Simple equations method (SEsM) and its connection with the inverse scattering transform method

2021 ◽  
Author(s):  
Nikolay K. Vitanov
Keyword(s):  
Inverse Scattering ◽  
Inverse Scattering Transform ◽  
Transform Method ◽  
Scattering Transform ◽  
Simple Equations

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.

Application of the inverse scattering transform method to stimulated Brillouin backscattering in a generator setup: The Zakharov–Manakov solution

1981 ◽  
Vol 59 (12) ◽  
pp. 1817-1828 ◽  
Author(s):  
S. S. Rangnekar ◽  
R. H. Enns
Keyword(s):  
Laser Pulse ◽  
Numerical Integration ◽  
Inverse Scattering ◽  
Inverse Scattering Transform ◽  
Transform Method ◽  
Scattering Transform ◽  
Direct Numerical Integration ◽  
Stimulated Brillouin

Making use of the inverse scattering transform method (ISTM), we have solved the Zakharov–Manakov problem for the stimulated Brillouin backscattering (SBBS) of a laser pulse by a fluctuation, the envelopes of both being rectangular. The results are consistent with those obtained by Kaup and co-workers using a combination of direct numerical integration and Zakharov–Shabat analysis.

scholarly journals Simple Equations Method (SEsM): Algorithm, Connection with Hirota Method, Inverse Scattering Transform Method, and Several Other Methods

Entropy ◽  
2020 ◽  
Vol 23 (1) ◽  
pp. 10
Author(s):  
Nikolay K. Vitanov ◽  
Zlatinka I. Dimitrova ◽  
Kaloyan N. Vitanov
Keyword(s):  
Differential Equations ◽  
Soliton Solution ◽  
Vries Equation ◽  
Expansion Method ◽  
Inverse Scattering Transform ◽  
Transform Method ◽  
Scattering Transform ◽  
Homogeneous Balance Method ◽  
Simple Equations ◽  
Homogeneous Balance

The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.

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