation can be obtained by the Wronskian form = M with ([3,6,7 13]) (2.3) )1()1(1)0()0(11:),...,(MMMMMrMffffffW, where and ninnixff/)( },...,1|),,({Mityxfi is a linearly independent set of M solutions of the equations (2.4) (2.5) yfi = 22 x fi tfi = 33 x fi for .1Mi For example, the 2-soliton solution of the KP equation is obtained by the set with ),(21ff (2.6) iieefi212 2,1 i where the phases j are given by linear functions of ),,(tyx ,),,(032jjjjjtkykxktyx 4,...,1 j (2.7) with and s being constant. 4321kkkk jk This ordering is sufficient for the solution to be nonsingular. The ordering is needed for the positivity of u 4321kkkk 2 Note, for example that if 4

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4321kkkk 2 takes zero and the solution blows up at some points in formula (2.6) can be extended to the M -soliton solution with {}[17]. Mff,...,1 On the other hand, it is also known that the solutions of the finite Toda lattice hierarchy are obtained by the set of tau-functions }...1|{MiM with the choice of -functions, f ,:,1,11)1(fefNMiffNiiii (2.8) where the phases i are given in the form (2.7)[ 8]. This implies that each tau-function Ni1 M gives a solution of the KP equation. If the -functions are chosen according to (2.8), the tau-functions are then given by the Hankel determinants f )22()()1()()2()1()1()1()0(MMMMMMfffffffff (2.9) for Note here that NM1 M = )...exp(1Nc with =constant, yielding the trivial solution. Note also that c M and MN produce the same solution with the symmetry due to duality of the determinants. ),,(),,(tyxtyx In this thesis, we are concerned with the behavior of the KP solutions (2.3) whose tau functions are given by (2.9). We describe the patterns of the solutions in the y x plane where each soliton solution of the KP equation is asymptotically expressed as a line, namely, ycx for y with appropriate constants and c for a fixed t. In particular, we find that all the solutions are resonant solitons in the sense that these solutions are different from ordinary multi-soliton solutions. The difference appears in the interaction patterns. Suppose the numbers of line solitons in asymptotic stages as y are denoted by respectively. Therefore, the number of incoming solitons is represented by and the number of outgoing solitons is represented by which is given by the size of the hankel determinant (2.9), i.e., = N N N N M Thus, the total number of exponential terms in the function in (2.8) gives the total number of solitons presented in both asymptotic limits, i.e., = (+ ). We call these solutions (, )-solitons. N f N N N N N 5

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In particular, if = 2= 2, N N N the solution describes an N soliton having the same set of line solitons in each asymptotics for y However these multi-soliton solutions also differ from the ordinary multi-soliton solutions of the KP equation. The ordinary -soliton solution of the KP equation is described by n intersecting line solitons with a phase shift at each interaction point. If we ignore the phase shifts, these lines form ( 1)( 2)/2 bounded regions in the generic situation. However, the number of bounded regions for the (resonant) -soliton solution with (2.9) is found to be (-1) n n n n N N 2 ; We will see later that for the case of a (, )-soliton solution, the number of bounded regions(holes) in the graph of the solution is given by (1)(1), except at finite values of t in the temporal evolution. N N N N These resonant [6] -soliton solutions are similar to some of the solitons of the coupled KP (cKP) hierarchy recently studied. These solutions are called spider-web-like [8] solutions. The analysis of finding web structure that we describe in this thesis may also be applied to the case of cKP hierarchy. N 2.2 Asymptotic analysis of the solutions In the plane, the solution u defined by (2.3), (2.4) and (2.8) with = 2 describes a plane wave as ),(yx N )]([tykxkuxx having the wavenumber vector ),(yxkkk and the frequency 2,1222121:),(kkkkkk 2,13231:kk Here ),( k satisfies the dispersion relation, (2.10) 03424yxxkkk We refer to the one-soliton solution as a line soliton, which can be expressed by a (contour) line, 1 = 2 in the plane. In this paper, since we discuss the pattern of soliton solutions in the plane, we refer to c= as the velocity of the line soliton in the direction; that is, = 0 indicates the direction of the positive -axis. The results of this section are from Ref [9]. We list and discuss them for later reference in chapter 3. ),(yx ),(yx dydx/ x c y Theorem 2.2.1 Let be given by with Then for =+ and 1 f Niief1 032iiiiitkykxk N N N N 1, the tau-function defined by the Hankel determinant has the form N )exp(),...,(...1111NiiNjNNNjieii where is the square of the VanderMonde determinant ),...,(1Nii 6

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.)(),...,(211ljiNljiNkkii Proof. Apply the Binet-Cauchy theorem for .112111121121111det212121NNNNNNNNNNNNNkkkkkkekekekekekekeeeNNN One should note from that the N -function contains all possible combinations of phases from the set N },,...,1|{Njj unlike the case of ordinary multi-soliton solutions of the KP equation. For example, the 2 -function for the 2-soliton solution with include only four terms, and is missing the combinations 21 and 43 This makes a crucial difference on the interaction patterns of soliton solutions, as explained in this thesis. In particular, we will see that the (,)-solitons are all of resonant type[15] in the sense that the local structure of each interaction point in those solitons consists of either (2, 1)or (1, 2)-solitons. N N Theorem 2.2.2 The N function given by (2.9) is positive definite, and therefore, the solution has no singularity. In general, the Wronskian takes zeros at some points in the flow parameters. u Let us now define a local coordinate frame ),(y in order to study the asymptotic behaviour for large || with y cyx (2.10) Then the phase functions i in of (2.10) becomes f 0)(iiiiyck for Ni,...,1 with )(:)(ckkciii Without loss of generality, we assume the ordering for the parameters as follows: },...,1|{Niki Nkkk ...21 Then one can easily show that the lines )(ci are in general position; that is, each line )(ci intersects with all other lines at 1 distinct points in the N ),( c plane; in other words, only two lines meet at each intersection point. Now the purpose is to find the dominant exponential terms in the N -function for y as a function of the velocity First note that if only one exponential is dominant, and then c Nx log1 is just a 7

