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arxiv: 2503.01093 · v2 · submitted 2025-03-03 · 🌊 nlin.PS · math-ph· math.MP· nlin.SI

Regularizations for shock and rarefaction waves in the perturbed solitons of the KP equation

Guangfu Han , Yuji Kodama , Chuanzhong Li , Lin Sun This is my paper

Pith reviewed 2026-05-23 02:18 UTC · model grok-4.3

classification 🌊 nlin.PS math-phmath.MPnlin.SI
keywords KP equationline solitonsmodulation systemshock wavesrarefaction wavesresonant interactionparabolic soliton
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The pith

Shock waves in the two-component modulation system for KP line solitons generate new solitons via resonant interaction, while rarefactions appear as parabolic solitons.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the KP equation with initial data formed from segments of exact line-soliton solutions. An asymptotic perturbation method reduces the slow evolution of the soliton parameters to a two-component quasi-linear dynamical system. In this reduced system, singular shock solutions describe the resonant creation of an additional soliton, and regular rarefaction solutions correspond to parabolic solitons. Numerical simulations of the full KP equation reproduce the evolution predicted by the modulation system.

Core claim

Using an asymptotic perturbation method, the slow modulation of soliton parameters for the KP equation with piecewise line-soliton initial data is described by a 2-component quasi-linear system. A singular shock-wave solution of the system leads to the generation of a new soliton resulting from resonant interaction of solitons. A regular rarefaction solution of the system is described by a parabola, termed a parabolic soliton. Numerical simulations confirm agreement with these analytic predictions.

What carries the argument

The 2-component quasi-linear system obtained from the asymptotic perturbation method, which governs the slow modulation of soliton parameters.

If this is right

  • Resonant soliton interactions are captured by singular shock solutions of the modulation system.
  • Rarefaction dynamics appear geometrically as parabolic solitons in the parameter space.
  • The perturbation method accurately describes the long-time behavior for these classes of initial data.
  • Direct numerical solutions of the KP equation match the predictions of the reduced modulation system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same modulation approach could be applied to other integrable equations to track resonant soliton processes.
  • Parabolic solitons might be used as a diagnostic signature for rarefaction events in physical realizations of the KP equation.
  • The regularization of shocks within the modulation system suggests how apparent singularities resolve in the underlying nonlinear dynamics.

Load-bearing premise

The initial data consisting of parts of exact line-soliton solutions admit a slow modulation that can be captured by the asymptotic perturbation method without rapid transients or higher-order corrections.

What would settle it

A numerical simulation of the KP equation showing that a shock predicted by the modulation system fails to generate the expected new soliton would disprove the claimed correspondence.

Figures

Figures reproduced from arXiv: 2503.01093 by Chuanzhong Li, Guangfu Han, Lin Sun, Yuji Kodama.

