Unconventional Schemes for a Class of Ordinary Differential Equations—With Applications to the Korteweg–de Vries Equation

1997 ◽  
Vol 134 (2) ◽  
pp. 316-331 ◽  
Author(s):  
William Kahan ◽  
Ren-Chang Li
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Vries Equation ◽  
De Vries

scholarly journals Symmetries and Reductions of Integrable Nonlocal Partial Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 884
Author(s):  
Linyu Peng
Keyword(s):  
Partial Differential Equations ◽  
General Theory ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Equations ◽  
Vries Equation ◽  
Lie Point Symmetries ◽  
Nonlocal Equations ◽  
De Vries ◽  
Nonlocal Nonlinear

In this paper, symmetry analysis is extended to study nonlocal differential equations. In particular, two integrable nonlocal equations are investigated, the nonlocal nonlinear Schrödinger equation and the nonlocal modified Korteweg–de Vries equation. Based on general theory, Lie point symmetries are obtained and used to reduce these equations to nonlocal and local ordinary differential equations, separately; namely, one symmetry may allow reductions to both nonlocal and local equations, depending on how the invariant variables are chosen. For the nonlocal modified Korteweg–de Vries equation, analogously to the local situation, all reduced local equations are integrable. We also define complex transformations to connect nonlocal differential equations and differential-difference equations.

scholarly journals Bäcklund transformations for several cases of a type of generalized KdV equation

2004 ◽  
Vol 2004 (63) ◽  
pp. 3369-3377
Author(s):  
Paul Bracken
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Bäcklund Transformations ◽  
Backlund Transformation ◽  
Generalized Kdv Equation ◽  
Backlund Transformations ◽  
De Vries ◽  
Derivative Term

An alternate generalized Korteweg-de Vries system is studied here. A procedure for generating solutions is given. A theorem is presented, which is subsequently applied to this equation to obtain a type of Bäcklund transformation for several specific cases of the power of the derivative term appearing in the equation. In the process, several interesting, new, ordinary, differential equations are generated and studied.

scholarly journals Time Fractional Generalized Korteweg-de Vries Equation: Explicit Series Solutions and Exact Solutions

2021 ◽  
Vol 2 (2) ◽  
pp. 62-77
Author(s):  
Rajeev Kumar ◽  
Sanjeev Kumar ◽  
Sukhneet Kaur ◽  
Shrishty Jain
Keyword(s):  
Differential Equations ◽  
Series Solution ◽  
Vries Equation ◽  
Convergent Series ◽  
De Vries ◽  
Conformable Derivatives ◽  
Graphical View ◽  
Conditionally Convergent Series ◽  
The Given ◽  
Explicit Series Solutions

In this article, an attempt is made to achieve the series solution of the time fractional generalized Korteweg-de Vries equation which leads to a conditionally convergent series solution. We have also resorted to another technique involving conversion of the given fractional partial differential equations to ordinary differential equations by using fractional complex transform. This technique is discussed separately for modified Riemann-Liouville and conformable derivatives. Convergence analysis and graphical view of the obtained solution are demonstrated in this work.

Finite Genus Solutions to the Generalised Kaup-Newell Hierarchy and Two 2+1 Dimensional Modified Korteweg-de Vries Equations

2017 ◽  
Vol 72 (7) ◽  
pp. 589-594
Author(s):  
Xiao Yang ◽  
Jiayan Han
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Hamiltonian Systems ◽  
Algebraic Curve ◽  
Inversion Theorem ◽  
First Order ◽  
Hamiltonian Flows ◽  
De Vries ◽  
Jacobi Inversion ◽  
Finite Genus

AbstractA generalised Kaup-Newell (gKN) hierarchy is introduced, which starts with a system of first-order ordinary differential equations and includes the Gerdjikov-Ivanov equation. By introducing an appropriate generating function, its related Hamiltonian systems and algebraic curve are given. The Hamiltonian systems are proved to be integrable, then the gKN hierarchy is solved by Hamiltonian flows. The algebraic curve is provided with suitable genus, then based on the trace formula and Riemann-Jacobi inversion theorem, finite genus solutions of the gKN hierarchy are obtained. Besides, two 2+1 dimensional modified Korteweg-de Vries (mKdV) equations are also solved.

scholarly journals Asymptotic analysis of the singularly perturbed Korteweg-de Vries equation

2019 ◽  
pp. 194-197
Author(s):  
V. H. Samoilenko ◽  
Yu. I. Samoilenko ◽  
V. S. Vovk
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Vries Equation ◽  
Singular Part ◽  
Variable Coefficients ◽  
Singularly Perturbed ◽  
Asymptotic Solutions ◽  
Partial Differential ◽  
De Vries

The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. An algorithm for constructing asymptotic one-phase soliton-like solutions of this equation is described. The algorithm is based on the nonlinear WKB technique. The constructed asymptotic soliton-like solutions contain a regular and singular part. The regular part of this solution is the background function and consists of terms, which are defined as solutions to the system of the first order partial differential equations. The singular part of the asymptotic solution characterizes the soliton properties of the asymptotic solution. These terms are defined as solutions to the system of the third order partial differential equations. Solutions of these equations are obtained in a special way. Firstly, solutions of these equations are considered on the so-called discontinuity curve, and then these solutions are prolongated into a neighborhood of this curve. The influence of the form of the coefficients of the considered equation on the form of the equation for the discontinuity curve is analyzed. It is noted that for a wide class of such coefficients the equation for the discontinuity curve has solution that is determined for all values of the time variable. In these cases, the constructed asymptotic solutions are determined for all values of the independent variables. Thus, in the case of a zero background, the asymptotic solutions are certain deformations of classical soliton solutions.

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