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constant, and therefore the solution 12 xu is zero. Then, nontrivial contributions to arise when one can find two exponential terms, which dominate over the others. Note that because the intersections of the u i s are always pair wise, three or more terms cannot make a dominant balance for large | y |. In the general case, 1, the N N -function involves exponential terms having combinations of phases, and two exponential terms that make a dominant balance can be found as follows. Let us first define the level of intersection of )(ci Definition 2.2.3 Let )(ci and )(cj intersect at the value jijikkcc , i.e., )()(,,jijjiicc The level of intersection, denoted by ji, is defined, as the number of other j s that at are larger than jicc, )()(,,jijjiicc That is, .|)}()()(|{|:,,,,jijjiijilljiccc We also define as the set of pairs having the level )(nI ),(jinn ji, = n, namely ,|),{(:)(,nnIjiji for }ji The level of intersection can take the range 0 ji, N 2. Then one can show: Theorem 2.2.4 The set is given by )(nI }.1,...,1|),{()(1 ninIinNi Proof. From the assumption Nkkk ...21 we have the following inequality at (i.e., jicc, ji ) for ji Njijiji ,...,,,...,,...,11111 Then taking 1 nNj leads to the assertion of the theorem 2.2.4. Note here that the total number of pairs is ),(jinn 20|)(|)1(212NnnINNN For the case of (,)-solitons, the following formulae are useful: N N }.,...,1|),{()1(},,...,1|),{()1(NiNINiNIiNiiNi Here recall that + =. These formulae indicate that, for each intersecting pair with the level 1 ( 1), there are 1 terms N N N ),(jinn N N N i s that are smaller (larger) than ji Then the sum of those 1 terms with either N j or i provides two dominant exponents in the N -function for (see more detail in the proof of Theorem 2.2.5). Note also that )(yy |)1(| NI N Now we can state our main theorem: Theorem 2.2.5 Let 1 be a function defined by ,log1 N 8

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with N being given by (2.10). Then 1 has the following asymptotic for: y For and y ycxiNi, for Ni,...,1 111;:),(,:),(iNijjiiNijjiaskKaskK and for y and ycxiNi, for Ni,...,1 ijiNjijNjiijiNjjNjiaskkKaskkK11111111;:),(,:),( where jijikkc, Proof. + i, i.e., I( 1), from the theorem 2.2.4 we have the inequality, N ),(jNinn N .,...,,1121iNiNNiii This implies that, for the following two exponential terms in the jNijikkc, N+ -function in theorem 2.2.1, ),exp(1iNijj ),exp(1iNijj provide the dominant terms for y Note that the condition iNi leads to Thus the function jNijNikkcc, 1 can be approximated by the following form along ycxiNi, for : y )),(),(log(~),(),(1iiKiKiee ),(),(),(),(),(),(),(),(),(),(iiiiKiKiKiiKiieeeKeK ,),(),(),(),(),(),()()(iKKiiiKKiiiiNiiNeKeK where )exp()1,...,(),(10iNijjiiNi ).exp(),...,1(),(10iNijjiiNi 9

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Now, from it is obvious that iNikk 1 has the desired asymptotics as for y Similarly, for the case of ( iNi ) I ( 1) we have the inequality N .,...,,,...,,11121 NNiNiiNi Then the dominant terms in the N -function on iNicx , y for are given by the exponential terms y ),exp(11iNijjNijj ).exp(1111iNijjNijj Then, following the previous argument, we obtain the desired asymptotics as for y For other values of c, i.e., iNicc and iNicc , just one exponential term becomes dominant, and thus 1 approaches a constant as ||y This completes the proof. Theorem 2.2.6 The above theorem can be summarized: i) As y the function 1 has jumps, moving with velocities for ; as N jNjc, Nj,...,1 y 1 has jumps, moving with velocities for N iNic, Ni,...,1 ii) Each jump represents a line soliton of the u-solution, and therefore the whole solution represents an (, )-soliton. N N iii) Each velocity of the asymptotic line solitons in the (, )-soliton is determined from the N N c graph of the levels of intersections. iv) This theorem determines the complete structure of asymptotic patterns of the solutions given by (2.3) for the Toda lattice equation. In the case of the ordinary multi-soliton solution of the KP equation, the tau-function (2.4) does not contain all the possible combinations of phases, and therefore the theorem should be modified. ),,(tyxu The key idea for the asymptotic analysis of using the levels of intersection is still applicable of ordinary multi-solitons. In fact, one can find from the same argument that the asymptotic velocities for the ordinary M -solitons are given by where the iiiikkc2122,12 M -function is the Wronskian with for iieefi212 Mi,...,1 and Note that the velocities are different from those of the resonant Mkkk221... M -soliton solution. Theorem 2.2.7 In the generic situation, the number of holes (bounded regions) in the graph of the (,)-soliton solution is ( 1)( 1). N N N N Proof. We use mathematical induction. The case = 1 corresponds to the Burgers equation, and it is immediate to show that the graph of the (, 1)-soliton solution has a tree shape; that is, no holes. Now suppose that the (, )-soliton has ( 1) ( N N N N N N 10

