NONLINEAR WAVE TRANSITIONS AND THEIR MECHANISMS OF THE (2+1)-DIMENSIONAL KORTEWEG-DE VRIES-SAWADA-KOTERA-RAMANI EQUATION

doi:10.1007/s10473-025-0410-5

数学物理学报(英文版) ›› 2025, Vol. 45 ›› Issue (4): 1405-1437.doi: 10.1007/s10473-025-0410-5

NONLINEAR WAVE TRANSITIONS AND THEIR MECHANISMS OF THE (2+1)-DIMENSIONAL KORTEWEG-DE VRIES-SAWADA-KOTERA-RAMANI EQUATION

Haolin WANG, Shoufu TIAN^*, Tiantian ZHANG^*

School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China

收稿日期:2024-03-13 出版日期:2025-10-10 发布日期:2025-10-10

NONLINEAR WAVE TRANSITIONS AND THEIR MECHANISMS OF THE (2+1)-DIMENSIONAL KORTEWEG-DE VRIES-SAWADA-KOTERA-RAMANI EQUATION

Haolin WANG, Shoufu TIAN^*, Tiantian ZHANG^*

School of Mathematics, China University of Mining and Technology, Xuzhou 221116, China

Received:2024-03-13 Online:2025-10-10 Published:2025-10-10
Contact: ^*Shoufu TIAN, E-mail: sftian@cumt.edu.cn and shoufu2006@126.com; Tiantian ZHANG, E-mail: ttzhang@cumt.edu.cn
About author:Haolin WANG, E-mail: TS23080024A31LD@cumt.edu.cn
Supported by:
National Natural Science Foundation of China (12371255, 11975306), the Xuzhou Basic Research Program Project (KC23048), the Six Talent Peaks Project in Jiangsu Province (JY-059) and the 333 Project in Jiangsu Province and the Fundamental Research

摘要/Abstract

摘要： In this work, we study wave state transitions of the (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani (2KdVSKR) equation by analyzing the characteristic line and phase shift. By converting the wave parameters of the N-soliton solution into complex numbers, the breath wave solution is constructed. The lump wave solution is derived through the long wave limit method. Then, by choosing appropriate parameter values, we acquire a number of transformed nonlinear waves whose gradient relation is discussed according to the ratio of the wave parameters. Furthermore, we reveal transition mechanisms of the waves by exploring the nonlinear superposition of the solitary and periodic wave components. Subsequently, locality, oscillation properties and evolutionary phenomenon of the transformed waves are presented. And we also prove the change in the geometrical properties of the characteristic lines leads to the phenomena of wave evolution. Finally, for higher-order waves, a range of interaction models are depicted along with their evolutionary phenomena. And we demonstrate that their diversity is due to the fact that the solitary and periodic wave components produce different phase shifts caused by time evolution and collisions.

关键词: the (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani equation, characteristic line, transformed nonlinear waves, phase shift, collision

Abstract: In this work, we study wave state transitions of the (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani (2KdVSKR) equation by analyzing the characteristic line and phase shift. By converting the wave parameters of the N-soliton solution into complex numbers, the breath wave solution is constructed. The lump wave solution is derived through the long wave limit method. Then, by choosing appropriate parameter values, we acquire a number of transformed nonlinear waves whose gradient relation is discussed according to the ratio of the wave parameters. Furthermore, we reveal transition mechanisms of the waves by exploring the nonlinear superposition of the solitary and periodic wave components. Subsequently, locality, oscillation properties and evolutionary phenomenon of the transformed waves are presented. And we also prove the change in the geometrical properties of the characteristic lines leads to the phenomena of wave evolution. Finally, for higher-order waves, a range of interaction models are depicted along with their evolutionary phenomena. And we demonstrate that their diversity is due to the fact that the solitary and periodic wave components produce different phase shifts caused by time evolution and collisions.

Key words: the (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani equation, characteristic line, transformed nonlinear waves, phase shift, collision

Haolin WANG, Shoufu TIAN, Tiantian ZHANG. NONLINEAR WAVE TRANSITIONS AND THEIR MECHANISMS OF THE (2+1)-DIMENSIONAL KORTEWEG-DE VRIES-SAWADA-KOTERA-RAMANI EQUATION[J]. 数学物理学报(英文版), 2025, 45(4): 1405-1437.

