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The Riemann-Bäcklund method to a quasiperiodic wave solvable generalized variable coefficient \((2+1)\)-dimensional KdV equation. (English) Zbl 1373.37165
Summary: In this paper, the Riemann-Bäcklund method is extended to a generalized variable coefficient \((2+1)\)-dimensional Korteweg-de Vries equation. The soliton and quasiperiodic wave solutions are investigated systematically. The relations between the quasiperiodic wave solutions and the soliton solutions are rigorously established by a limiting procedure. It is proved that the periodic wave solutions tend to the soliton solutions under a small amplitude limit. Furthermore, the propagation characteristics of the soliton solutions and periodic wave solutions are discussed through the graphical analysis.

MSC:
37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions
34C25 Periodic solutions to ordinary differential equations
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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[1] Olver, P.J.: Application of Lie Groups to Differential Equations. Springer, New York (1993) · Zbl 0785.58003
[2] Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2002) · Zbl 1013.34004
[3] Hirota, R; Bullough, RK (ed.); Caudrey, PJ (ed.), Direct methods in soliton theory, 157-176, (1980), Berlin
[4] Hrota, R, Exact N-soliton solutions of the wave equation of long wave in shallow-water and in nonlinear lattice, J. Math. Phys., 14, 810-814, (1973) · Zbl 0261.76008
[5] Ablowitz, M.J., Clarkson, P.A.: Solitons; Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991) · Zbl 0762.35001
[6] Boiti, M; Pempinelli, F; Pogrebkov, AK; Prinari, B, Inverse scattering theory of the heat equation for a perturbed one-soliton potential, J. Math. Phys., 43, 1044, (2002) · Zbl 1059.35112
[7] 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, 112102, (2014) · Zbl 1319.35217
[8] Rangwala, AA; Rao, JA, Bäcklund transformations, soliton solutions and wave functions of Kaup-Newell and wadati-konno-ichikawa systems, J. Math. Phys., 31, 1126, (1990) · Zbl 0704.58024
[9] Karasu, A; Sakovich, SY, Bäcklund transformation and special solutions for the Drinfeld-Sokolov-Satsuma-Hirota system of coupled equations, J. Phys. A: Math. Gen., 34, 7355-7358, (2001) · Zbl 0983.35116
[10] Matveev, V.A., Salle, M.A.: Darboux Transformation and Soliton. Springer, Berlin (1991) · Zbl 0744.35045
[11] Zhao, HH; Zhao, XJ; Hao, HQ, Breather-to-soliton conversions and nonlinear wave interactions in a coupled Hirota system, Appl. Math. Lett., 61, 8-12, (2016) · Zbl 1347.35068
[12] Guo, R; Hao, HQ, Breathers and localized solitons for the Hirota-Maxwell-Bloch system on constant backgrounds in erbium doped fibers, Ann. Phys., 344, 10-16, (2014) · Zbl 1343.81265
[13] Guo, R; Hao, HQ, Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul., 18, 2426-2435, (2013) · Zbl 1304.35641
[14] Gesztesy, F., Holden, H.: Soliton Equations and Their Algebro-Geometric Solutions I: (\(1+1\))-Dimensional Continuous Models. Cambridge University Press, New York (2003) · Zbl 1061.37056
[15] Gesztesy, F., Holden, H.: Soliton Equations and Their Algebro-Geometric Solutions II: (\(1+1\))-Dimensional Discrete Models. Cambridge University Press, New York (2008) · Zbl 1151.37056
[16] Nakamura, A, A direct method of calculating periodic wave solutions to nonlinear evolution equations. I. exact two-periodic wave solution, J. Phys. Soc. Jpn., 47, 1701-1705, (1979) · Zbl 1334.35006
[17] Nakamura, A, A direct method of calculating periodic wave solutions to nonlinear evolution equations. II. exact one- and two-periodic wave solution of the coupled bilinear equations, J. Phys. Soc. Jpn., 48, 1365-1370, (1980) · Zbl 1334.35250
[18] Fan, EG; Hon, YC, Quasiperiodic waves and asymptotic behaviour for bogoyavlenskii’s breaking soliton equation in (\(2+1\)) dimensions, Phys. Rev. E, 78, 036607, (2008)