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1) holes. Add a new phase 1N with satisfying 1Nk 11... NNkkk which produces a new, fastest, incoming [+ 1, + 1] soliton, and assume that this solution intersects with the [1, +1] soliton, which is the slowest outgoing soliton. Then the resonant process of those solitons generates a [1, + 1] soliton as a (2, 1) process, which then intersects with the new slowest [1, + 2] soliton to generate an intermediate [+2, +1] soliton. This intermediate soliton interacts with the second slowest outgoing soliton, the [2, +2] soliton, to generate N N N N N N N N [2, +3] and [+3, +1] solitons, and so on. It is obvious that there are 1 newly created holes; that is, if (,) (+ 1, ), the number of holes increases as N N N N N N N N )1()1()1()1()1)(1( NNNNNNN The case of the (, + 1) solution can be analyzed in the same way using the duality of the determinants. This completes the proof. N N Theorem 2.2.8 In the generic situation for + = 3, the total numbers of intersection points and intermediate solitons in a (, )-soliton solution are respectively given by 2 and 3 N N N N N N N N N N 2. N Proof. By applying mathematical induction one can easily find that the number of new vertices (intersection points) is 2 1 and that of new intermediate solitons is 3 2. This yields the desired results. N N One should compare these numbers with the case of ordinary M -soliton solution, where the total numbers of holes and intersection points are 1/2 ( M 1)( M 2) and 1/2 M ( M 1), respectively. The resonant process blows up each vertex in an ordinary M -soliton solution to create a hole, so that the total number of holes in a resonant M -soliton solution is given by 2)1()1(21)2)(1(21MMMMM Note also that the total number of vertices in a resonant M -soliton is four times of the vertices of an ordinary M -soliton, i.e., each vertex is blown up to make 4 vertices with one hole. Finally, we would like to point out that the KP equation has a large variety of multi-soliton-type solutions. Among those solutions, we found that, since the M -function of the resonant (,)-soliton for the Toda lattice hierarchy contains all possible combinations of phase terms N N },...,1|{Nii the interaction process for these solutions results in a fully resonant situation. On the other hand, the ordinary M -soliton solutions display a nonresonant case; that is, resonant triangles representing either (2, 1)or (1, 2)11

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solitons cannot be formed because of the missing exponential terms in the tau-function [17,19]. One can then find a partially resonant case consisting of ordinary multi-soliton interaction with the addition of some resonant interactions; one such example is the case having and for the 3211eeef 432eef 2 -function where the ordinary 2-soliton interaction coexists with resonant interactions. We will discuss the details of the general patterns for 2 specific classes of multi-soliton solutions for the KP equation in the next chapter. 12

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3. Interaction Patterns and Web-structures First we would like to start with the structure of (1, 2)-soliton solution case. We describe the intermediate patterns of the resonant solitons in the plane. The key idea is to consider the pattern as a collection of fundamental resonances. The fundamental resonance consists of three parameters: that is, the case of ),(yx },,{321kkk N = 3 with | | = 1. Without loss of generality, let us take = 1 and = 2, i.e., a (1,2)-soliton. Then, with the pattern of the fundamental resonance is a N N N N 321kkk Y -shape graph. Here and in the following we denote with the asymptotic line soliton with. Notice that ],[ji jijikkcc, )1( NI = = { )0(I ),(21 } and = = { )1(NI )1(I ),(21 ,) ,(31 }. One should note that at the vertex of the Y -shape graph each index appears exactly twice as the result of resonance, and in figure those vertices form a triangle, which we refer to as a resonant triangle. The resonant triangle is equivalent to the resonance condition for the wavenumber vectors. Since the vertex of the Y -shape graph consists of three line solitons, i = j 31 ji the location of the vertex is obtained from the solution of the equations 1 = 2 = 3 Note here that the coefficient matrix is nonsingular for 321kkk and the location is uniquely determined by a function of This implies that there always exists a ),(yx t Y -shape graph if there are three line solitons satisfying the resonance conditions given in chapter 2. Since the N -function (2.9) contains all possible combinations of phases, all the vertices in the graph form N Y -shape intersections as a result of dominant balance of three exponential terms in the N at each vertex. One should also note that a vertex with 4 or more line solitons is not generic: A vertex with distinct line solitons is obtained from the system of equations, m m },...,1,|{mkjikkjkik in which at least 1 equations are linearly independent. Then for 4, this system in m m x and y is overdetermined so that the solution exists only for specific choices of 0i for fixed values of In the cases of both ordinary and resonant 2-soliton solutions, the two pairs of solitons as are the same, and therefore there are only two independent equations. t y 13

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The ordinary 2-soliton solution needs a balance of four exponential terms to realize an X -shape vertex. It was shown that the X -shape vertex of an ordinary 2-soliton solution is blown up into a hole with four Y -shape vertices for the resonant 2-soliton solution. Now we discuss the asymptotic analysis of the (1, 2)-soliton solution. First, we consider the point 2ii i.e., )0(),(2Iii Then we get the inequality from the theorem (2.2.4) as iii21 and 2 iikkc Therefore the 2 exponential terms ),exp(12iijj ),exp(21iijj which are dominant in the 2 function for y Also the condition 2ii leads to Thus the function 22,iiiikkcc 1 will be approximated by the following form along ycxii2, for y )),(),(log(~),(),(1iiKiKiee ),(),(),(),(),(),(),(),(),(),(iiiiKiKiKiiKiieeeKeK ,),(),(),(),(),(),()()(22iKKiiiKKiiiiiieKeK where ),exp()12,...,(),(12iijjiii ).exp()2,...,1(),(21iijjiii Since iikk2 1 has the asymptotics as mentioned in the theorem 2.2.5. Now we proceed similarly for the case )1(),(1Iii and we have the inequality 11 ii on ycxii1, for y and So the dominant terms are ),exp(2131iijjijj ).exp(121311iijjijj 14

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For all the other values of just one exponential term becomes dominant, so c 1 approaches a constant as Therefore the function y 1 has one jump moving with velocities for as jjc2, 1j y When y 1 has two jumps moving with velocities for Every jump represents a line soliton solution which represents the whole solution of a (1, 2)-soliton. The velocity of each line soliton in [1, 2] soliton is found by the graph of levels of interactions. So here we have 1 incoming solitons having the velocities corresponding to the set and 2 outgoing solitons with the velocities corresponding to the set iic1, 2,1i u 3,1c )0(I 2,1c 3,2c )1(I Now we construct the graph of the levels of interaction for the (1, 2)-soliton case. The level of interaction will take the range 30, ji a) The circle at the level corresponds to the incoming soliton, and the diamonds at the level correspond to the outgoing solitons. )0(I )1(I Figure 1. Levels of interaction of (1, 2)-soliton case b) 3-dimensional pattern 15

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Figure 2. 3-dimensional pattern of (1, 2)-soliton The Y-shape graph illustrates the fundamental resonance with the parameters () where 321,,kkk 321kkk Now we shall see the effect of parameters on the shape and region of the interaction pictures. b) If () = (-1, 0, 1) at t= (-20, 0, 20) then the interaction pictures are: 321,,kkk -100 -50 0 50 100 150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -100 -50 0 50 100 150 16

