Analytic Solitary-wave Solutions for Modified Korteweg - de Vries Equation with t-dependent Coefficients

2001 ◽  
Vol 56 (5) ◽  
pp. 366-370 ◽  
Author(s):  
Woo-Pyo Hong ◽  
Myung-Sang Yoona
Keyword(s):  
Solitary Wave ◽  
Symbolic Computation ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Modified Kdv Equation ◽  
De Vries ◽  
Truncated Painlevé Expansion

Abstract We find analytic solitary wave solutions for a modified KdV equation with t-dependent coefficients of the form ut - 6α(t)uux + ß (t) uxxx -6γu2ux = 0. We make use of both the application of the truncated Painleve expansion and symbolic computation to obtain an auto-Bäcklund transformation. We show that kink-type analytic solitary-wave solutions exist under some constraints on α (t), ß (t) and γ.

New Non-traveling Solitary Wave Solutions for a Second-order Korteweg-de Vries Equation

1999 ◽  
Vol 54 (6-7) ◽  
pp. 375-378 ◽  
Author(s):  
Woo-Pyo Hong ◽  
Young-Dae Jung
Keyword(s):  
Solitary Wave ◽  
Symbolic Computation ◽  
Traveling Wave ◽  
Vries Equation ◽  
Kdv Equation ◽  
Traveling Wave Solutions ◽  
Second Order ◽  
Solitary Wave Solutions ◽  
Current Interest ◽  
Wave Solutions

Abstract Modeling the propagation of two different wave modes simultaneously, the second-order KdV equation is of current interest. Applying a tanh-typed method with symbolic computation, we have found certain new analytic soliton-typed solutions which go beyond the the previously obtained traveling wave solutions.

Auto-Bäcklund Transformation and Solitary-wave Solutions to Non- integrable Generalized Fifth-order Nonlinear Evolution Equations

1999 ◽  
Vol 54 (8-9) ◽  
pp. 549-553 ◽  
Author(s):  
Woo-Pyo Hong ◽  
Young-Dae Jung
Keyword(s):  
Solitary Wave ◽  
Symbolic Computation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
New Class ◽  
Wave Solutions ◽  
Fifth Order ◽  
Truncated Painlevé Expansion

We show that the application of the truncated Painlevé expansion and symbolic computation leads to a new class of analytical solitary-wave solutions to the general fifth-order nonlinear evolution equations which include Lax, Sawada-Kotera (SK), Kaup-Kupershmidt (KK), and Ito equations. Some explicit solitary-wave solutions are presented.

scholarly journals The stability of solitary waves

1972 ◽  
Vol 328 (1573) ◽  
pp. 153-183 ◽  
Keyword(s):  
Solitary Wave ◽  
Spectral Theory ◽  
Solitary Waves ◽  
Vries Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Long Time ◽  
Stability Of Solitary Waves ◽  
The Stability

The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary waves. The present analysis establishes that these solutions are stable, confirming a property that has for a long time been presumed. The demonstration of stability hinges on two nonlinear functionals which for solutions of the Korteweg-de Vries equation are invariant with time: these are introduced in § 2, where it is recalled that Boussinesq recognized their significance in relation to the stability of solitary waves. The principles upon which the stability theory is based are explained in § 3, being supported by a few elementary ideas from functional analysis. A proof that solitary wave solutions are stable is completed in § 4, the most exacting steps of which are accomplished by means of spectral theory. In appendix A a method deriving from the calculus of variations is presented, whereby results needed for the proof of stability may be obtained independently of spectral theory as used in § 4. In appendix B it is shown how the stability analysis may readily be adapted to solitary-wave solutions of the ‘regularized long-wave equation’ that has recently been advocated by Benjamin, Bona & Mahony as an alternative to the Korteweg-de Vries equation. In appendix C a variational principle is demonstrated relating to the exact boundaryvalue problem for solitary waves in water: this is a counterpart to a principle used in the present work (introduced in §2) and offers some prospect of proving the stability of exact solitary waves.

Exact Periodic Solitary Wave and Double Periodic Wave Solutions for the (2+1)-Dimensional Korteweg-de Vries Equation

2009 ◽  
Vol 64 (9-10) ◽  
pp. 609-614 ◽  
Author(s):  
Changfu Liu ◽  
Zhengde Dai
Keyword(s):  
Solitary Wave ◽  
Function Method ◽  
Kdv Equation ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Trial Function Method ◽  
A New Technique

A new technique, the extended ansatz function method, is proposed to seek periodic solitary wave solutions of integrable systems. Exact periodic solitary wave solutions for the (2+1)-dimensional Korteweg-de Vries (KdV) equation are obtained by using this technique. By using the trial function method, Jacobi elliptic function double periodic solutions are also constructed for this equation. This result shows that there exist periodic solitary waves in the different directions for the (2+1)- dimensional KdV equation

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