Breather wave solutions on the Weierstrass elliptic periodic background for the \((2 + 1)\)-dimensional generalized variable-coefficient KdV equation. (English) Zbl 07855213
References:
| [1] | Craik, A. D., Wave Interactions and Fluid Flows, 1988, Cambridge University Press: Cambridge University Press, London · Zbl 0712.76002 |
| [2] | Hasegawa, A., Optical Soliton Theory and its Applications in Communication, 2002, Springer: Springer, Berlin |
| [3] | Malomed, B. A., Nonlinear optics: Symmetry breaking in laser cavities, Nat. Photonics, 9, 5, 287, 2015 · doi:10.1038/nphoton.2015.66 |
| [4] | Pitaevskii, L.; Stringari, S., Bose-Einstein Condensation and Superfluidity, 2016, Oxford University Press: Oxford University Press, Oxford · Zbl 1347.82004 |
| [5] | Bailung, H.; Sharma, S. K.; Nakamura, Y., Observation of Peregrine solitons in a multicomponent plasma with negative ions, Phys. Rev. Lett., 107, 25, 255005, 2011 · doi:10.1103/PhysRevLett.107.255005 |
| [6] | Madsen, P. A.; Bingham, H. B.; Liu, H., A new Boussinesq method for fully nonlinear waves from shallow to deep water, J. Fluid Mech., 462, 1-30, 2002 · Zbl 1061.76009 · doi:10.1017/S0022112002008467 |
| [7] | Yan, Z. Y., Vector financial rogue waves, Phys. Lett. A, 375, 48, 4274-4279, 2011 · Zbl 1254.91190 · doi:10.1016/j.physleta.2011.09.026 |
| [8] | Hirota, R., Direct Methods in Soliton Theory, 2004, Springer: Springer, Berlin |
| [9] | Boiti, M.; Pempinelli, F.; Pogrebkov, A. K.; Prinari, B., Inverse scattering theory of the heat equation for a perturbed one-soliton potential, J. Math. Phys., 43, 2, 1044-1062, 2002 · Zbl 1059.35112 · doi:10.1063/1.1427410 |
| [10] | Ganguly, A.; Das, A., Generalized Korteweg-de Vries equation induced from position-dependent effective mass quantum models and mass-deformed soliton solution through inverse scattering transform, J. Math. Phys., 55, 11, 112102, 2014 · Zbl 1319.35217 · doi:10.1063/1.4900895 |
| [11] | Zhang, G. Q.; Yan, Z. Y., Inverse scattering transforms and soliton solutions of focusing and defocusing nonlocal mKdV equations with non-zero boundary conditions, Physica D, 402, 132170, 2020 · Zbl 1453.37069 · doi:10.1016/j.physd.2019.132170 |
| [12] | Rogers, C.; Schief, W. K., Bäcklund and Darboux Transformations: Geometry and Modern Applications in Soliton Theory, 2002, Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1019.53002 |
| [13] | Tu, J. M.; Tian, S. F.; Xu, M. J.; Song, X. Q.; Zhang, T. T., Bäcklund transformation, infinite conservation laws and periodic wave solutions of a generalized \((3+1)\)-dimensional nonlinear wave in liquid with gas bubbles, Nonlinear Dyn., 83, 1199-1215, 2016 · Zbl 1351.37249 · doi:10.1007/s11071-015-2397-2 |
| [14] | Matveev, V. B.; Salle, M. A., Darboux Transformation and Solitons, 1991, Springer-Verlag: Springer-Verlag, Berlin · Zbl 0744.35045 |
| [15] | Wen, X. Y.; Yang, Y. Q.; Yan, Z. Y., Generalized perturbation (n, M)-fold Darboux transformations and multi-rogue-wave structures for the modified self-steepening nonlinear Schrödinger equation, Phys. Rev. E, 92, 012917, 2015 · doi:10.1103/PhysRevE.92.012917 |
| [16] | Wen, X. Y.; Yan, Z. Y.; Zhang, G. Q., Nonlinear self-dual network equations: Modulation instability, interactions of higher-order discrete vector rational solitons and dynamical behaviours, Proc. R. Soc. A, 476, 20200512, 2020 · Zbl 1472.78029 · doi:10.1098/rspa.2020.0512 |
| [17] | Gesztesy, F.; Holden, H., Soliton Equations and their Algebro-Geometric Solutions: Volume I, \((1+1)\)-Dimensional Continuous Models, 2003, Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1061.37056 |