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-150 -100 -50 0 50 100 -100 -50 0 50 100 150 Figure 3. Contour plot of the interaction pattern of (1, 2)-soliton with () = (-1, 0, 1) at t=(-20, 0, 20) 321,,kkk c) Suppose we decrease the value of one of the parameters, for example, we set = -3 keeping t=20 then the interaction picture is : 1k -200 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 Figure 4. Contour plot of the interaction pattern of (1, 2)-soliton with = -3 at =20 1k t d) Suppose we increase the value of one of the parameters, for example, we set = 3 keeping t=20 then the interaction picture is: 3k 17

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-150 -100 -50 0 50 -50 0 50 100 150 Figure 5. Contour plot of the interaction pattern of (1, 2)-soliton with = 3 at =20 3k t e) Suppose we set all the values of the parameters negative, for example, we set () = (-1,-2,-3) keeping =20 then the interaction picture is: 321,,kkk t -100 0 100 200 300 -200 -100 0 100 200 Figure 6. Contour plot of the interaction pattern of (1, 2)-soliton with () = (-1,-2,-3) at =20 321,,kkk t f) Suppose we set all the values of the parameters positive, for example, we set () = (1, 2, 3) keeping =20 then the interaction picture is: 321,,kkk t 18

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0 100 200 300 0 25 50 75 100 125 150 Figure 7. Contour plot of the interaction pattern of (1, 2)-soliton with () = (1, 2, 3) at =20 321,,kkk t 3.1 Asymptotic analysis of the (2, 3)-soliton solution By using the Theorem 2.2.5 we analyze the solution as follows: For and y ycxii3, for 2,1 i 13311.:),(,:),(iijjiiijjiaskKaskK For and y ycxii2, for 3,2,1i ijijjjiijijjjiaskkKaskkK1311511131151.:),(,:),( Therefore the function 1 has two jumps moving with velocities jjc3, for as When 2,1j y y 1 has three jumps moving with velocities jjc2, for 19

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3,2,1i Every jump represents a line soliton of the solution u which represents the whole solution of a (2, 3)-soliton. The velocity of each line soliton in (2, 3)-soliton is found by the graph of levels of interactions. So here we have two incoming solitons having the velocities 4,1c 5,2c corresponding to the set and three outgoing solitons with the velocities ,, corresponding to the set )1(I 3,1c 4,2c 5,3c )2(I The level of interaction can take the range 30, ji Then the set is given by )(nI = { )(nI ),(15ini for }1,...,1 ni and 3, 2, 1,0n Therefore the total number of pairs of ),(ji is (1/2.5.4)=10, i.e., = )0(I ) ,{(51 }, = )1(I ),{(41 )},(52 = )2(I )} ,(),,(),,{(534231 = )3(I )} ,(),,(),,(),,{(54433221 The number of holes in the graph of the (23)-soliton solution is 2. The total number of intersection points and intermediate solitons are 7 and 8 respectively. Now we construct the graph of the levels of interaction for the (2 3)-soliton case: Figure 8. Levels of interaction of (2, 3)-soliton In the above levels of interaction picture the level stars represent the incoming line solitons and the level -boxes represent the outgoing line solitons. )1(I )2(I Now we shall see the effect of parameters on the shape and region of the interaction picture of the (2,3)-soliton solution: 20

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b) When () = (-3, -2, -1, 1, 2) at t= (-20, 0) then the interaction pictures are: 54321,,,,kkkkk -200 -100 0 100 200 -100 -50 0 50 100 150 -200 -100 0 100 -150 -100 -50 0 50 100 150 Figure 9. Contour plot of the interaction pattern of (2, 3)-soliton with () = (-3, -2, -1, 1, 2) at 54321,,,,kkkkk t = (-20, 0) [1,2] -200 -100 0 100 -150 -100 -50 0 50 100 150 [3,4][2,4][1,4][1,5][3,5][1,3][2,5][2,3][4,5][3,5][2,4][2,5][1,2] -200 -100 0 100 -150 -100 -50 0 50 100 150 -200 -100 0 100 -150 -100 -50 0 50 100 150 [3,4][2,4][1,4][1,5][3,5][1,3][2,5][2,3][4,5][3,5][2,4][2,5] Figure 10. The interaction pattern of the intermediate solitons of the (2, 3)-soliton solution We can observe the following facts: 1) The region of interaction widens on increasing the variable t from -20 to 0 keeping all the parameters () constant. ik 21

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2) The interaction becomes weak and the interaction pattern vanishes as if the value of one of the parameter is decreased. 3) If all the value are of parameters () are negative or positive then interaction pattern vanishes. ik 3.2 Asymptotic analysis of the (2, 4)-soliton solution By using the Theorem 2.2.5 we can analyze the solution as follows: For and y ycxii4, for 2,1i 14411.:),(,:),(iijjiiijjiaskKaskK For and y ycxii2, for 4,3,2,1i ijijjjiijijjjiaskkKaskkK1411611141161.:),(,:),( Therefore the function 1 has two jumps moving with velocities jjc4, for as When 2,1j y y 1 has four jumps moving with velocities iic2, for Every jump represents a line soliton solution which represents the whole solution of a (2, 4)-soliton. The velocity of each line soliton in (2, 4)-soliton is found by the graph of levels of interactions. So here we have two incoming solitons having the velocities 4,3,2,1i u 5,1c 6,2c corresponding to the set and four outgoing solitons with the velocities )1(I 3,1c 4,2c 5,3c 6,4c corresponding to the set )3(I The level of interaction can take the range 40, ji Then the set is given by )(nI = for )(nI ),{(16ini }1,...,1 ni and 4,3,2,1,0n Therefore the total number of pairs of ),(ji is 15: = )0(I )},{(61 = )1(I )},(),,{(6251 = )2(I )},(),,(),,{(635241 = )3(I )},(),,(),,(),,{(64534231 = )4(I )},(),,(),,(),,(),,{(6554433221 The number of holes in the graph of the (2, 4)-soliton solution is 3. The total number of intersection points and intermediate solitons are 10 and 12 respectively. Now we construct the graph of the levels of interaction for the [2, 4] case: 22