Figure 1
Figure 1. Figure 1: The left panel shows the contour plot of the [i, j]-soliton solution (2.1) with κi = −1, κj = 2 at t = 0. The dotted line is the crest of the soliton. The middle panel shows the corresponding permutation (transposition i ↔ j), which we call the chord diagram of the soliton. The diagram indicates the asymptotic structure of KP soliton, that is, the upper (lower) part of the diagram shows the [i, j]-soliton … view at source ↗
Figure 2
Figure 2. Figure 2: The parameters in the matrix A are a = 1 a′ = q 3·105 2 , b = 1 b ′ = q 2 3·105 (i.e., ab = 1). The κ-parameters are given by (κ1, κ2, κ3, κ4) = (− 3 2 , −10−5 , 10−5 , 3 2 ). The left panel shows the contour plot of the solution u(x, y, 0). The middle panel is the chord diagram for the O-type soliton. The right panel shows the colored κ-graph in the slow scale Y = ǫy, and note that the phase shift in the … view at source ↗
Figure 3
Figure 3. Figure 3: shows the resonant interaction at y = y+ in (2.11) for y ≫ 0 in the limit κ3 → κ2. Similarly, we have the resonant interaction at y = y− = −y+ as shown also in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Y-soliton and the colored κ-graph with the singular point C [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The initial data (4.1). Each bold face line shows a semi-infinite line-soliton (half-soliton). In the right panel, the amplitude of soliton 2 (u − 0 ) is fixed to be 2, and that of soliton 1 (u + 0 ) is a variable A0. 4.1. Half-line initial data. We first consider the initial data consisting of a half line-soliton for y > 0, (4.2) u(x, y, 0) = u0(x, y)H(y), where u0(x, y) is a line-soliton with the paramet… view at source ↗
Figure 6
Figure 6. Figure 6: Example of a half-soliton initial data, incomplete chord diagram, and the κ-graph. In the right panel, the red line and the blue line represent the initial values of κ1 and κ2, i.e., κ1(Y, 0) = κ 0 1 and κ2(Y, 0) = κ 0 2 for Y > 0. Here we take κ 0 1 = −1 and κ 0 2 = 2, so that we have tan Ψ0 = 1 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The Initial data corresponding to the half-line soliton in [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Numerical simulation and theoretical prediction: the contour plots of the numerical simulation, and the red line represents the theoretical result (4.7) for T = 1, T = 2, and T = 3. In a similar way, we can solve the initial value problem with a half-line initial soliton for y < 0, i.e., the initial data is given by u(x, y, 0) = u0(x, y)H(−y) with u0 in (4.3). The initial κ-parameters are given by κ1 = ( κ… view at source ↗
Figure 9
Figure 9. Figure 9: Example of lower semi (1, 2)-soliton solution, incomplete chord diagram, and the κ-graph [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Numerical simulation and theoretical comparison: the main part is the numerical simulation results, and the red line represents the peak trajectory function. We take (κ 0 1 , κ0 2 ) = (− 5 4 , 3 4 ), and the figures are taken at T = 1, 2 and 3. Definition 4.2. For the initial κ-parameters, we define the following: (a) An initial parameter κ 0 i is a fixed point, if κi = κ 0 i =constant for all T > 0. (b) … view at source ↗
Figure 11
Figure 11. Figure 11: The κ-graph for an initial half-soliton in Y < 0. The left panel shows the incomplete chord diagram for the initial value problem, where the κ2 marked by • represents the fixed point. The right panel shows the κ-graph at T > 0 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: All possible V -shaped initial data. Each region is parametrized by a unique incomplete chord diagram. Each • in the diagram marks the fixed point. The length (amplitude) of the lower chord is fixed as 2. These cases in [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: V -shaped soliton of the case (a). Note that κ 0 3 is a fixed point (i.e., constant for all Y at T > 0). The system gives a simple wave, which depends only on κ1(Y, T ), ∂κ1 ∂T + (2κ1 + κ 0 3 ) ∂κ1 ∂Y = 0. The characteristic velocity is then given by V (κ1, κ2) = 2κ1 + κ 0 3 . Since the initial data of κ1 increases in Y , we have a global solution, (5.3) κ1(Y, T ) =    κ 0 1 , Y < Yb(T ), κ 0 1 + κ… view at source ↗
Figure 14
Figure 14. Figure 14: Numerical simulation for the case (a). We take (κ 0 1 , κ0 2 , κ0 3 ) = (− 5 4 , − 1 4 , 3 4 ), and the figures are taken at T = 1, 2 and 3. The peak trajectories are shown as the solid curves, and the curve between points a and b is a parabola connecting upper and lower solitons. Since κ1 = κ 0 1 for all Y , the κ-system gives a simple wave solution. Note that the initial data κ2 increases in Y , and the… view at source ↗
Figure 15
Figure 15. Figure 15: ). Note first that the initial data for the κ-system shown in [PITH_FULL_IMAGE:figures/full_fig_p016_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Regularized initial data for the case (c). The right panel shows the initial conditions corresponding to the κ-graph [PITH_FULL_IMAGE:figures/full_fig_p017_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The κ-graphs and the numerical simulations for the case (c). The counter plots are obtained at T = 0, 1, 2, 3. The peak trajectories in the intervals (Yb, Ya) and (Yd, Yc) are parabola. One should note that the solution of this initial value problem depends on the ε, say κi(Y, T ; ε), and the limit ε ↓ 0 of the solution is well-defined. Explicit form of the peak trajectory is given by (5.7) X(Y ) =  … view at source ↗
Figure 18
Figure 18. Figure 18: The V -shaped initial data of the case (d). The middle panel shows the corresponding incomplete chord diagram with fixed points marked by •. Y , we have a simple wave system for κ1, (5.8) ∂κ1 ∂Y + (2κ1 + κ 0 2 ) ∂κ1 ∂Y = 0, with κ1(Y, 0) =  κ 0 2 , Y < 0, κ 0 1 , Y > 0. This initial value problem is not well-posed, because the initial data is not increasing (see Lemma 3.1) and the characteristic velociti… view at source ↗
Figure 19