| [18] | Gesztesy, F.; Holden, H., Soliton Equations and their Algebro-Geometric Solutions: Volume II, \((1+1)\)-Dimensional Continuous Models, 2008, Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1151.37056 |
| [19] | Pu, J. C.; Chen, Y., Data-driven forward-inverse problems for Yajima-Oikawa system using deep learning with parameter regularization, Commun. Nonlinear Sci. Numer. Simul., 118, 107051, 2023 · Zbl 1523.76019 · doi:10.1016/j.cnsns.2022.107051 |
| [20] | Zhou, Z. J.; Wang, L.; Yan, Z. Y., Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitons, Comput. Math. Appl., 151, 164-171, 2023 · Zbl 1541.35473 · doi:10.1016/j.camwa.2023.09.047 |
| [21] | Li, J. H.; Chen, J. C.; Li, B., Gradient-optimized physics-informed neural networks (GOPINNs): A deep learning method for solving the complex modified KdV equation, Nonlinear Dyn., 107, 781-792, 2022 · doi:10.1007/s11071-021-06996-x |
| [22] | Cao, C. W.; Geng, X. G., C Neumann and Bargmann systems associated with the coupled KdV soliton hierarchy, J. Phys. A: Math. Gen., 23, 18, 4117, 1990 · Zbl 0719.35082 · doi:10.1088/0305-4470/23/18/017 |
| [23] | Chen, J. B.; Pelinovsky, D. E., Rogue periodic waves of the focusing nonlinear Schrödinger equation, Proc. R. Soc. A: Math. Phys., 474, 2210, 20170814, 2018 · Zbl 1402.35256 · doi:10.1098/rspa.2017.0814 |
| [24] | Chen, J. B.; Zhang, R. S., The complex Hamiltonian systems and quasi-periodic solutions in the derivative nonlinear Schrödinger equations, Stud. Appl. Math., 145, 2, 153-178, 2020 · Zbl 1452.35180 · doi:10.1111/sapm.12311 |
| [25] | Chen, J. B.; Pelinovsky, D. E., Rogue waves on the background of periodic standing waves in the derivative nonlinear Schrödinger equation, Phys. Rev. E, 103, 6, 062206, 2021 · doi:10.1103/PhysRevE.103.062206 |
| [26] | Chen, J. B.; Pelinovsky, D. E., Rogue periodic waves of the modified KdV equation, Nonlinearity, 31, 5, 1955, 2018 · Zbl 1393.35201 · doi:10.1088/1361-6544/aaa2da |
| [27] | Chen, J. B.; Pelinovsky, D. E., Periodic travelling waves of the modified KdV equation and rogue waves on the periodic background, J. Nonlinear Sci., 29, 6, 2797-2843, 2019 · Zbl 1427.35226 · doi:10.1007/s00332-019-09559-y |
| [28] | Zhang, H. Q.; Chen, F.; Pei, Z. J., Rogue waves of the fifth-order Ito equation on the general periodic travelling wave solutions background, Nonlinear Dyn., 103, 1023-1033, 2021 · Zbl 1516.35173 · doi:10.1007/s11071-020-06153-w |
| [29] | Zhang, H. Q.; Chen, F., Rogue waves for the fourth-order nonlinear Schrödinger equation on the periodic background, Chaos, 31, 2, 023129, 2021 · Zbl 1467.35273 · doi:10.1063/5.0030072 |
| [30] | Li, R. M.; Geng, X. G., Rogue periodic waves of the sine-Gordon equation, Appl. Math. Lett., 102, 106147, 2020 · Zbl 1440.35030 · doi:10.1016/j.aml.2019.106147 |
| [31] | Li, R. M.; Geng, X. G., Rogue waves and breathers of the derivative Yajima-Oikawa long wave-short wave equations on theta-function backgrounds, J. Math. Anal. Appl., 527, 1, 127399, 2023 · Zbl 1530.35255 · doi:10.1016/j.jmaa.2023.127399 |
| [32] | Geng, X. G.; Li, R. M.; Xue, B., A vector Geng-Li model: New nonlinear phenomena and breathers on periodic background waves, Physica D, 434, 133270, 2022 · Zbl 1490.35093 · doi:10.1016/j.physd.2022.133270 |
| [33] | Li, R. M.; Geng, J. R.; Geng, X. G., Rogue-wave and breather solutions of the Fokas-Lenells equation on theta-function backgrounds, Appl. Math. Lett., 142, 108661, 2023 · Zbl 1530.35285 · doi:10.1016/j.aml.2023.108661 |
| [34] | Kedziora, D. J.; Ankiewicz, A.; Akhmediev, N., Rogue waves and solitons on a cnoidal background, Eur. Phys. J. Spec. Top., 223, 1, 43-62, 2014 · doi:10.1140/epjst/e2014-02083-4 |