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Figure 11. Levels of interaction of (2, 4)-soliton In the above levels of interaction picture the level -stars represent the incoming line solitons and the level -diamonds represent the outgoing line solitons. )1(I )3(I -300 -200 -100 0 100 200 300 -150 -100 -50 0 50 100 150 200 [1,5][2,5][3,5][5,6][4,6][3,5][1,2][2,6][1,6][2,3][1,3][2,4][4,6][3,6][3,4][3,6][2,6] -300 -200 -100 0 100 200 300 -150 -100 -50 0 50 100 150 200 -300 -200 -100 0 100 200 300 -150 -100 -50 0 50 100 150 200 -300 -200 -100 0 100 200 300 -150 -100 -50 0 50 100 150 200 [1,5][2,5][3,5][5,6][4,6][3,5][1,2][2,6][1,6][2,3][1,3][2,4][4,6][3,6][3,4][3,6][2,6] Figure 12. The interaction pattern of the intermediate solitons of the (2, 4)-soliton solution Now we consider the effect of parameters in the interaction patterns of the (2, 4)-soliton solution: a) When () = (-3,-2,-1, 1, 2, 3) at 654321,,,,,kkkkkk 23

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t = (-20, 0, 20) -200 -100 0 100 200 -150 -100 -50 0 50 100 150 200 -200 -100 0 100 200 -150 -100 -50 0 50 100 150 200 -300 -200 -100 0 100 200 300 -150 -100 -50 0 50 100 150 200 Figure 13. Contour plot of the interaction pattern of (2, 4)-soliton with () = (-3,-2,-1, 1, 2, 3) at 654321,,,,,kkkkkk = (-20, 0, 20) t b) If we decrease the value of one of the parameters, i.e., if we set = -5, then we will see the following interaction picture: 1k 24

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-300 -200 -100 0 100 200 300 -150 -100 -50 0 50 100 150 200 Figure 14. Contour plot of the interaction pattern of (2, 4)-soliton with = -5 at =20 1k t c) If we increase the value of one of the parameters, i.e., if we set = 5, then we will see the following interaction picture: 6k -400 -200 0 200 400 -100 0 100 200 Figure 15. Contour plot of the interaction pattern of (2, 4)-soliton with = 5 at =20 6k t We can observe the following facts: 1) The region of interaction widens on increasing the variable t from -20 to 20 keeping all the parameters (s) constant. ik 2) The region of interaction widens on increasing or decreasing one of the parameter values. 3) The interaction becomes weak and the interaction pattern vanishes if we increase or decrease more than one parameter value. 25

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4) If all the value are of parameters () are negative or positive then the interaction pattern vanishes. ik 3.3 Asymptotic analysis of the (2, 5)-soliton solution By using the theorem 2.2.5 we analyze the solution as follows: For and y ycxii5, for 2,1i 15511.:),(,:),(iijjiiijjiaskKaskK For and y ycxii2, for 5,...,1 i ijijjjiijijjjiaskkKaskkK1511711151171.:),(,:),( Therefore the function 1 has two jumps moving with velocities jjc5, for as 2,1j y When y 1 has five jumps moving with velocities iic5, for Every jump represents a line soliton solution which represents the whole solution of a (2, 5)-soliton. The velocity of each line soliton in (2 5)-soltion is found by the graph of levels of interactions. So here we have two incoming solitons having the velocities 5,4,3,2,1i u 6,1c 7,2 c corresponding to the set and five outgoing solitons with the velocities )1(I 3,1c 4,2c 5,3c 6,4c corresponding to the set 7,5c )4(I The level of interaction can take the range 50, ji .Then the set is given by )(nI = for )(nI ),{(17ini }1,...,1 ni and 5,4,3,2,1,0n Therefore the total number of pairs of ),(ji is 10, i.e., = )0(I )},{(71 = )1(I )},(),,{(7261 = )2(I )},(),,(),,{(736251 = )3(I )},(),,(),,(),,{(74635241 = )4(I )},(),,(),,(),,(),,{(7564534231 = )5(I )},(),,(),,(),,(),,(),,{(766554433221 The number of holes in the graph of the (2 5)-soliton solution is 4. The total number of intersection points and intermediate solitons are 13 and 16, respectively. 26

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Now we construct the graph of the levels of interaction for the (2, 5)-soliton case: Figure 16. Levels of interaction of (2, 5)-soliton In the above levels of interaction picture the level diamonds represent the incoming line solitons and the level stars represent the outgoing line solitons. Now we construct the graph of the levels of interaction for the (2, 5)-soliton case. )1(I )4(I -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 [2,7][1,6][6,7][2,6][5,7][1,7][4,6][1,5][1,5][3,5][1,2][1,4][3,4][2,4][2,3][1,3][3,4][1,5][1,6][5,6][1,5][2,4] [2,5] -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 [2,7][1,6][6,7][2,6][5,7][1,7][4,6][1,5][1,5][3,5][1,2][1,4][3,4][2,4][2,3][1,3][3,4][1,5][1,6][5,6][1,5][2,4] [2,5] Figure 17. The interaction pattern of the intermediate solitons 27

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of the (2, 5)-soliton solution Now we consider the effect of parameters in the interaction patterns of the (2 5)-) When () = (-3-2-1 0 1 2 3) and 00) we have the intr soliton solution: a 7654321,,,,,,kkkkkkk , , , t= (-10, 2eraction pictue as: -150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -200 -100 0 100 200 -100 -50 0 50 100 150 200 -40 -20 0 20 40 -20 -10 0 10 20 30 Figure 18. Contour plot of the interaction pattern of (2, 5)-soliton nd 20 0) with (7654321,,,,,,kkkkkkk) = (-3,-2,-1, 0, 1, 2, 3) a t= (-10 , 28

