Geometric theory of local and non-local conservation laws for the sine-Gordon equation

1981 ◽  
Vol 376 (1766) ◽  
pp. 401-433 ◽  
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Gordon Equation ◽  
Vries Equation ◽  
Geometric Theory ◽  
Nonlinear Evolution Equations ◽  
Sine Gordon Equation ◽  
Local Conservation ◽  
Complete Set ◽  
Non Local

In previous work one of the authors gave a geometric theory of those nonlinear evolution equations (n. e. es) that can be solved by the Zakharov & Shabat (1972) inverse scattering scheme as generalized by Ablowitz, Kaup, Newell & Segur (1973 b , 1974). In this paper we extend the geo­metric theory to include the Hamiltonian structure of those n. e. es solvable by the method, and we indicate the connection between the geometric theory and the theory of prolongation structures and pseudopotentials due to Wahlquist & Estabrook (1975, 1976). We exploit a ‘gauge’ in­variance of the geometric theory to derive both the well known polynomial conserved densities of the sine-Gordon equation and a non-local set of conserved densities. These act as Hamiltonian densities for a hierarchy of sine-Gordon equations which is analogous to that found by Lax (1968) for the Korteweg-de Vries equation and appears to be new. In an Appendix we derive an expression for the equation of motion for an arbitrary member of the sine-Gordon hierarchy by methods which can be applied in larger context. The results for the sine-Gordon equation lead to the conclusion that a complete set of conserved densities for an arbitrary n. e. e. solvable by the A. K. N. S. –Z. S. scheme must include non-local conserved densities.

Conservation laws and Hamiltonian structures of the generalized sine-Gordon hierarchy

2014 ◽  
Vol 28 (32) ◽  
pp. 1450248
Author(s):  
Fang Li ◽  
Bo Xue ◽  
Yan Li
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Nonlinear Evolution ◽  
Infinite Sequence ◽  
Nonlinear Evolution Equations ◽  
Conserved Quantities ◽  
Trace Identity ◽  
Sine Gordon Equation ◽  
Hamiltonian Structures ◽  
Matrix Spectral Problem

By introducing a 2 × 2 matrix spectral problem, a new hierarchy of nonlinear evolution equations is proposed. A typical equation in this hierarchy is the generalization of sine-Gordon equation. With the aid of trace identity, the Hamiltonian structures of the hierarchy are constructed. In addition, the infinite sequence of conserved quantities of the generalized sine-Gordon equation are obtained.

An inverse scattering study of the radiation solution of the sine–Gordon equation

1984 ◽  
Vol 62 (7) ◽  
pp. 701-713
Author(s):  
R. H. Enns ◽  
S. S. Rangnekar
Keyword(s):  
Inverse Scattering ◽  
Evolution Equations ◽  
Gordon Equation ◽  
Wave Interaction ◽  
Nonlinear Evolution Equations ◽  
Sine Gordon Equation ◽  
Transform Method ◽  
Scattering Transform ◽  
Scattering Study ◽  
Interaction Problem

Making use of the diagrammatic approach to the inverse scattering transform method that we pioneered on the 3-wave interaction problem, we have studied the complete temporal and spatial evolution of the radiation solution of the sine–Gordon equation. The analytic results are consistent with numerical simulations as well as qualitative ideas prevalent in the literature. The extension of the diagrammatic approach to the sinh–Gordon and other nonlinear evolution equations of physical significance is also briefly discussed.

scholarly journals A note on improved F -expansion method combined with Riccati equation applied to nonlinear evolution equations

2014 ◽  
Vol 1 (2) ◽  
pp. 140038 ◽  
Author(s):  
Md. Shafiqul Islam ◽  
Kamruzzaman Khan ◽  
M. Ali Akbar ◽  
Antonio Mastroberardino
Keyword(s):  
Riccati Equation ◽  
Wave Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Wave Equations ◽  
Travelling Wave Solutions

The purpose of this article is to present an analytical method, namely the improved F -expansion method combined with the Riccati equation, for finding exact solutions of nonlinear evolution equations. The present method is capable of calculating all branches of solutions simultaneously, even if multiple solutions are very close and thus difficult to distinguish with numerical techniques. To verify the computational efficiency, we consider the modified Benjamin–Bona–Mahony equation and the modified Korteweg-de Vries equation. Our results reveal that the method is a very effective and straightforward way of formulating the exact travelling wave solutions of nonlinear wave equations arising in mathematical physics and engineering.

Noncommutative coupled complex modified Korteweg–de Vries equation: Darboux and binary Darboux transformations

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950054
Author(s):  
H. Wajahat A. Riaz
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Vries Equation ◽  
Soliton Solutions ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Order Effects ◽  
Darboux Transformations ◽  
De Vries ◽  
Higher Order Effects

Higher-order nonlinear evolution equations are important for describing the wave propagation of second- and higher-order number of fields in optical fiber systems with higher-order effects. One of these equations is the coupled complex modified Korteweg–de Vries (ccmKdV) equation. In this paper, we study noncommutative (nc) generalization of ccmKdV equation. We present Darboux and binary Darboux transformations (DTs) for the nc-ccmKdV equation and then construct its Quasi-Grammian solutions. Further, single and double-hump soliton solutions of first- and second-order are given in commutative settings.