| [35] | Chin, S. A.; Ashour, O. A.; Nikolić, S. N.; Belić, M. R., Peak-height formula for higher-order breathers of the nonlinear Schrödinger equation on nonuniform backgrounds, Phys. Rev. E, 95, 1, 012211, 2017 · doi:10.1103/PhysRevE.95.012211 |
| [36] | Lou, S. Y.; Cheng, X. P.; Tang, X. Y., Dressed dark solitons of the defocusing nonlinear Schrödinger equation, Chin. Phys. Lett., 31, 7, 070201, 2014 · doi:10.1088/0256-307X/31/7/070201 |
| [37] | Lin, J.; Jin, X. W.; Gao, X. L.; Lou, S. Y., Solitons on a periodic wave background of the modified KdV-Sine-Gordon equation, Commun. Theor. Phys., 70, 2, 119, 2018 · Zbl 1451.35168 · doi:10.1088/0253-6102/70/2/119 |
| [38] | Feng, B. F.; Ling, L. M.; Takahashi, D. A., Multi-breather and high-order rogue waves for the nonlinear Schrödinger equation on the elliptic function background, Stud. Appl. Math., 144, 1, 46-101, 2020 · Zbl 1454.35338 · doi:10.1111/sapm.12287 |
| [39] | Peng, W. Q.; Pu, J. C.; Chen, Y., PINN deep learning method for the Chen-Lee-Liu equation: Rogue wave on the periodic background, Commun. Nonlinear Sci., 105, 106067, 2022 · Zbl 1497.35440 · doi:10.1016/j.cnsns.2021.106067 |
| [40] | Yang, Y. Q.; Dong, H. H.; Chen, Y., Darboux-Bäcklund transformation and localized excitation on the periodic wave background for the nonlinear Schrödinger equation, Wave Motion, 106, 102787, 2021 · Zbl 1524.35612 · doi:10.1016/j.wavemoti.2021.102787 |
| [41] | Davis, R. E.; Acrivos, A., Solitary internal waves in deep water, J. Fluid Mech., 29, 593-607, 1967 · Zbl 0147.46503 · doi:10.1017/S0022112067001041 |
| [42] | Farmer, D. M.; Smith, J. D., Tidal interaction of stratified flow with a sill in Knight Inlet, Deep-Sea Res. A, 27, 239-254, 1980 · doi:10.1016/0198-0149(80)90015-1 |
| [43] | Xu, G.; Chabchoub, A.; Pelinovsky, D. E.; Kibler, B., Observation of modulation instability and rogue breathers on stationary periodic waves, Phys. Rev. Res., 2, 3, 033528, 2020 · doi:10.1103/PhysRevResearch.2.033528 |
| [44] | Shultz, J. L.; Salamo, G. J., Experimental observation of the continuous pulse-train soliton solution to the Maxwell-Bloch equations, Phys. Rev. Lett., 78, 5, 855, 1997 · doi:10.1103/PhysRevLett.78.855 |
| [45] | Mo, C.; Singh, J.; Raney, J. R.; Purohit, P. K., Cnoidal wave propagation in an elastic metamaterial, Phys. Rev. E, 100, 1, 013001, 2019 · doi:10.1103/PhysRevE.100.013001 |
| [46] | Yan, Z. Y.; Dai, C. Q., Optical rogue waves in the generalized inhomogeneous higher-order nonlinear Schrödinger equation with modulating coefficients, J. Opt., 15, 064012, 2013 · doi:10.1088/2040-8978/15/6/064012 |
| [47] | Yang, Y. Q.; Wang, X.; Yan, Z. Y., Optical temporal rogue waves in the generalized inhomogeneous nonlinear Schrödinger equation with varying higher-order even and odd terms, Nonlinear Dyn., 81, 833-842, 2015 · Zbl 1347.37120 · doi:10.1007/s11071-015-2033-1 |
| [48] | Peng, Y. Z., A new \((2+1)\)-dimensional KdV equation and its localized structures, Commun. Theor. Phys., 54, 5, 863, 2010 · Zbl 1220.35154 · doi:10.1088/0253-6102/54/5/17 |
| [49] | Toda, K.; Yu, S. J., The investigation into the Schwarz-Korteweg-de Vries equation and the Schwarz derivative in \((2+1)\) dimensions, J. Math. Phys., 41, 7, 4747-4751, 2000 · Zbl 1031.37062 · doi:10.1063/1.533374 |
| [50] | Wang, Y.; Chen, Y., Binary Bell polynomial manipulations on the integrability of a generalized \((2+1)\)-dimensional Korteweg-de Vries equation, J. Math. Anal. Appl., 400, 2, 624-634, 2013 · Zbl 1258.35180 · doi:10.1016/j.jmaa.2012.11.028 |