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b ) If we decrease the value of one of the parameters, i.e., if we set = -10, then we will 1k see the following interaction picture: -200 -100 0 100 200 -100 -50 0 50 100 150 200 Figure 19. Contour plot of the interaction pattern of (2, 5)-soliton ) If we increase the value of one of the parameters, i.e., if we set = 5, then we will see with 1k= -10 c 7k the following interaction picture: -400 -200 0 200 400 -100 0 100 200 Figure 20. Contour plot of the interaction pattern of (2, 5)-soliton e can observe the following facts: ) The region of interaction widens on increasing the variable from -10 to 20 keeping all the parameters () the parameter values constant. with 7k= 5 W t ik 1 29

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2) The region of interaction widens on increasing or decreasing one of the parameter values. 3) The interaction becomes weak and the interaction pattern vanishes if we increase ordecrease more than one parameter value. rn totic analysis of the (3, 2)-soliton solution By using the theorem 2.2.5 we e solution as follows: 4) If all the values of parameters (ik) are negative or positive, then the interaction pattevanishes. 3.4 Asympnalyze th a For y and ycxii2, for 3,2,1i, 12211.:),(,:),(iijjiiijjiaskKaskK For and y ycxii3, for 2,1 i Therefore the function ijijjjiijijjjiaskkKaskkK1211511121151,:),(,:),( 1 has three jumps moving with velocities as jjc2, for 3,2,1j y. When y, 1 has two jumps moving with velocities iic2, for 2,. E rep 1ivery jumpresliton solution The v fou the gr ionn take the range ents a line so u which represents the whole solution of a (3, 2)soliton.ocy of each line soliton in (3, 2)-soliton isnd byaph of levels of interactions. So here we have three incoming solitons having the velocities 3,1c el it 4,2c, 5,3c corresponding to the set )2(Iand two outgoing solitons with the velocities 4,1c 5,2c corresponding to the set )1(I. The level of interact ca 30, ji .Then the set is given by = )(nI )(nI ),{( for }1,...,1 ni and 3,2,1,0n. 15ini Therefore the total number of pairs of ),(j i is 10, i.e., { )0(I = )} ,( 51 )1(I= )},(),,{(5241 = )2(I )},(),,(),,{(534231 = )3(I )},(),,(),,(),,{(54433221 30

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The number of holes in the graph of (3 2)-soliton solution is 2. The total number of tersection points and intermediate solitons are 7 and 8 respectively. : oliton ing itons. in Now we construct the graph of the levels of interaction for the (3, 2)-soliton case Figure 21. Levels of interaction of (3, 2)-s In the above levels of interaction picture the level )2(Iboxes represent the incomine solitons and the level )1(Istars represent the outgoing line sol l [3,6] -150 -100 -50 0 50 100 150 -100 -50 0 50 100 150 [3,5][4,5][2,4][3,4][2,5][2,3][1,5][1,2][2,4][2,5][1,4][1,3][2,5][3,6] -150 -100 -50 0 50 100 150 -100 -50 0 50 100 150 -150 -100 -50 0 50 100 150 -100 -50 0 50 100 150 [3,5][4,5][2,4][3,4][2,5][2,3][1,5][1,2][2,4][2,5][1,4][1,3][2,5] Figure 22. The interaction pattern of the intermediate solitons of the (3, 2)-soliton solution 31

PAGE 40

arameters in the interaction patterns of the (3, 2)-liton solution: ,,kk ) = (-3, -2, -1, 1, 2) at = (-20, 0, 20) we get the following tion picture: Now we consider the effect of p so a) When (21,,kkk 543 t interac -150 -100 -50 0 50 100 150 -100 -50 0 50 100 150 -100 -50 0 50 100 1 00 50 0 50 1 00 -200 -100 0 100 200 -150 -100 -50 0 50 100 Figure 23. Contour plot of the interaction pattern of (3, 2)-soliton () = (-3, -2, -1, 1, 2) at = (-20, 0, 20) ) If we decrease the value of one of the parameters, i.e., if we set = -5, at = -20 then we will see the following interaction picture: 54321 ,,,,kkkkk t 1k b t 32

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-200 -100 0 100 200 150 100 50 0 -5 0 100 200 Fig4. Cr plotf the inctionn of (3, 2)-soliton = -5 at = -20 fewo of the valuess decreased; suppose we set = -5 ndk= -9/2 at = -20 then we get the interaction picture as: ure 2 ontou o tera patte r 1k t ski'i 1k c) If the difrence between t a 2 t -400 -200 0 200 400 -200 -100 0 100 200 300 Figure 25. Contour plot of the interaction pattern of (3, 2)-soliton = -5 and = -9/2 at = -20 of the param t= 5, then we will see e following interaction picture: 1k 2k t 6k d) If we increase the value of oneeters, i.e.,if we se th 33

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-400 -200 0 200 400 -200 -100 0 100 200 Figure 26. Contour plot of the interaction pattern of (3, 2)-soliton = 5 at = -20 e can observe the following facts: n increasing the variable from -20 to 20 keeping ll the parameters () the parameter values constant. ease or nalysis of the (3, 3)-soliton solution By using the theorem 2.2.5 we on as follows: 6k t W t 1) The region of interaction widens o a ik 2) The region of interaction widens on increasing or decreasing one or two of the parameter values. 3) The interaction becomes weak and the interaction pattern vanishes as if we incrdecrease more than two parameter values. 4) If all the value are of parameters (ik) are negative or positive then the interaction pattern vanishes. 3.4 Asymptotic analyze the soluti a For y and ycxii3, for 3,2,1i, 13311.:),(,:(iijjiiijjiaskKkK ),as 34

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For and y ycxii3, for iijjjiijijjjiaskkaskkK1311611131161.:),(,:),( Therefore the function 3,2,1i, jK 1 has three jumps moving with velocities as Wh jjc3, for 3,2,1j y en y, 1 has three jumps moving with velocities. Every jump represents a line soliton solution which represents the whole solution of a 3)-soliton. The velocity of each line soliton in (3, 3)-soliton is found by the graph of hlocities ocities iic3, for 3,2,1i u (3 levels of interactions. Soere we have three incoming solitons having the ve4 ,1c corresponding to the set )2(Iand three outgoing solitons with the vel4 5,2c, 6,3c ,1c 5,2c 6,3c corresponding to the set )2(I. The level of interaction can take the range 30, ji .Then the set )(nIis given )(nI= ),{(16ini for}1,...,1 by n and 4,3,2,1,0n. Therefore the total number of pairs oi if is 10, i.e., ),(j )0(I = )},{( 61 )1(I= )},(),,{(251 6 )2= )},52 (I ,(), (),,{(63 41 53 )3(I= )},(),,(),,{(643231 (),, )4(I = )},(),,(),,(),,(),,{(6554433221 The number of holes in the graph of (3, 3)-soliton solution is 4.The total number of intersection points and intermediate solitons are 12 and 15. Now we construct the graph of the levels of interaction for the (3, 3)-soliton case: 35

