Quasi-periodic Waves and Solitary Waves to a Generalized KdV-Caudrey-Dodd-Gibbon Equation from Fluid Dynamics

In this paper, a generalized KdV-Caudrey-Dodd-Gibbon (KdV-CDG) equation is investigated, which describes certain situations in the fluid mechanics, ocean dynamics and plasma physics. By using Bell polynomials, a lucid and systematic approach is proposed to systematically study its Hirota's bilinear form and $N$-soliton solution, respectively. Furthermore, based on the Riemann theta function, the one-quasi- and two-quasi-periodic wave solutions are also constructed. Finally, an asymptotic relation of the quasi-periodic wave solutions are strictly analyzed to reveal the relations between quasi-periodic wave solutions and soliton solutions.

[1]  Wen-Xiu Ma,et al.  EXACT ONE-PERIODIC AND TWO-PERIODIC WAVE SOLUTIONS TO HIROTA BILINEAR EQUATIONS IN (2+1) DIMENSIONS , 2008, 0812.4316.

[2]  S. Lou.  Extended Painlevé Expansion, Nonstandard Truncation and Special Reductions of Nonlinear Evolution Equations , 1998 .

[3]  J. Nimmo,et al.  On the combinatorics of the Hirota D-operators , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[4]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[5]  G. Bluman,et al.  Symmetries and differential equations , 1989 .

[6]  Chen Yong,et al.  Binary Bell Polynomials, Bilinear Approach to Exact Periodic Wave Solutions of (2 + 1)-Dimensional Nonlinear Evolution Equations , 2011 .

[7]  Colin Rogers,et al.  On reciprocal properties of the Caudrey-Dodd-Gibbon and Kaup-Kupershmidt hierarchies , 1987 .

[8]  Wenxiu Ma,et al.  Bilinear Equations and Resonant Solutions Characterized by Bell Polynomials , 2013 .

[9]  Juncheng Wei,et al.  Symbiotic bright solitary wave solutions of coupled nonlinear Schrödinger equations , 2006, math/0610133.

[10]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[11]  Johan Springael,et al.  Classical Darboux transformations and the KP hierarchy , 2001 .

[12]  Shou-Fu Tian,et al.  Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations , 2010 .

[13]  Engui Fan,et al.  Quasi-periodic waves and an asymptotic property for the asymmetrical Nizhnik–Novikov–Veselov equation , 2009 .

[14]  Akira Nakamura,et al.  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 , 1980 .

[15]  S. Sawada,et al.  A Method for Finding N-Soliton Solutions of the KdV and KdV-Like Equation , 1974 .

[16]  Shou-Fu Tian,et al.  Lie symmetries and nonlocally related systems of the continuous and discrete dispersive long waves system by geometric approach , 2015 .

[17]  Peter A. Clarkson,et al.  THE DIRECT METHOD IN SOLITON THEORY (Cambridge Tracts in Mathematics 155) , 2006 .

[18]  Xing-Biao Hu,et al.  An integrable symmetric (2+1)-dimensional Lotka–Volterra equation and a family of its solutions , 2005 .

[19]  Yong Chen,et al.  PDEBellII: A Maple package for finding bilinear forms, bilinear Bäcklund transformations, Lax pairs and conservation laws of the KdV-type equations , 2014, Comput. Phys. Commun..

[20]  Shou-Fu Tian,et al.  On the Lie algebras, generalized symmetries and darboux transformations of the fifth-order evolution equations in shallow water , 2015 .

[21]  広田 良吾,et al.  The direct method in soliton theory , 2004 .

[22]  Wenxiu Ma,et al.  Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions , 2004, nlin/0503001.

[23]  M. Wadati,et al.  Relationships among Inverse Method, Bäcklund Transformation and an Infinite Number of Conservation Laws , 1975 .

[25]  Engui Fan,et al.  Quasiperiodic waves and asymptotic behavior for Bogoyavlenskii's breaking soliton equation in (2+1) dimensions. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  Kwok Wing Chow,et al.  A class of exact, periodic solutions of nonlinear envelope equations , 1995 .

[27]  Zhengde Dai,et al.  Exact soliton solutions for the fifth-order Sawada-Kotera equation , 2008, Appl. Math. Comput..

[28]  Ping Zhang,et al.  Incompressible and Compressible Limits of Coupled Systems of Nonlinear Schrödinger Equations , 2006 .

[29]  Shou-Fu Tian,et al.  A kind of explicit Riemann theta functions periodic waves solutions for discrete soliton equations , 2011 .

[30]  V. Matveev,et al.  Darboux Transformations and Solitons , 1992 .

[31]  Jie Ji,et al.  Soliton scattering with amplitude changes of a negative order AKNS equation , 2009 .