| [51] | Pu, J. C.; Chen, Y., Integrability and exact solutions of the \((2+1)\)-dimensional KdV equation with Bell polynomials approach, Acta Math. Appl. Sin-E, 38, 4, 861-881, 2022 · Zbl 1501.35351 · doi:10.1007/s10255-022-1020-9 |
| [52] | Wazwaz, A. M., A new \((2+1)\)-dimensional Korteweg-de Vries equation and its extension to a new \((3+1)\)-dimensional Kadomtsev-Petviashvili equation, Phys. Scr., 84, 3, 035010, 2011 · Zbl 1260.35191 · doi:10.1088/0031-8949/84/03/035010 |
| [53] | Zhao, Z.; Han, B., The Riemann-Bäcklund method to a quasiperiodic wave solvable generalized variable coefficient \((2+1)\)-dimensional KdV equation, Nonlinear Dyn., 87, 2661-2676, 2017 · Zbl 1373.37165 · doi:10.1007/s11071-016-3219-x |
| [54] | Lü, X.; Lin, F.; Qi, F., Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions, Appl. Math. Model., 39, 12, 3221-3226, 2015 · Zbl 1443.35135 · doi:10.1016/j.apm.2014.10.046 |
| [55] | Chen, J. C.; Ma, Z. Y., Consistent Riccati expansion solvability and soliton-cnoidal wave interaction solution of a \((2+1)\)-dimensional Korteweg-de Vries equation, Appl. Math. Lett., 64, 87-93, 2017 · Zbl 1354.35128 · doi:10.1016/j.aml.2016.08.016 |
| [56] | Pastras, G., The Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics: A Primer for Advanced Undergraduates, 2020, Springer Nature: Springer Nature, Berlin · Zbl 1454.33001 |
| [57] | Krichever, I. M., Elliptic solutions of nonlinear integrable equations and related topics, Acta Appl. Math., 36, 7-25, 1994 · Zbl 0811.35123 · doi:10.1007/BF01001540 |
| [58] | Brezhnev, Y. V., Elliptic solitons and Gröbner bases, J. Math. Phys., 45, 2, 696-712, 2004 · Zbl 1070.37050 · doi:10.1063/1.1633353 |
| [59] | Li, X.; Zhang, D. J., Elliptic soliton solutions: \( \tau\) functions, vertex operators and bilinear identities, J. Nonlinear Sci., 32, 70, 2020 · Zbl 1503.37075 · doi:10.1007/s00332-022-09835-4 |
| [60] | Nijhoff, F. W.; Sun, Y. Y.; Zhang, D. J., Elliptic solutions of Boussinesq type lattice equations and the elliptic \(N\) th root of unity, Commun. Math. Phys., 399, 599-650, 2023 · Zbl 1518.39003 · doi:10.1007/s00220-022-04567-8 |
| [61] | Gesztesy, F.; Weikard, R., Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies-an analytic approach, Bull. Amer. Math. Soc., 35, 271-317, 1998 · Zbl 0909.34073 · doi:10.1090/S0273-0979-98-00765-4 |
| [62] | Zakharov, V. E., Description of the \(n\)-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type, I: Integration of the Lamé equations, Duke Math. J., 94, 1, 103-139, 1998 · Zbl 0963.37068 · doi:10.1215/S0012-7094-98-09406-6 |
| [63] | Eilbeck, J. C.; Enolskii, V. Z.; Previato, E., Varieties of elliptic solitons, J. Phys. A: Math. Gen., 34, 2215-2227, 2001 · Zbl 0988.35138 · doi:10.1088/0305-4470/34/11/314 |
| [64] | Matveev, V. B.; Smirnov, A. O., Elliptic solitons and “Freak waves”, St. Petersburg Math. J., 33, 3, 523-551, 2022 · Zbl 1497.37080 · doi:10.1090/spmj/1713 |
| [65] | Brezhnev, Y. V., Elliptic solitons with free constants and their isospectral deformations, Rep. Math. Phys., 48, 1-2, 39-46, 2001 · Zbl 1011.37044 · doi:10.1016/S0034-4877(01)80062-8 |
| [66] | Sun, Y. Y.; Sun, W. Y., An update of a Bäcklund transformation and its applications to the Boussinesq system, Appl. Math. Comput., 421, 126964, 2022 · Zbl 1510.37110 · doi:10.1016/j.amc.2022.126964 |
| [67] | Rudneva, D.; Zabrodin, A., Dynamics of poles of elliptic solutions to the BKP equation, J. Phys. A: Math. Theor., 53, 7, 075202, 2020 · Zbl 1514.37090 · doi:10.1088/1751-8121/ab63a8 |
| [68] | Sun, W. Y.; Sun, Y. Y., The degenerate breather solutions for the Boussinesq equation, Appl. Math. Lett., 128, 107884, 2022 · Zbl 1484.35133 · doi:10.1016/j.aml.2021.107884 |
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