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Figure 27 Levels of interaction of (3, 3)-soliton In the above levels of interaction picture the level boxes represent the incoming line solitons and the level boxes represent the outgoing line solitons. )2(I )2(I -100 -50 0 50 100 150 -100 -50 0 50 100 [2,5][3,6][1,4][4,6][1,6][3,6][5,6][2,6][2,4][2,5][1,4][1,3][2,5][1,5][1,2][4,5][2,3][2,4][3,5][2,3][3,4] -100 -50 0 50 100 150 -100 -50 0 50 100 -100 -50 0 50 100 150 -100 -50 0 50 100 -100 -50 0 50 100 150 -100 -50 0 50 100 [2,5][3,6][1,4][4,6][1,6][3,6][5,6][2,6][2,4][2,5][1,4][1,3][2,5][1,5][1,2][4,5][2,3][2,4][3,5][2,3][3,4] Figure 28. Interaction pattern of (3, 3)-soliton Now we consider the effect of parameters in the interaction patterns of the (3, 3)-soliton soluiton: 36

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a) When () = (-3, -2, -1, 1, 2, 3) and t = (-10, 0,10) we will see the following interaction pictures: 654321,,,,,kkkkkk -100 -50 0 50 100 150 -100 -50 0 50 100 -100 -50 0 50 100 -100 -50 0 50 100 -150 -100 -50 0 50 100 150 -100 -50 0 50 100 Figure 29. Contour plot of the interaction pattern of (3, 3)-soliton with () = (-3,-2,-1, 1, 2, 3) and 654321,,,,,kkkkkk = (-10 0 10) t , b) If we decrease the value of one of the parameters, i.e., if we set = -10, at t=10 then we will see the following interaction picture: 1k 37

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-150 -100 -50 0 50 100 -100 -50 0 50 100 Figure 30. Contour plot of the interaction pattern of (3, 3)-soliton with = -10 at = 10 1k t c) If we increase the value of one of the parameters, i.e., if we set = 6, at =10 then we will see the following interaction picture: 6k t -300 -200 -100 0 100 200 300 -200 -100 0 100 200 Figure 31. Contour plot of the interaction pattern of (3, 3)-soliton with = -10 at = 10 1k t We can observe the following facts: 1) The region of interaction widens on increasing the variable t from -10 to 10 keeping all the parameters () the parameter values constant. ik 2) The region of interaction widens on increasing or decreasing one or two of the parameter values. 3) The interaction becomes weak and the interaction pattern vanishes as if we increase or decrease more than two parameter values. 38

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4) If all the value are of parameters () are negative or positive then the interaction pattern vanishes. ik 3.5 Asymptotic analysis of the (3, 4)-soliton solution By using the theorem 2.2.5 we analyze the solution as follows: For and y ycxii4, for 3,2,1i 14411.:),(,:),(iijjiiijjiaskKaskK For and y ycxii3, for 3,2,1i ijijjjiijijjjiaskkKaskkK1411711141171.:),(,:),( Therefore, the function 1 has three jumps moving with velocities jjc4, for as 3,2,1j y When y 1 has four jumps moving with velocities iic3, for 4,3,2,1i Every jump represents a line soliton solution which represents the whole solution of a (3, 4)-soliton. The velocity of each line soliton in [3, 4] soliton is found by the graph of levels of interactions. So here we have three incoming solitons having the velocities u 5,1c 6,2c 7,3c corresponding to the set and four outgoing solitons with the velocities )2(I 4,1c 5,2c 6,3c 7,4c corresponding to the set )3(I The level of interaction can take the range 50, ji .Then the set is given by )(nI )(nI = for ),{(17ini }1,...,1 ni and 5,4,3,2,1,0n Therefore the total number of pairs of ),(ji are: )0(I = )},{(71 )1(I = )},(),,{(7261 )2(I = )},(),,(),,{(736251 )3(I = )},(),,(),,(),,{(74635241 )4(I = )},(),,(),,(),,(),,{(7564534231 )5(I = )},(),,(),,(),,(),,(),,{(766554433221 The number of holes in the graph of (3, 4)-soliton solution is 6. The total number of intersection points and intermediate solitons are 17 and 22. 39

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Now we construct the graph of the levels of interaction for the (3, 3)-soliton case: Figure 32. Levels of interaction of (3, 4)-soliton In the above levels of interaction picture the level boxes represent the incoming line solitons and the level triangles represent the outgoing line solitons. )2(I )3(I -200 -100 0 100 200 -150 -100 -50 0 50 100 150 200 [3,7][2,6][1,5][6,7][1,7][2,7][5,7][4,7][3,6][1,3][1,2][2,3][4,5][3,5][3,6][2,5][1,4][1,2][1,6][1,4][4,6][2,4][3,4][5,6][4,5][2,4][2,4][2,6][4,6] -200 -100 0 100 200 -150 -100 -50 0 50 100 150 200 -200 -100 0 100 200 -150 -100 -50 0 50 100 150 200 -200 -100 0 100 200 -150 -100 -50 0 50 100 150 200 [3,7][2,6][1,5][6,7][1,7][2,7][5,7][4,7][3,6][1,3][1,2][2,3][4,5][3,5][3,6][2,5][1,4][1,2][1,6][1,4][4,6][2,4][3,4][5,6][4,5][2,4][2,4][2,6][4,6] Figure 33. Levels of interaction of (3, 4)-soliton 40