Figure 19. Figure 19: Regularization of the initial data (5.8). is to add a small piece of soliton so that the intersection point forms a resonant Y-soliton. To show that this regularized initial value problem has a global solution, we first recall that the intersection point propagates with the speed Ca = κ 0 1 + κ 0 2 + κ 0 3 (see (2.13) with the slow scales, i.e., Ya(T ) = CaT ). For the evolution of the small soliton with … view at source ↗
Figure 20
Figure 20. Figure 20: The κ-graph and the numerical simulation for the case (d). We take (κ1, κ2, κ3) = (−17 8 , −5 8 , 11 8 ). The counter plot at the right panel is at T = 1. We have Vc < Vb < Va. The peak trajectory in the region between the points b and c is given by a parabola, X(Y ) = − 1 4T (Y + κ 0 1T ) 2 + (κ 0 1 ) 2T. Note here that κ 0 1 is a fixed point. Before closing the case (d), we remark that the incomplete ch… view at source ↗
Figure 21
Figure 21. Figure 21: The V -shaped initial data of the case (e). In the incomplete chord diagram (middle panel), black dots are fixed points, while the red dot is singular point. the κ-system (5.12) and the numerical simulation [PITH_FULL_IMAGE:figures/full_fig_p020_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The global solution of the κ-system (5.12) and the numerical simulation with the peak trajectory. We take (κ 0 1 = κ 0 2 , κ0 3 , κ0 4 ) = (− 3 4 , 1 4 , 5 4 ), and the numerical result are at T = 0, T = 2 and T = 4. Here the asymptotic solution u0(x, y, t) in (1.4) is given by the Y-soliton of type π = (2, 3, 1) (see Section 2.2.2). 5.4. The case (f). The initial half-solitons of this case are [1, 3]-sol… view at source ↗
Figure 23
Figure 23. Figure 23: The initial profile and the regularized initial data for the case (f) [PITH_FULL_IMAGE:figures/full_fig_p020_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: shows the evolution of the κ-graph and the solution u(x, y, t) of the numerical simulation [PITH_FULL_IMAGE:figures/full_fig_p021_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: shows the initial data for symmetric choice of the parameters (κ 0 1 , κ0 2 , κ0 3 , κ0 4 ) = (−1, − 1 2 , 1 2 , 1). Note that the κ1 is increasing, but κ2 is decreasing. This implies that the κ1 is a rarefaction wave, and [PITH_FULL_IMAGE:figures/full_fig_p022_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: The regularized initial data for the case (g). The characteristic velocities at the points from a through e are given by Va = 3κ 0 4 > Vb = 2κ 0 3 + κ 0 4 > Vc = κ 0 2 + κ 0 3 + κ 0 (5.15) 4 > > Vd = 2κ 0 2 + κ 0 4 > Ve = 2κ 0 1 + κ 0 4 . Thus all points are separated with increasing distances between them. This implies that the system with this regularized initial data has a global solution (see [PITH_F… view at source ↗
Figure 27
Figure 27. Figure 27: The κ-graphs and numerical simulation for the case (g). We take (κ 0 1 , κ0 2 , κ0 3 , κ0 4 ) = (−1, −1 2 , 1 2 , 1), and the κ-graphs are obtained at T = 0, T = 2, and T = 4. 5.6. The case (i). The initial half-solitons are [1, 2]-soliton for Y > 0 and [3, 4]-soliton for Y < 0. We first note that κ 0 1 and κ 0 4 are fixed points. Then we take the regularized initial data as shown in [PITH_FULL_IMAGE:fig… view at source ↗
Figure 29
Figure 29. Figure 29: The κ-graphs and numerical simulation for the case (i). We take (κ 0 1 , κ0 2 , κ0 3 , κ0 4 ) = (−11 5 , −1 5 , 1 5 , 11 5 ), and the simulations are at T = 0, 1 and T = 2. The asymptotic solution u0(x, y, t) in (1.4) is given by O-soliton (see Section 2.2.1). 5.7. The case (j). The initial half-solitons are [1, 2]-soliton for Y < 0 and [3, 4]-soliton for Y > 0. In this case, κ 0 2 and κ 0 3 are the fixed… view at source ↗
Figure 30
Figure 30. Figure 30: The regularized initial data for the case (j). velocities at the points a, b, c and d are given by (5.17) Va = 2κ 0 4 + κ 0 3 > Vb = 3κ 0 3 > Vc = 3κ 0 2 > Vd = 2κ 0 1 + κ 0 2 . Again we note that all points are separated with increasing distances between them. The κ-graphs and the results of numerical simulation are shown in [PITH_FULL_IMAGE:figures/full_fig_p024_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: The κ-graphs and the numerical simulation for the case (j). We take (κ 0 1 , κ0 2 , κ0 3 , κ0 4 ) = (−11 5 , −1 5 , 1 5 , 11 5 ), and the simulations are at T = 0, 1 and T = 2. 5.8. Summary. In this section, we studied the initial value problem of the κ-system with V-shape initial data. We summarize our results as follows. For given initial V-shape data, we started with the following steps: (1) Draw an in… view at source ↗
Figure 32
Figure 32. Figure 32: The complete chord diagrams for the initial value problem of the κ-system with the initial V -shape data in [PITH_FULL_IMAGE:figures/full_fig_p026_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: The solutions of the initial value problems with the initial data consisting of two half-solitons with the same amplitude A0 but the different angles tan Ψ± 0 . Acknowledgements. The authors would like to thank Harry Yeh for critical reading of the manuscript. They also appreciate a research fund from Shandong University of Science and Technology. One of the authors (C.L) is supported by National Natural … view at source ↗
Figure 34
Figure 34. Figure 34: Example of the KP soliton with π = (4, 5, 1, 2, 6, 3). Appendix B. The κ-system In this appendix, we derive the κ-system (1.5) for the parameters {κ1, κ2} of [1, 2]-soliton using an asymptotic perturbation theory. Although the system has been derived in [22, 4], we here give much elementary derivation using a standard perturbation method. We also emphasize that our slow variables are just Y = ǫy and T = ǫ… view at source ↗
Figure 35
Figure 35. Figure 35: The regularization in the Whitham-KdV equation. The left panel is u(x, 0) in (C.5). The middle panel is the regularized initial data for (r1, r2, r3). The right panel shows the solution for T > 0. Declarations. • Data sharing not applicable to this article as no datasets were generated or analyzed during the current study. • The authors declare no conflicts of interest associated with this manuscript. Ref… view at source ↗
read the original abstract