Representations of affine Lie algebras and soliton equations

1985 ◽  
Vol 315 (1533) ◽  
pp. 391-391
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Gordon Equation ◽  
Vries Equation ◽  
Affine Lie Algebras ◽  
Loop Groups ◽  
Sine Gordon Equation ◽  
Kyoto School ◽  
Hidden Symmetries ◽  
Soliton Equations

A few years ago the 'hidden symmetries’ of the soliton equations had been identified as affine Lie groups, also known as loop groups. The first extensive use of the representation theory of affine Lie algebras for the soliton equations have been developed in a series of works by mathematicians of the Kyoto school. We will review some of their results and develop them further on the basis of the representation theory. Thus an orbit of the simplest affine Lie group SL(2, C)^ in the fundamental representation V will provide the solutions of the Korteweg-de Vries equation, and similarly the solutions of the sine-Gordon equation will come from an orbit of the group (SL(2, C) x SL(2, C)) ^ in V x V*.

Symbolic Computation and Non-travelling Wave Solutions of the (2+1)-Dimensional Korteweg de Vries Equation

2005 ◽  
Vol 60 (4) ◽  
pp. 221-228 ◽  
Author(s):  
Dengshan Wang ◽  
Hong-Qing Zhang
Keyword(s):  
Symbolic Computation ◽  
Evolution Equations ◽  
Vries Equation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Nonlinear Evolution Equations ◽  
Jacobi Elliptic Function ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
De Vries

Abstract In this paper, with the aid of symbolic computation we improve the extended F-expansion method described in Chaos, Solitons and Fractals 22, 111 (2004) to solve the (2+1)-dimensional Korteweg de Vries equation. Using this method, we derive many exact non-travelling wave solutions. These are more general than the previous solutions derived with the extended F-expansion method. They include the Jacobi elliptic function, soliton-like trigonometric function solutions, and so on. Our method can be applied to other nonlinear evolution equations.

scholarly journals Third-order differential equations and local isometric immersions of pseudospherical surfaces

2016 ◽  
Vol 18 (06) ◽  
pp. 1650021 ◽  
Author(s):  
Tarcísio Castro Silva ◽  
Niky Kamran
Keyword(s):  
Differential Equations ◽  
Fundamental Form ◽  
Evolution Equations ◽  
Gordon Equation ◽  
Second Fundamental Form ◽  
Isometric Immersions ◽  
Sine Gordon Equation ◽  
Third Order ◽  
The Second Fundamental Form ◽  
Pseudospherical Surfaces

The class of differential equations describing pseudospherical surfaces enjoys important integrability properties which manifest themselves by the existence of infinite hierarchies of conservation laws (both local and nonlocal) and the presence of associated linear problems. It thus contains many important known examples of integrable equations, like the sine-Gordon, Liouville, KdV, mKdV, Camassa–Holm and Degasperis–Procesi equations, and is also home to many new families of integrable equations. Our paper is concerned with the question of the local isometric immersion in E3 of the pseudospherical surfaces defined by the solutions of equations belonging to the class introduced by Chern and Tenenblat [Pseudospherical surfaces and evolution equations, Stud. Appl. Math. 74 (1986) 55–83]. In the case of the sine-Gordon equation, it is a classical result that the second fundamental form of the immersion depends only on a jet of finite order of the solution of the partial differential equation. A natural question is therefore to know if this remarkable property extends to equations other than the sine-Gordon equation within the class of differential equations describing pseudospherical surfaces. In a pair of earlier papers [N. Kahouadji, N. Kamran and K. Tenenblat, Second-order equations and local isometric immersions of pseudo-spherical surfaces, to appear in Comm. Anal. Geom., arXiv: 1308.6545; Local isometric immersions of pseudo-spherical surfaces and evolution equations, in Hamiltonian Partial Differential Equations and Applications, eds. P. Guyenne, D. Nichols and C. Sulem, Fields Institute Communications, Vol. 75 (Springer-Verlag, 2015), pp. 369–381], it was shown that this property fails to hold for all [Formula: see text]th-order evolution equations [Formula: see text] and all other second-order equations of the form [Formula: see text], except for the sine-Gordon equation and a special class of equations for which the coefficients of the second fundamental form are universal, that is functions of [Formula: see text] and [Formula: see text] which are independent of the choice of solution [Formula: see text]. In this paper, we consider third-order equations of the form [Formula: see text], [Formula: see text], which describe pseudospherical surfaces with the Riemannian metric given in [T. Castro Silva and K. Tenenblat, Third order differential equations describing pseudospherical surfaces, J. Differential Equations 259 (2015) 4897–4923]. This class contains the Camassa–Holm and Degasperis–Procesi equations as special cases. We show that whenever there exists a local isometric immersion in E3 for which the coefficients of the second fundamental form depend on a jet of finite order of [Formula: see text], then these coefficients are universal in the sense of being independent on the choice of solution [Formula: see text]. This result further underscores the special place that the sine-Gordon equations seem to occupy amongst integrable partial differential equations in one space variable.

NONLINEAR EVOLUTION EQUATIONS AND THE PAINLEVÉ TEST

1992 ◽  
Vol 07 (08) ◽  
pp. 1669-1683 ◽  
Author(s):  
W.-H. STEEB ◽  
N. EULER
Keyword(s):  
Evolution Equations ◽  
Gordon Equation ◽  
Nonlinear Evolution ◽  
Energy Eigenvalue ◽  
Nonlinear Evolution Equations ◽  
The Self ◽  
Painlevé Test ◽  
Yang Mills ◽  
Petviashvili Equation ◽  
Klein Gordon

A survey is given of new results of the Painlevé test and nonlinear evolution equations where ordinary- and partial-differential equations are considered. We study the semiclassical Jaynes-Cumming model, the energy-eigenvalue-level-motion equation, the Kadomtsev-Petviashvili equation, the nonlinear Klein-Gordon equation and the self-dual Yang-Mills equation.

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