[19] Fan, EG; Hon, YC, On a direct procedure for the quasi-periodic wave solutions of the supersymmetric ito’s equation, Rep. Math. Phys., 66, 355-365, (2010) · Zbl 1236.81114
[20] Fan, EG, Supersymmetric KdV-Sawada-Kotera-Ramani equation and its quasi-periodic wave solutions, Phys. Lett. A, 374, 744-749, (2010) · Zbl 1235.35242
[21] Ma, WX; Zhou, RG; Gao, L, Exact one-periodic and two-periodic wave solutions to Hirota bilinear equations in (\(2+1\)) dimensions, Mod. Phys. Lett. A, 21, 1677-1688, (2009) · Zbl 1168.35426
[22] Fan, EG, Quasi-periodic waves and an asymptotic property for the asymmetrical Nizhnik-Novikov-Veselov equation, J. Phys. A: Math. Theor., 42, 095206, (2009) · Zbl 1165.35044
[23] Chen, YR; Song, M; Liu, ZR, Soliton and Riemann theta function quasi-periodic wave solutions for a (\(2+1\))-dimensional generalized shallow water wave equation, Nonlinear Dyn., 82, 333-347, (2015) · Zbl 1348.76038
[24] Qiao, ZJ; Fan, EG, Negative-order Korteweg-de Vries equtions, Phys. Rev. E, 86, 016601, (2012)
[25] Zhao, ZL; Han, B, Quasiperiodic wave solutions of a (\(2+1\))-dimensional generalized breaking soliton equation via bilinear Bäcklund transformation, Eur. Phys. J. Plus, 131, 128, (2016)
[26] Wang, YH; Chen, Y, Binary Bell polynomial manipulations on the integrability of a generalized (\(2+1\))-dimensional Korteweg-de Vries equation, J. Math. Anal. Appl., 400, 624-634, (2013) · Zbl 1258.35180
[27] Kovalyov, M, Uncertainty principle for the nonlinear waves of the Korteweg-de Vries equation, Chaos Solitons Fractals, 32, 431-444, (2007) · Zbl 1139.35087
[28] Toda, K; Yu, SJ, The investigation into the Schwarz-Korteweg-de Vries equation and the Schwarz derivative in (\(2+1\)) dimensions, J. Math. Phys., 41, 4747-4751, (2000) · Zbl 1031.37062
[29] Peng, YZ, A new (\(2+1\))-dimensional KdV equation and its localized structures, Commun. Theor. Phys., 54, 863-865, (2010) · Zbl 1220.35154
[30] Lü, X; Tian, B; Zhang, HQ; Xu, T; Li, H, Generalized (\(2+1\))-dimensional gardner model: bilinear equations, Bäcklund transformation, Lax representation and interaction mechanisms, Nonlinear Dyn., 67, 2279-2290, (2012) · Zbl 1247.35107
[31] Wang, YP; Tian, B; Wang, M; Wang, YF; Sun, Y; Xie, XY, Bäcklund transformations and soliton solutions for a (\(2+1\))-dimensional Korteweg-de Vries-type equation in water waves, Nonlinear Dyn., 81, 1815-1821, (2015) · Zbl 1348.37108
[32] Lü, X, New bilinear Bäcklund transformation with multisoliton solutions for the (\(2+1\))-dimensional Sawada-Kotera model, Nonlinear Dyn., 76, 161-168, (2014) · Zbl 1319.35222
[33] Huang, Y, New no-traveling wave solutions for the Liouville equation by Bäcklund transformation method, Nonlinear Dyn., 72, 87-90, (2013)
[34] Fan, EG; Chow, KW, Darboux covariant Lax pairs and infinite conservation laws of the (\(2+1\))-dimensional breaking soliton equation, J. Math. Phys., 52, 023504, (2011) · Zbl 1314.35138
[35] Fan, EG, The integrability of nonisospectral and variable-coefficient KdV equation with binary Bell polynomials, Phys. Lett. A, 375, 493-497, (2011) · Zbl 1241.35176
[36] Fan, EG; Hon, YC, Super extension of Bell polynomials with applications to supersymmetric equations, J. Math. Phys., 53, 013503, (2012) · Zbl 1273.81107
[37] Wadati, M; Sanuki, H; Konno, K, Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws, Prog. Theor. Phys., 53, 419-436, (1975) · Zbl 1079.35506
[38] Konno, K., Wadati, M.: Simple derivation of Bäcklund transformation from Riccati form of inverse method. Prog. Theor. Phys. 53, 1652-1656 (1975) · Zbl 1079.35505
[39] Ma, WX, Lump solutions to the Kadomtsev-Petviashvili equation, Phys. Lett. A, 379, 1975-1978, (2015) · Zbl 1364.35337
[40] Ma, WX; Qin, ZY; Lü, X, Lump solutions to dimensionally reduced p-gkp and p-gbkp equations, Nonlinear Dyn., 84, 923-931, (2016) · Zbl 1354.35127
[41] Lü, X; Ma, WX, Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation, Nonlinear Dyn., 85, 1217-1222, (2016) · Zbl 1355.35159
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