Using an asymptotic perturbation method, we study the initial value problem for the KP equation with initial data consisting of parts of exact line-soliton solutions. We consider a slow modulation of the soliton parameters, described by a dynamical system obtained via the perturbation method. {The dynamical system is given by a $2$-component quasi-linear system.} In particular, we show that a singular solution (\emph{shock wave}) {of the system} leads to the generation of a new soliton as a result of the resonant interaction of solitons. We also show that a regular solution corresponding to a rarefaction wave {of the system} can be described by a parabola (which we call a \emph{parabolic soliton}). We then perform numerical simulations of the initial value problem and show that they are in excellent agreement with the results obtained by the perturbation method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript applies an asymptotic perturbation method to the KP equation initialized with segments of exact line-soliton solutions. It derives a two-component quasi-linear modulation system governing the slow evolution of soliton parameters, then analyzes its singular (shock) solutions, which are claimed to produce resonant generation of a new soliton, and its regular (rarefaction) solutions, which are claimed to correspond to a parabolic soliton. Numerical simulations of the initial-value problem are reported to agree with these predictions from the modulation system.

Significance. If the central claims hold, the work extends Whitham-type modulation theory to resonant soliton interactions in the KP equation and identifies a novel parabolic-soliton structure arising from rarefaction waves. The absence of free parameters in the modulation system and the provision of direct numerical comparisons constitute concrete strengths that would make the results falsifiable and potentially useful for modeling soliton dynamics in integrable systems.

major comments (2)
  1. [Abstract] Abstract: the derivation of the 2-component quasi-linear modulation system is stated to rest on a slow-modulation assumption for the soliton parameters. The central claims, however, concern precisely the singular (shock) solutions of this system, which contain discontinuous jumps and therefore violate the slow-variation hypothesis by construction. No diagnostic (e.g., ratio of modulation length scale to soliton width near the jump) is supplied to justify the extrapolation.
  2. [Abstract] Abstract: the reported numerical agreement with the perturbation predictions is asserted without error bars, convergence checks, or a description of how the modulation system is obtained from the perturbation method. This leaves the verification of the shock-induced soliton generation and the parabolic-soliton rarefaction incomplete.
minor comments (1)
  1. [Abstract] The abstract introduces the term 'parabolic soliton' for the rarefaction solution but does not clarify whether this is a new exact solution of the KP equation or only an approximate description within the modulation system.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the derivation of the 2-component quasi-linear modulation system is stated to rest on a slow-modulation assumption for the soliton parameters. The central claims, however, concern precisely the singular (shock) solutions of this system, which contain discontinuous jumps and therefore violate the slow-variation hypothesis by construction. No diagnostic (e.g., ratio of modulation length scale to soliton width near the jump) is supplied to justify the extrapolation.

