scholarly journals Miura-Reciprocal Transformation and Symmetries for the Spectral Problems of KdV and mKdV

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 926
Author(s):  
Paz Albares ◽  
Pilar Garcia Estévez
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Lie Symmetries ◽  
Reciprocal Transformation ◽  
Lax Pairs ◽  
Spectral Problems ◽  
De Vries ◽  
Similarity Reductions

We present reciprocal transformations for the spectral problems of Korteveg de Vries (KdV) and modified Korteveg de Vries (mKdV) equations. The resulting equations, RKdV (reciprocal KdV) and RmKdV (reciprocal mKdV), are connected through a transformation that combines both Miura and reciprocal transformations. Lax pairs for RKdV and RmKdV are straightforwardly obtained by means of the aforementioned reciprocal transformations. We have also identified the classical Lie symmetries for the Lax pairs of RKdV and RmKdV. Non-trivial similarity reductions are computed and they yield non-autonomous ordinary differential equations (ODEs), whose Lax pairs are obtained as a consequence of the reductions.

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.

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 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.

Symmetries and conservation laws of a damped Boussinesq equation

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640012 ◽  
Author(s):  
María Luz Gandarias ◽  
María Rosa
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Source Term ◽  
Boussinesq Equation ◽  
Lie Symmetries ◽  
Multiplier Method ◽  
Independent Source ◽  
Damped Equation

In this work, we consider a damped equation with a time-independent source term. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. We also present some exact solutions. Conservation laws for this equation are constructed by using the multiplier method.

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