scholarly journals On Lax pairs and matrix extended simple Toda systems

2005 ◽  
Vol 2005 (17) ◽  
pp. 2735-2747
Author(s):  
M. Legaré
Keyword(s):  
Lax Pair ◽  
Lax Pairs ◽  
Toda System ◽  
Toda Systems

AA1Toda system is extended via Lax pair formulations in order to probe noncommutative variables extensions. Systems, some solvable, are built using matrix generalizations.

On the Full-Discrete Extended Generalised q-Difference Toda System

2017 ◽  
Vol 72 (8) ◽  
pp. 703-709
Author(s):  
Chuanzhong Li ◽  
Anni Meng
Keyword(s):  
Difference Equation ◽  
Hamiltonian Structure ◽  
Soliton Solutions ◽  
Toda Equation ◽  
Lax Pairs ◽  
Toda System ◽  
Toda Hierarchy

AbstractIn this paper, we construct a full-discrete integrable difference equation which is a full-discretisation of the generalised q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of an extended generalised full-discrete q-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the extended full-discrete generalised q-Toda hierarchy are given.

scholarly journals Lax Pairs and First Integrals for Autonomous and Non-Autonomous Differential Equations Belonging to the Painlevé – Gambier List

2020 ◽  
Vol 16 (4) ◽  
pp. 637-650
Author(s):  
P. Guha ◽  
◽  
S. Garai ◽  
A.G. Choudhury ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
First Integral ◽  
Lax Pair ◽  
First Integrals ◽  
Second Order ◽  
Lax Representation ◽  
Lax Pairs ◽  
Second Order Differential Equations ◽  
Autonomous Version ◽  
Autonomous Differential Equations

Recently Sinelshchikov et al. [1] formulated a Lax representation for a family of nonautonomous second-order differential equations. In this paper we extend their result and obtain the Lax pair and the associated first integral of a non-autonomous version of the Levinson – Smith equation. In addition, we have obtained Lax pairs and first integrals for several equations of the Painlevé – Gambier list, namely, the autonomous equations numbered XII, XVII, XVIII, XIX, XXI, XXII, XXIII, XXIX, XXXII, XXXVII, XLI, XLIII, as well as the non-autonomous equations Nos. XV and XVI in Ince’s book.

scholarly journals Symmetry Reductions of (2 + 1)-Dimensional CDGKS Equation and Its Reduced Lax Pairs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Na Lv ◽  
Xuegang Yuan ◽  
Jinzhi Wang
Keyword(s):  
Direct Method ◽  
Lax Pair ◽  
Group Method ◽  
Symmetry Groups ◽  
Invariant Solutions ◽  
Lax Pairs ◽  
Group Invariant ◽  
Lie Group Method ◽  
Group Invariant Solutions ◽  
Symmetry Transformations

With the aid of symbolic computation, we obtain the symmetry transformations of the (2 + 1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation by Lou’s direct method which is based on Lax pairs. Moreover, we use the classical Lie group method to seek the symmetry groups of both the CDGKS equation and its Lax pair and then reduce them by the obtained symmetries. In particular, we consider the reductions of the Lax pair completely. As a result, three reduced (1 + 1)-dimensional equations with their new Lax pairs are presented and some group-invariant solutions of the equation are given.

scholarly journals Vanishing Estimates for Fully Bubbling Solutions of SU (n + 1) Toda Systems at a Singular Source

2018 ◽  
Vol 2020 (18) ◽  
pp. 5774-5795
Author(s):  
Lei Zhang
Keyword(s):  
Riemann Surface ◽  
Gauss Curvature ◽  
Toda System ◽  
Curvature Equation ◽  
Singular Source ◽  
Toda Systems

AbstractFor Gauss curvature equation (or more general Toda systems) defined on 2D spaces, the vanishing rate of certain curvature functions on blowup points is a key estimate for numerous applications. However, if these equations have singular sources, very few vanishing estimates can be found. In this article we consider a Toda system with singular sources defined on a Riemann surface and we prove a very surprising vanishing estimates and a reflection phenomenon for certain functions involving the Gauss curvature.

scholarly journals A Note on Lax Pairs of the Sawada-Kotera Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Sergei Sakovich
Keyword(s):  
Spectral Parameter ◽  
Lax Pair ◽  
Lax Pairs ◽  
Zero Curvature

We prove that the new Lax pair of the Sawada-Kotera equation, discovered recently by Hickman, Hereman, Larue, and Göktaş, and the well-known old Lax pair of this equation, considered in the form of zero-curvature representations, are gauge equivalent to each other if and only if the spectral parameter is nonzero, while for zero spectral parameter a nongauge transformation is required.

scholarly journals Quantum Toda systems and Lax pairs

1987 ◽  
Vol 109 (1) ◽  
pp. 23-32 ◽  
Author(s):  
N. Ganoulis
Keyword(s):  
Lax Pairs ◽  
Toda Systems

scholarly journals Orthogonal functions related to Lax pairs in Lie algebras

2021 ◽  
Author(s):  
Wolter Groenevelt ◽  
Erik Koelink
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Lax Pair ◽  
Standard Basis ◽  
Lax Pairs ◽  
Orthogonal Functions

AbstractWe study a Lax pair in a 2-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of L and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Eigenfunctions for the operator L for a Lax pair for $$\mathfrak {sl}(d+1,\mathbb {C})$$ sl ( d + 1 , C ) is studied in certain representations.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

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