    Authors: We recognize that the shock solutions introduce discontinuities that challenge the slow-modulation assumption underlying the derivation. However, this is inherent to the use of modulation equations in capturing rapid transitions via shocks, as is standard in Whitham modulation theory for integrable systems. The numerical simulations provide empirical validation of the predictions. In the revised version, we will include a diagnostic measure, such as the ratio of the modulation length scale to the soliton width in the vicinity of the shock, to better justify the applicability. revision: yes

  2. Referee: [Abstract] Abstract: the reported numerical agreement with the perturbation predictions is asserted without error bars, convergence checks, or a description of how the modulation system is obtained from the perturbation method. This leaves the verification of the shock-induced soliton generation and the parabolic-soliton rarefaction incomplete.

    Authors: The full manuscript provides the derivation of the modulation system via the perturbation method in the main text. Regarding the numerical aspects, we agree that additional details would strengthen the verification. We will revise the manuscript to include error bars on the numerical comparisons, describe convergence checks performed, and ensure the abstract or a dedicated section outlines the derivation process more clearly. revision: yes

Circularity Check

0 steps flagged

No circularity: modulation system derived independently; shock/rarefaction analysis is downstream application

full rationale

The paper obtains the 2-component quasi-linear system via asymptotic perturbation applied to the KP equation with line-soliton initial data. It then solves that derived system for its singular (shock) and regular (rarefaction) solutions and interprets the outcomes as soliton generation or parabolic solitons. Numerical simulations of the original IVP supply an external check. No quoted step equates a claimed prediction to a fitted parameter, self-citation chain, or input by construction; the slow-variation premise is an explicit modeling assumption, not a result smuggled back in. This is the normal non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The central claims rest on the validity of the asymptotic perturbation reduction to a 2-component quasi-linear system and on the assumption that the initial data are sufficiently close to exact line solitons for slow modulation to dominate. No explicit free parameters, new axioms, or invented entities are stated in the abstract.

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Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

  1. [1]

    A. M. Bloch and Y. Kodama, The Whitham equation and shocks in the Toda lattice. In Singular limits of dispersive waves, Eds. N. M. Ercolani, I.R. Gabitov, C.D. Levermore, D. Serre. 1994, 1-19

    work page 1994

  2. [2]

    Chakravarty and Y

    S. Chakravarty and Y. Kodama, Classification of the line- soliton solutions of KP II. J. Phys. A: Math. Theor. 41 (2008) 275209

    work page 2008

  3. [3]

    Chakravarty and Y

    S. Chakravarty and Y. Kodama, Soliton solutions of the KP equation and application to shallow water waves, Stud. Appl. Math. 123, (2009) 83-151

    work page 2009

  4. [4]

    Grava, C

    T. Grava, C. Klein and G. Pitton, Numerical study of the Ka domtsev–Petviashvili equation and dispersive shock waves. Proc. R. Soc. A 474, 20170458

    work page

  5. [5]

    Gurevich and L.P

    A.V. Gurevich and L.P. Pitayevsky. Nonstationary struc ture of a collisionless shock wave. Sov. J. Exp. Theor. Phys 38 (1974) 291–297. 32 GUANGFU HAN 1, YUJI KODAMA 1,2, CHUANZHONG LI 1, LIN SUN 1

    work page 1974

  6. [6]

    Jia, Numerical study of the KP solitons and higher order Miles the ory of the Mach reflection in shallow water , Ph.D

    Y. Jia, Numerical study of the KP solitons and higher order Miles the ory of the Mach reflection in shallow water , Ph.D. Thesis, The Ohio State University (2014)

    work page 2014

  7. [7]

    Kamchatnov, Nonlinear periodic waves and their modulations: an introdu ctory course , (W orld Scientific, Singapore, 2000)

