On the local fractional variable‐coefficient Ablowitz–Kaup–Newell–Segur hierarchy: Hamiltonian structure, localization of nonlocal symmetries, and exact solution of reduced equations

2021 ◽  
Author(s):  
Chao Yue ◽  
Xiaoyan Wang
Keyword(s):  
Exact Solution ◽  
Variable Coefficient ◽  
Hamiltonian Structure ◽  
Nonlocal Symmetries ◽  
Reduced Equations

scholarly journals Exact solution of variable coefficient mixed hyperbolic partial differential problems

2003 ◽  
Vol 16 (3) ◽  
pp. 309-312 ◽  
Author(s):  
M.J. Rodriguez-Alvarez ◽  
G. Rubio ◽  
L. Jódar ◽  
A.E. Posso
Keyword(s):  
Exact Solution ◽  
Variable Coefficient ◽  
Partial Differential

On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves

1994 ◽  
Vol 272 ◽  
pp. 1-20 ◽  
Author(s):  
Vladimir P. Krasitskii
Keyword(s):  
Power Series ◽  
Surface Waves ◽  
Gravity Waves ◽  
Hamiltonian Structure ◽  
Classical Method ◽  
Weakly Nonlinear ◽  
Integral Power ◽  
Reduced Equations ◽  
Nonlinear Surface Waves ◽  
Nonlinear Surface

Many studies of weakly nonlinear surface waves are based on so-called reduced integrodifferential equations. One of these is the widely used Zakharov four-wave equation for purely gravity waves. But the reduced equations now in use are not Hamiltonian despite the Hamiltonian structure of exact water wave equations. This is entirely due to shortcomings of their derivation. The classical method of canonical transformations, generalized to the continuous case, leads automatically to reduced equations with Hamiltonian structure. In this paper, attention is primarily paid to the Hamiltonian reduced equation describing the combined effects of four- and five-wave weakly nonlinear interactions of purely gravity waves. In this equation, for brevity called five-wave, the non-resonant quadratic, cubic and fourth-order nonlinear terms are eliminated by suitable canonical transformation. The kernels of this equation and the coefficients of the transformation are expressed in explicit form in terms of expansion coefficients of the gravity-wave Hamiltonian in integral-power series in normal variables. For capillary–gravity waves on a fluid of finite depth, expansion of the Hamiltonian in integral-power series in a normal variable with accuracy up to the fifth-order terms is also given.

scholarly journals NONLOCAL SYMMETRIES AND EXACT SOLUTIONS OF A VARIABLE COEFFICIENT AKNS SYSTEM

2020 ◽  
Vol 10 (6) ◽  
pp. 2669-2681
Author(s):  
Xiangpeng Xin ◽  
◽  
Lihua Zhang ◽  
Yarong Xia ◽  
Hanze Liu ◽  
...  
Keyword(s):  
Exact Solutions ◽  
Variable Coefficient ◽  
Nonlocal Symmetries ◽  
Akns System

scholarly journals Hamiltonian formulation for the description of interfacial solitary waves

1998 ◽  
Vol 5 (1) ◽  
pp. 3-12 ◽  
Author(s):  
R. Grimshaw ◽  
S. R. Pudjaprasetya
Keyword(s):  
Solitary Waves ◽  
Variable Coefficient ◽  
Hamiltonian Structure ◽  
Vries Equation ◽  
Hamiltonian Formulation ◽  
Constant Density ◽  
Two Fluids ◽  
De Vries ◽  
Lower Fluid ◽  
New Equation

Abstract. We consider solitary waves propagating on the interface between two fluids, each of constant density, for the case when the upper fluid is bounded above by a rigid horizontal plane, but the lower fluid has a variable depth. It is well-known that in this situation, the solitary waves can be described by a variable-coefficient Korteweg-de Vries equation. Here we reconsider the derivation of this equation and present a formulation which preserves the Hamiltonian structure of the underlying system. The result is a new variable-coefficient Korteweg-de Vries equation, which conserves energy to a higher order than the more conventional well-known equation. The new equation is used to describe the transformation of an interfacial solitary wave which propagates into a region of decreasing depth.

scholarly journals Lie symmetry and exact solution of (2+1)-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients

2013 ◽  
Vol 17 (5) ◽  
pp. 1490-1493
Author(s):  
Hong-Cai Ma ◽  
Zhen-Yun Qin ◽  
Ai-Ping Deng
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Variable Coefficient ◽  
Direct Method ◽  
Symmetry Transformation ◽  
Solution Procedure ◽  
Variable Coefficients ◽  
Lie Symmetry ◽  
Petviashvili Equation ◽  
Non Linear Systems

The simple direct method is adopted to find Non-Auto-Backlund transformation for variable coefficient non-linear systems. The (2+1)-dimensional generalized Kadomtsev-Petviashvili equation with variable coefficients is used as an example to elucidate the solution procedure, and its symmetry transformation and exact solutions are obtained.

Slowly varying solitary waves. II. Nonlinear Schrödinger equation

1979 ◽  
Vol 368 (1734) ◽  
pp. 377-388 ◽  
Keyword(s):  
Exact Solution ◽  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Variable Coefficient ◽  
Schrodinger Equation ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Scale Method

The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient nonlinear Schrodinger equation. A multiple scale method is used to determine the amplitude and phases of the wave to the second order in the perturbation parameter. The method is similar to that used in (I) (R. Grimshaw 1979 Proc. R. Soc. Lond . A 368, 359). The results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.

Slowly varying solitary waves. I. Korteweg-de Vries equation

1979 ◽  
Vol 368 (1734) ◽  
pp. 359-375 ◽  
Keyword(s):  
Exact Solution ◽  
Solitary Wave ◽  
Asymptotic Solution ◽  
Solitary Waves ◽  
Variable Coefficient ◽  
Vries Equation ◽  
Multiple Scale ◽  
Perturbation Parameter ◽  
Scale Method ◽  
De Vries

The slowly varying solitary wave is constructed as an asymptotic solution of the variable coefficient Korteweg-de Vries equation. A multiple scale method is used to determine the amplitude and phase of the wave to the second order in the perturbation parameter. The structure ahead and behind the solitary wave is also determined, and the results are interpreted by using conservation laws. Outer expansions are introduced to remove non-uniformities in the expansion. Finally, when the coefficients satisfy a certain constraint, an exact solution is constructed.

Hamiltonian structure of the reduced equations for the ponderomotive interaction between a whistler and fast magnetic sound waves

1989 ◽  
Vol 139 (8) ◽  
pp. 423-425 ◽  
Author(s):  
V.I. Karpman ◽  
J.Juul Rasmussen ◽  
S.K. Turitsyn
Keyword(s):  
Hamiltonian Structure ◽  
Sound Waves ◽  
Reduced Equations ◽  
Magnetic Sound
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