Haolin WANG, Shoufu TIAN, Tiantian ZHANG. NONLINEAR WAVE TRANSITIONS AND THEIR MECHANISMS OF THE (2+1)-DIMENSIONAL KORTEWEG-DE VRIES-SAWADA-KOTERA-RAMANI EQUATION[J]. Acta mathematica scientia,Series B, 2025, 45(4): 1405-1437.

参考文献

[1] Nicolis G.Introduction to Nonlinear Science. Cambridge: Cambridge University Press, 1995
[2] Lam L.Introduction to Nonlinear Physics. Berlin: Springer, 2003
[3] Chai J, Tian B, Sun W R, et al.Solitons and rouge waves for a generalized (3+1)-dimensional variable-coefficient Kadomtsev-Petviashvili equation in fluid mechanics. Comput Math Appl, 2016, 71(10): 2060-2068
[4] Manikandan K, Muruganandam P, Senthilvelan M, et al. Manipulating matter rogue waves and breathers in Bose-Einstein condensates. Phys Rev E, 2014, 90(6): Art 062905
[5] Ding C C, Gao Y T, Li L Q. Breathers and rogue waves on the periodic background for the Gerdjikov-Ivanov equation for the Alfv$\acute{\text{e}}$n waves in an astrophysical plasma. Chaos, Solitons & Fractals, 2019, 120: 259-265
[6] Sakkaravarthi K, Mareeswaran R B, Kanna T. Engineering optical rogue waves and breathers in a coupled nonlinear Schr$\ddot{\text{o}}$dinger system with four-wave mixing effect. Phys Scr, 2020, 95(9): Art 095202
[7] Dudley J M, Dias F, Erkintalo M, et al. Instabilities, breathers and rogue waves in optics. Nat Photonics, 2014, 8(10): 755-764
[8] Salman H. Breathers on quantized superfluid vortices. Phys Rev Lett, 2013, 111(16): Art 165301
[9] Zhao F, Li Z D, Li Q Y, et al.Magnetic rogue wave in a perpendicular anisotropic ferromagnetic nanowire with spin-transfer torque. Ann Phys, 2012, 327(9): 2085-2095
[10] Liu J G, Zhu W H. Breather wave solutions for the generalized shallow water wave equation with variable coefficients in the atmosphere, rivers, lakes and oceans. Comput Math Appl, 2019, 78(3): 848-856
[11] Onorato M, Residori S, Bortolozzo U, et al. Rogue waves and their generating mechanisms in different physical contexts. Phys Rep, 2013, 528(2): 47-89
[12] Akhmediev N,Soto-Crespo J M, Ankiewicz A. Extreme waves that appear from nowhere: on the nature of rogue waves. Phys Lett A, 2009, 373(25): 2137-2145
[13] Flach S, Willis C R. Discrete breathers. Phys Rep, 1998, 295: 181-264
[14] Dysthe K B, Trulsen K. Note on breather type solutions of the NLS as models for freak-waves. Phys Scr, 1999, 1999(T82): Art 48
[15] Ten I, Tomita H. Simulation of the ocean waves and appearance of freak waves. Reports of RIAM Symposium, 2006, 17SP1-2: 10-11
[16] Zhao L C, Ling L, Yang Z Y. Mechanism of kuznetsov-ma breathers. Phys Rev E, 2018, 97(2): Art 022218
[17] Guo B L, Ling L M. Rogue wave, breathers and bright-dark-rogue solutions for the coupled Schr$\ddot{\text{o}}$dinger equations. Chin Phys Lett, 2011, 28(11): 110202-110202
[18] Chowdury A, Ankiewicz A, Akhmediev N. Moving breathers and breather-to-soliton conversions for the Hirota equation. Proc Math Phys Eng Sci, 2015, 471(2180): Art 20150130
[19] Liu C, Yang Z Y, Zhao L C, et al. State transition induced by higher-order effects and background frequency. Phys Rev E, 2015, 91(2): Art 022904
[20] Wang L, Zhang J H, Wang Z Q, et al. Breather-to-soliton transitions, nonlinear wave interactions,modulational instability in a higher-order generalized nonlinear Schr$\ddot{\text{o}}$dinger equation. Phys Rev E, 2016, 93(1): Art 012214
[21] Wu Z J, Tian S F. Breather-to-soliton conversions and their mechanisms of the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation. Math Comput Simul, 2023, 210: 235-259
