Verifying the Soliton Solutions to the Fifth Order KdV Equation by Ordinary Inverse Scattering Method

2014 ◽  
Vol 1051 ◽  
pp. 1000-1003 ◽  
Author(s):  
Chao Ma ◽  
Jin Chun He
Keyword(s):  
Inverse Scattering ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Fifth Order ◽  
Jost Solution

The Jost solution of the fifth order KdV equation derived from inverse scattering transformation in Gel’fand-Levitan-Marchenko formalism satisfy the both two compatibility equations. Therefore, the soliton solutions to the fifth order KdV equation can be verified theoretically.

SUPERSYMMETRY, REFLECTIONLESS SYMMETRIC POTENTIALS AND THE INVERSE METHOD

1990 ◽  
Vol 05 (09) ◽  
pp. 1763-1772 ◽  
Author(s):  
B. BAGCHI
Keyword(s):  
Inverse Scattering ◽  
Schrodinger Equation ◽  
Inverse Method ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Wave Functions ◽  
Scattering Method ◽  
De Vries

The role of inverse scattering method is illustrated to examine the connection between the multi-soliton solutions of Korteweg-de Vries (KdV) equation and discrete eigenvalues of Schrödinger equation. The necessity of normalization of the Schrödinger wave functions, which are constructed purely from a supersymmetric consideration is pointed out.

EXACT SOLUTIONS OF NONLINEAR SU(N) PRINCIPAL σ-MODELS

1988 ◽  
Vol 03 (05) ◽  
pp. 1147-1154
Author(s):  
TIBOR KISS-TOTH
Keyword(s):  
Exact Solutions ◽  
Inverse Scattering ◽  
Soliton Solutions ◽  
Static Solution ◽  
Finite Energy ◽  
Single Step ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Axially Symmetric ◽  
Symmetric Static Solution

The superpotential for n-step soliton solution is derived in the case of an arbitrary dimensional projector for axially symmetric, static solution of nonlinear principal SU (N) σ-models. This was done by using an inverse scattering method developed by Belinski and Zakharov. Finite energy solutions are constructed for all SU (N) one soliton solutions generated by a single step.

Double-Alekseev inverse scattering method in the stationary axi-symmetric vacuum gravitation field equations

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Chun-Yan Wang ◽  
Yuan-Xing Gui ◽  
Ya-Jun Gao
Keyword(s):  
Inverse Scattering ◽  
Function Method ◽  
Complex Function ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Field Equations ◽  
Double Complex ◽  
Gravitation Field ◽  
Symmetric Vacuum

AbstractWe present a new improvement to the Alekseev inverse scattering method. This improved inverse scattering method is extended to a double form, followed by the generation of some new solutions of the double-complex Kinnersley equations. As the double-complex function method contains the Kramer-Neugebauer substitution and analytic continuation, a pair of real gravitation soliton solutions of the Einstein’s field equations can be obtained from a double N-soliton solution. In the case of the flat Minkowski space background solution, the general formulas of the new solutions are presented.

Explicit N -Soliton solutions in the Inverse Scattering Method

1989 ◽  
Vol 12 (3) ◽  
pp. 327-332 ◽  
Author(s):  
Chen Zong–yun ◽  
Huang Nian–ning ◽  
Xiao Yi
Keyword(s):  
Inverse Scattering ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Scattering Method

Polynomial conserved densities for the sine-Gordon equations

1977 ◽  
Vol 352 (1671) ◽  
pp. 481-503 ◽  
Keyword(s):  
Inverse Scattering ◽  
Wave Equations ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Bäcklund Transformations ◽  
Scattering Method ◽  
Sine Gordon Equation ◽  
Backlund Transformations

Like a number of other nonlinear dispersive wave equations the sine–Gordonequation z , xt = sin z has both multi-soliton solutions and an infinity of conserved densities which are polynomials in z , x , z , xx , etc. We prove that the generalized sine–Gordon equation z , xt = F ( z ) has an infinity of such polynomial conserved densities if, and only if, F ( z ) = A e αz + B e – αz for complex valued A, B and α ≠ 0. If F ( z ) does not take the form A e αz + B e βz there is no p. c. d. of rank greater than two. If α ≠ – β there is only a finite number of p. c. ds. If α = – β then if A and B are non-zero all p. c. ds are of even rank; if either A or B vanishes the p. c. ds are of both even and odd ranks. We exhibit the first eleven p. c. ds in each case when α = – β and the first eight when α ≠ – β . Neither the odd rank p. c. ds in the case α = – β , nor the particular limited set of p. c. ds in the case when α ≠ – β have been reported before. We connect the existence of an infinity of p. c. ds with solutions of the equations through an inverse scattering method, with Bäcklund transformations and, via Noether’s theorem, with infinitesimal Bäcklund transformations. All equations with Bäcklund transformations have an infinity of p. c. ds but not all such p. c. ds can be generated from the Bäcklund transformations. We deduce that multiple sine–Gordon equations like z , xt = sin z + ½ sin ½ z , which have applications in the theory of short optical pulse propagation, do not have an infinity of p. c. ds. For these equations we find essentially three conservation laws: one and only one of these is a p. c. d. and this is of rank two. We conclude that the multiple sine–Gordons will not be soluble by present formulations of the inverse scattering method despite numerical solutions which show soliton like behaviour. Results and conclusions are wholly consistent with the theorem that the generalized sine–Gordon equation has auto-Bäcklund transformations if, and only if Ḟ ( z ) – α 2 F ( z ) = 0.

Singular solutions of the KdV equation and the inverse scattering method

1985 ◽  
Vol 31 (6) ◽  
pp. 3264-3279 ◽  
Author(s):  
V. A. Arkad'ev ◽  
A. K. Pogrebkov ◽  
M. K. Polivanov
Keyword(s):  
Inverse Scattering ◽  
Kdv Equation ◽  
Inverse Scattering Method ◽  
Singular Solutions ◽  
Scattering Method ◽  
The Kdv Equation

Inverse scattering method for the Kundu-Eckhaus equation with zero/nonzero boundary conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Guixian Wang ◽  
Xiu-Bin Wang ◽  
Bo Han ◽  
Qi Xue
Keyword(s):  
Boundary Conditions ◽  
Inverse Scattering ◽  
Hilbert Problem ◽  
Soliton Solutions ◽  
Rogue Waves ◽  
Higher Order ◽  
Inverse Scattering Method ◽  
Scattering Method ◽  
Nonzero Boundary ◽  
Wave Phenomena

Abstract In this paper, the inverse scattering approach is applied to the Kundu-Eckhaus equation with two cases of zero boundary condition (ZBC) and nonzero boundary conditions (NZBCs) at infinity. Firstly, we obtain the exact formulae of soliton solutions of three cases of N simple poles, one higher-order pole and multiple higher-order poles via the associated Riemann-Hilbert problem (RHP). Moreover, given the initial data that allow for the presence of discrete spectrum, the higher-order rogue waves of the equation are presented. For the case of NZBCs, we can construct the infinite order rogue waves through developing a suitable RHP. Finally, by choosing different parameters, we aim to show some prominent characteristics of this solution and express them graphically in detail. Our results should be helpful to further explore and enrich the related nonlinear wave phenomena.

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