scholarly journals Remarks on an integrable evolution equation

2002 ◽  
Vol 55 ◽  
pp. 77-85
Author(s):  
A. Alexandrou Himonas ◽  
Gerard Misiołek
Keyword(s):  
Evolution Equation ◽  
Integrable Evolution Equation ◽  
Integrable Evolution

scholarly journals Well-posedness and behaviors of solutions to an integrable evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Sen Ming ◽  
Shaoyong Lai ◽  
Yeqin Su
Keyword(s):  
Evolution Equation ◽  
Existence And Uniqueness ◽  
Blow Up ◽  
Analytic Solutions ◽  
Well Posedness ◽  
Initial Values ◽  
Persistence Property ◽  
Integrable Evolution Equation ◽  
Integrable Evolution ◽  
And Behaviors

Abstract This work is devoted to investigating the local well-posedness for an integrable evolution equation and behaviors of its solutions, which possess blow-up criteria and persistence property. The existence and uniqueness of analytic solutions with analytic initial values are established. The solutions are analytic for both variables, globally in space and locally in time. The effects of coefficients λ and β on the solutions are given.

scholarly journals On a Generalized Fifth-Order Integrable Evolution Equation and its Hierarchy

2006 ◽  
Vol 61 (1-2) ◽  
pp. 7-15
Author(s):  
Amitava Choudhuri ◽  
Benoy Talukdar ◽  
S. B. Dattab
Keyword(s):  
Evolution Equation ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Hamiltonian Structures ◽  
Lagrangian Equations ◽  
Integrable Evolution Equation ◽  
Integrable Evolution ◽  
Homogeneous Balance Method ◽  
Fifth Order ◽  
Homogeneous Balance

A general form of a fifth-order nonlinear evolution equation is considered. The Helmholtz solution of the inverse variational problem is used to derive conditions under which this equation admits an analytic representation. A Lennard type recursion operator is then employed to construct a hierarchy of Lagrangian equations. It is explicitly demonstrated that the constructed system of equations has a Lax representation and two compatible Hamiltonian structures. The homogeneous balance method is used to derive analytic soliton solutions of the third- and fifth-order equations. - PACS numbers: 47.20.Ky, 42.81.Dp, 02.30.Jr

BILINEARIZATION AND SOLUTIONS OF THE KdV6 EQUATION

2008 ◽  
Vol 06 (04) ◽  
pp. 401-412 ◽  
Author(s):  
A. RAMANI ◽  
B. GRAMMATICOS ◽  
R. WILLOX
Keyword(s):  
Evolution Equation ◽  
Arbitrary Function ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Singularity Analysis ◽  
Backlund Transformation ◽  
Integrable Evolution Equation ◽  
The Kdv Equation ◽  
Integrable Evolution ◽  
Kdv6 Equation

We examine the recently proposed KdV6 integrable evolution equation. Starting from solutions suggested by singularity analysis and using the auto-Bäcklund transformation, we construct solutions of the KdV6 which involve one arbitrary function of time. Next, we proceed to bilinearize the equation and derive a new, simpler, auto-Bäcklund transformation. Starting from the solutions of the KdV equation we construct those of the KdV6 in the form of M kinks and N poles and which indeed involve an arbitrary function of time.

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