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.

An inverse scattering study of the radiation solution of the Korteweg–de Vries equation: the generic and nongeneric cases

1986 ◽  
Vol 64 (1) ◽  
pp. 53-64
Author(s):  
R. H. Enns ◽  
S. S. Rangnekar
Keyword(s):  
Inverse Scattering ◽  
Evolution Equations ◽  
Kdv Equation ◽  
Expansion Method ◽  
Nonlinear Evolution Equations ◽  
Zero Eigenvalue ◽  
Transform Method ◽  
Scattering Transform ◽  
De Vries ◽  
Generic Class

Although the asymptotic (t → ∞) behaviour of radiation solutions of the Korteweg–de Vries (KdV) equation has been investigated in the literature using the inverse scattering transform method (ISTM), the complete temporal (and spatial) evolution has not been studied in detail using this method. In this paper we discuss the application of the inverse scattering expansion method, a method that we have successfully applied to a wide variety of nonlinear evolution equations of physical interest, to both the generic and nongeneric cases. Using model inputs as illustrative examples, we find that (a) unlike all other problems studied to date, the "natural" expansion parameter is not the (dimensionless) area of the input potential in the direct (eigenvalue) problem, and (b) the value of the reflection coefficient R(k) at zero eigenvalue, i.e., k = 0, plays a crucial role in the success or failure of what we will call the "standard" expansion method. The standard expansion method works for input potentials belonging to the "nongeneric" class, [Formula: see text] but breaks down beyond lowest order in the expansion for potentials belonging to the "generic" class, R(0) = −1. Re-examination of the generic problem in the vicinity of k = 0 leads to a "renormalization" in each order of the expansion, which enables the generic case to be correctly solved. Unlike the nongeneric case, the generic solution is not found to be asymptotically valid.

ANALYTIC METHODS FOR SOLITON SYSTEMS

1993 ◽  
Vol 03 (01) ◽  
pp. 3-17 ◽  
Author(s):  
M. LAKSHMANAN
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Direct Methods ◽  
Nonlinear Evolution Equations ◽  
Initial Value ◽  
Mathematical Concepts ◽  
Transform Method ◽  
Analytic Methods ◽  
Scattering Transform

The study of soliton systems continues to be a highly rewarding exercise in nonlinear dynamics, even though it has been almost thirty years since the introduction of the soliton concept by Zabusky & Kruskal. Increasingly sophisticated mathematical concepts are being identified with integrable soliton systems, while newer applications are being made frequently. In this pedagogical review, after introducing solitons and their (2+1)-dimensional generalizations, we give an elementary discussion on the various analytic methods available for investigation of the soliton possessing nonlinear evolution equations. These include the inverse scattering transform method and its generalization, namely the d-bar approach, for solving the Cauchy initial value problem, as well as direct methods for obtaining N-soliton solutions. We also indicate how the Painlevé singularity structure analysis is useful for the detection of soliton systems.

Zakharov–Manakov solution of the 3-wave explosive interaction problem

1983 ◽  
Vol 61 (10) ◽  
pp. 1386-1400 ◽  
Author(s):  
R. H. Enns ◽  
S. S. Rangnekar
Keyword(s):  
Inverse Scattering ◽  
Temporal Evolution ◽  
Inverse Scattering Transform ◽  
Transform Method ◽  
Scattering Transform ◽  
Interaction Problem

The inverse scattering transform method has been applied to the on-resonance 3-wave explosive interaction problem. In particular, the Zakharov–Manakov problem has been solved to yield the complete spatial and temporal evolution of the envelopes of the three waves involved. A comparison with numerically derived envelope shapes is made and the results are discussed.

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.

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