    A.M. Kamchatnov, Nonlinear periodic waves and their modulations: an introdu ctory course , (W orld Scientific, Singapore, 2000)

    work page 2000

  8. [8]

    Kao and Y

    C.Y. Kao and Y. Kodama, Numerical study of the KP equation for non-periodic waves, Math. Comput. Simulation 82 (2012) 1185–1218

    work page 2012

  9. [9]

    Kodama, The Whitham equations for optical communicat ions: Mathematical theory of NRZ

    Y. Kodama, The Whitham equations for optical communicat ions: Mathematical theory of NRZ. SIAM J. Appl. Math. 59 (1999) 2162-2192

    work page 1999

  10. [10]

    Kodama, M

    Y. Kodama, M. Oikawa and H. Tsuji, Soliton solutions of t he KP equation with V-shape initial waves. J. Phys. A: Math. Theor. 42 (2009) 312001

    work page 2009

  11. [11]

    Kodama, KP solitons in shallow water, J

    Y. Kodama, KP solitons in shallow water, J. Phys. A: Math . Theor. 43 (2010) 434004 (54pp)

    work page 2010

  12. [12]

    Kodama, Solitons in two-dimensional shallow water , CBMS-NSF, 92, (SIAM, Philadelphia, 2018)

    Y. Kodama, Solitons in two-dimensional shallow water , CBMS-NSF, 92, (SIAM, Philadelphia, 2018)

    work page 2018

  13. [13]

    Kodama and L

    Y. Kodama and L. Williams, The Deodhar decomposition of the Grassmannian and the regularity of KP solitons, Adv. Math. 244 (2013) 979-1032

    work page 2013

  14. [14]

    Kodama and L

    Y. Kodama and L. Williams, KP solitons and total positiv ity for the Grassmannian, Invent. Math. 198 (2014) 637-699

    work page 2014

  15. [15]

    Kodama and H

    Y. Kodama and H. Yeh, The KP theory and Mach reflection, J. Fluid Mech. 800 (2016) 766-786

    work page 2016

  16. [16]

    W. Li, H. Yeh and Y. Kodama, On the Mach reflection of a soli tary wave: revisited, J. Fluid. Mech. 672 (2011) 326-357

    work page 2011

  17. [17]

    McDowell, M

    T. McDowell, M. Osborne, S. Chakravarty and Y. Kodama, O n a class of initial value problems and solitons for the KP equation: A numerical study. W ave Motion 72 (2017) 201-227

    work page 2017

  18. [18]

    J.W. Miles. Resonantly interacting solitary waves. J. Fluid Mech. 79 (1977) 171-179

    work page 1977

  19. [19]

    Mizumachi, Stability of line solitons for the KP II equation in R2

    T. Mizumachi, Stability of line solitons for the KP II equation in R2. Memoirs of the American Mathematical Society, 238 (AMS, Providence, RI, 2015)

    work page 2015

  20. [20]

    A. V. Porubov, H. Tsuji, I. L. Lavrenov and M. Oikawa, For mation of the rogue wave due to non-linear two- dimensional waves interaction, W ave Motion 42 (2005) 202-210

    work page 2005

  21. [21]

    Ryskamp, M.D

    S.J. Ryskamp, M.D. Maiden, G. Biondini and M A. Hoefer, E volution of truncated and bent gravity wave solitons: the Mach expansion problem. J. Fluid Mech. 909 (2021) A24

    work page 2021

  22. [22]

    Ryskamp, M.A

    S.J. Ryskamp, M.A. Hoefer and G. Biondini, Modulation t heory for soliton resonance and Mach reflection. Proc. Royal Soc. A 478 (2022) 20210823

    work page 2022

  23. [23]

    W ang and R

    C. W ang and R. Pawlowicz, Oblique wave-wave interactio ns of nonlinear near-surface internal waves in the Strait of Georgia. J. Geophy. Res.: Oceans 117 (2012)

    work page 2012

  24. [24]

    C.X. Yuan, R. Grimshaw, E. Johnson and Z. W ang, Topograp hic effect on oblique internal wave-wave interactions. J. Fluid Mech. 856 (2018) 36-60

    work page 2018

  25. [25]

    G. B. Whitham, Linear and nonlinear waves , (Wiley-Interscience, New York, 1974). 1 College of Mathematics and Systems Science, Shandong Univer sity of Science and Technology, Qingdao, 266590, China 2Department of Mathematics, The Ohio State University, Colum bus, OH 43210

    work page 1974