113 research outputs found
On a Generalized Fifth-Order Integrable Evolution Equation and its Hierarchy
A general form of the fifth-order nonlinear evolution equation is considered.
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.Comment: 16 pages, 1 figur
On Completely Integrability Systems of Differential Equations
In this note we discuss the approach which was given by Wazwaz for the proof
of the complete integrability to the system of nonlinear differential
equations. We show that his method presented in [Wazwaz A.M. Completely
integrable coupled KdV and coupled KP systems, Commun Nonlinear Sci Simulat 15
(2010) 2828-2835] is incorrect.Comment: 14 pages. This paper was sent to the Communications in Nonlinear
Science and Numerical Simulatio
Higher dimensional integrable deformations of the modified KdV equation
The derivation of nonlinear integrable evolution partial differential
equations in higher dimensions has always been the holy grail in the field of
integrability. The well-known modified KdV equation is a prototypical example
of integrable evolution equations in one spatial dimension. Do there exist
integrable analogs of modified KdV equation in higher spatial dimensions? In
what follows, we present a positive answer to this question. In particular,
rewriting the (1+1)-dimensional integrable modified KdV equation in
conservation forms and adding deformation mappings during the process allow one
to construct higher dimensional integrable equations. Further, we illustrate
this idea with examples from the modified KdV hierarchy, also present the Lax
pairs of these higher dimensional integrable evolution equations.Comment: 7 pages, 3 figure
Seven common errors in finding exact solutions of nonlinear differential equations
We analyze the common errors of the recent papers in which the solitary wave
solutions of nonlinear differential equations are presented. Seven common
errors are formulated and classified. These errors are illustrated by using
multiple examples of the common errors from the recent publications. We show
that many popular methods in finding of the exact solutions are equivalent each
other. We demonstrate that some authors look for the solitary wave solutions of
nonlinear ordinary differential equations and do not take into account the well
- known general solutions of these equations. We illustrate several cases when
authors present some functions for describing solutions but do not use
arbitrary constants. As this fact takes place the redundant solutions of
differential equations are found. A few examples of incorrect solutions by some
authors are presented. Several other errors in finding the exact solutions of
nonlinear differential equations are also discussed.Comment: 42 page
Generalized (2+1)−dimensional breaking soliton equation
In this work, a general (2+1)-dimensional breaking soliton equation is investigated. The Hereman’s simplified method is applied to derive multiple soliton solutions, hence to confirm the model integrability.Publisher's Versio
Generalized integrable evolution equations with an infinite number of free parameters
Evolution equations such as the nonliear Schrödinger equation (NLSE) can be extended to include an infinite number of free parameters. The extensions are not unique. We give two examples that contain the NLSE as the lowest-order PDE of each set. Such representations provide the advantage of modelling a larger variety of physical problems due to the presence of an infinite number of higher-order terms in this equation with an infinite number of arbitrary parameters. An example of a rogue wave solution for one of these cases is presented, demonstrating the power of the technique
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