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Now we consider the effect of parameters in the interaction patterns of the (3, 4)-soliton soluiton: a) When () = (-3,-2,-1, 0, 1, 2, 3) at 7654321,,,,,,kkkkkkk t = (-20, 0, 20) we have the following interaction pictures: -200 -100 0 100 200 -150 -100 -50 0 50 100 150 200 -100 -50 0 50 100 -40 -20 0 20 40 -200 -100 0 100 200 300 -200 -100 0 100 200 Figure 34. Contour plot of the interaction pattern of (3, 4)-soliton ( 7654321,,,,,,kkkkkkk ) = (-3, -2, -1, 0, 1, 2, 3) at t = (-20, 0, 20) b) If we decrease the value of one of the parameters, i.e., if we set = -6, at t=20 then we will see the following interaction picture: 1k 41

PAGE 50

-400 -300 -200 -100 0 100 200 -200 -150 -100 -50 0 50 100 150 Figure 35. Contour plot of the interaction pattern of (3, 4)-soliton = -6 at 1k t = 20 c) If we increase the value of one of the parameters, i.e., if we set = 6, at =20 then we will see the following interaction picture: 1k t -400 -300 -200 -100 0 100 200 300 -150 -100 -50 0 50 100 150 200 Figure 36. Contour plot of the interaction pattern of (3, 4)-soliton = 6 at t = 20 1k 42

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We can observe the following facts: 1) The region of interaction widens on increasing the variable t from -10 to 10 keeping all the parameters () the parameter values constant. ik 2) The region of interaction widens on increasing or decreasing one or two of the parameter values. 3) The interaction becomes weak and the interaction pattern vanishes as if we increase or decrease more than two parameter values. 4) If all the value are of parameters () are negative or positive then the interaction pattern vanishes. ik 43

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References [1] Ablowitz M J and Segur H: 1981, Solitons and Inverse scattering Transform, SIAM, Philadelphia. [2] Bernal J, Aguero M A, Romero E F: 2003, Soliton-like structures and the connection between the Bq and KP equations, Chaos, Solitons & Fractals, 17, 951-966. [3] Freeman N C and Nimmo J J C: 1983, Soliton solutions of the Korteweg-de Vries and Kadomtsev-Petviashvili equations: the Wronskian technique, Physics Letters A, 95, 1-3. [4] Hirota R: 1976, Direct methods of finding solutions of nonlinear evolution equations, Backlund Transformations, Lecture Notes in Mathematics, Vol 515, ed Miura R M, New York Springer. [5] Hirota R: 1974, A new form of Bcklund transformation and its relation to the inverse scattering problem, Progress of Theoretical Physics, 52, 1498-1512. [6] Hirota R and Ito M: 1988, Resonance of solitons in one dimension, Journal of the Physics Society of Japan, 52, 744-748. [7] Hirota R, Ohta Y and Satsuma J: 1988, Wronskian structures of solitons for soliton equations, Progress of Theoretical Physics Supplement, No. 94, 59-72. [8] Isojima S, Willox R and Satsuma J: 2003, Spider-web solution of the coupled KP equation, Journal of Physics A: Mathematical and General, 36, 9533-9552. [9] Kodama Y and Biondini G: 2003, On a family of solutions of the KP equation which also satisfy the toda lattice hierarchy, Journal of Physics A, 42, 10519-10536. [10] Ma W X: 2002, Complexiton solutions to the Korteweg-de Vries equation, Physics Letters A, 301, 35-44. [11] Ma W X: 2004, Wronskians, generalized Wronskians and solutions to the Korteweg-de Vries equation, Chaos, Solitons & Fractals, 19, 163-170. [12] Ma W X: 2005, Integrability, Encyclopedia of Nonlinear Science, pp450-453, ed Scott A, New York and London, Routledge. 44

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[13] Ma W X and You Y: 2005, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Transactions of the American Mathematical Society, 357, 1753-1778. [14] Matveev V B: 1992, Generalized Wronskian formula for solutions of the KdV equations: first applications, Physics Letters A, 166, 205-208. [15] Medina E: 2002, An N soliton resonance solution for the KP equation: interaction with change of form and velocity, Letters in Mathematical Physics, 62, 91-99. [16] Miles J W: 1977, Diffraction of solitary waves, Journal of Fluid Mechanics, 79, 171-179. [17] Nakamura Y and Kodama Y: 1995, Moment problem of Hamburger, hierarchies of integrable systems, and the positivity of tau-functions, Acta Applied Mathematics, 39, 435-443. [18] Ohta Y, Satsuma J, Takahashi D and Tokihiro T: 1998, Elementary introduction to the Sato theory, Progress of Theoretical Physics Supplement, No. 94, 210-241. [19] Satsuma J: 1979, A Wronskian representation of N-soliton solutions of nonlinear evolution equations, Journal of the Physics Society of Japan, 46, 359-360. [20] Whitham G B: 1974, Linear and Nonlinear Waves, pp110-112, New York, Willey. 45


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Tippabhotla, Anupama.
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Interaction patterns and web-structures of resonant solitons of the kadomtsev-petviashvili equation
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by Anupama Tippabhotla.
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[Tampa, Fla.] :
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2005.
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Thesis (M.A.)--University of South Florida, 2005.
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ABSTRACT: In this thesis, the interaction pattern for a class of soliton solutions of the Kadomtsev- Petviashvili (KP) equation is analyzed. The complete asymptotic properties of the soliton solutions for are determined. The resonance characteristic of two sub-classes of the soliton solutions, in which incoming line solitons for interact to form outgoing line solitons for is described. These two specific sub-classes of -soliton solutions are the following: 1) [(2, 3), (2, 4), (2, 5)], 2) [(3, 2), (3, 3), (3, 4)]. The intermediate solitons and the interaction regions of the above soliton solutions are determined, and their various interaction patterns are explored. Maple and Mathematica are used to get the 3 dimensional plots and contour plots of the soliton solutions to show their interaction patterns. Finally, the spider-web-structures of the discussed solitons of the KP equation are displayed.
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Adviser: Dr.Wen Xiu Ma.
Co-adviser: Dr.Youcheng You
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The kp equation.
Solitons.
Interaction patterns.
Spider-web-like structures.
Levels of intersection.
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Dissertations, Academic
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