[22] Zhang J H, Wang L, Liu C. Superregular breathers, characteristics of nonlinear stage of modulation instability induced by higher-order effects. Proc R Soc A, 2017, 473(2199): Art 20160681
[23] Wang L, Zhang J H, Liu C, et al. Breather transition dynamics, Peregrine combs and walls,modulation instability in a variable-coefficient nonlinear Schr$\ddot{\text{o}}$dinger equation with higher-order effects. Phys Rev E, 2016, 93(6): Art 062217
[24] Wang L, Wu X, Zhang H Y. Superregular breathers and state transitions in a resonant erbium-doped fiber system with higher-order effects. Phys Lett A, 2018, 382(37): 2650-2654
[25] Yu F, Li L. Dynamics of some novel breather solutions and rogue waves for the PT-symmetric nonlocal soliton equations. Nonlinear Dyn, 2019, 95: 1867-1877
[26] Yuan F, Cheng Y, He J. Degeneration of breathers in the Kadomttsev-Petviashvili I equation. Commun Nonlinear Sci, 2020, 83: Art 105027
[27] Hu W Q, Gao Y T, Jia S L, et al. Periodic wave, breather wave and travelling wave solutions of a (2+1)-dimensional B-type Kadomtsev-Petviashvili equation in fluids or plasmas. Eur Phys J Plus, 2016, 131: 1-19
[28] Duan C, Yu F, Tian M. Some novel solitary wave characteristics for a generalized nonlocal nonlinear Hirota (GNNH) equation. Int J Nonlinear Sci Numer Simul, 2019, 20(3/4): 441-448
[29] Ma H, Yue S, Gao Y, et al. Lump solution, breather soliton and more soliton solutions for a (2+1)-dimensional generalized Benjamin-Ono equation. Qual Theory Dyn Syst, 2023, 22(2): Art 72
[30] Sun W R, Wang L, Xie X Y. Vector breather-to-soliton transitions and nonlinear wave interactions induced by higher-order effects in an erbium-doped fiber. Phys A, 2018, 499: 58-66
[31] Zhang X, Wang L, Liu C, et al. High-dimensional nonlinear wave transitions and their mechanisms. Chaos, 2020, 30(11): Art 113107
[32] Yin Z Y, Tian S F. Nonlinear wave transitions and their mechanisms of (2+1)-dimensional Sawada-Kotera equation. Physica D, 2021, 427: Art 133002
[33] Zhu C, Long C X, Zhou Y T, et al. Dynamics of multi-solitons, multi-lumps and hybrid solutions in (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani equation. Results Phys, 2022, 34: Art 105248
[34] Wei P F, Long C X, Zhu C, et al. Soliton molecules, multi-breathers and hybrid solutions in (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani equation. Chaos, Solitons & Fractals, 2022, 158: Art 112062
[35] Ma H, Gao Y, Deng A. Novel y-type and hybrid solutions for the (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani equation. Nonlinear Dyn, 2023, 111(5): 4645-4656
[36] Chen W, Tang L, Tian L. New interaction solutions of the KdV-Sawada-Kotera-Ramani equation in various dimensions. Phys Scr, 2023, 98(5): Art 055217
[37] Li L, Dai Z, Cheng B, et al. Nonlinear superposition between lump soliton and other nonlinear localized waves for the (2+1)-dimensional Korteweg-de Vries-Sawada-Kotera-Ramani equation. Results Phys, 2023, 49: Art 106516
[38] Osborne A R.Nonlinear ocean wave and the inverse scattering transform//Pike E R, Sabatier P C. Scattering: Scattering and Inverse Scattering in Pure and Applied Science. London: Academic Press, 2002: 637-666
[39] Wang D S, Zhu X. Direct and inverse scattering problems of the modified Sawada-Kotera equation: Riemann-Hilbert approach. Proc Math Phys Eng Sci, 2022, 478(2268): Art 20220541

NONLINEAR WAVE TRANSITIONS AND THEIR MECHANISMS OF THE (2+1)-DIMENSIONAL KORTEWEG-DE VRIES-SAWADA-KOTERA-RAMANI EQUATION

NONLINEAR WAVE TRANSITIONS AND THEIR MECHANISMS OF THE (2+1)-DIMENSIONAL KORTEWEG-DE VRIES-SAWADA-KOTERA-RAMANI EQUATION

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