On Nonlinear Equations Amenable to the Inverse Scattering Method

1975 ◽  
Vol 53 (1) ◽  
pp. 58-61 ◽  
Author(s):  
J. G. Kingston ◽  
C. Rogers
Keyword(s):  
Inverse Scattering ◽  
Initial Value Problem ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Inverse Scattering Method ◽  
Nonlinear Evolution Equations ◽  
Scattering Method ◽  
Initial Value ◽  
Physical Importance ◽  
Extensive Class

The inverse scattering method can be used to solve the initial value problem for various nonlinear evolution equations of physical importance. Here an extensive class of equations for which the technique is available is delimited.

scholarly journals Multi-soliton rational solutions for some nonlinear evolution equations

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 26-36 ◽  
Author(s):  
Mohamed S. Osman
Keyword(s):  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Inverse Scattering Method ◽  
Nonlinear Evolution Equations ◽  
Scattering Method ◽  
Mathematical Tool ◽  
Rational Solutions ◽  
Generalized Unified Method ◽  
The Generalized Unified Method ◽  
Unified Method

AbstractThe Korteweg-de Vries equation (KdV) and the (2+ 1)-dimensional Nizhnik-Novikov-Veselov system (NNV) are presented. Multi-soliton rational solutions of these equations are obtained via the generalized unified method. The analysis emphasizes the power of this method and its capability of handling completely (or partially) integrable equations. Compared with Hirota’s method and the inverse scattering method, the proposed method gives more general exact multi-wave solutions without much additional effort. The results show that, by virtue of symbolic computation, the generalized unified method may provide us with a straightforward and effective mathematical tool for seeking multi-soliton rational solutions for solving many nonlinear evolution equations arising in different branches of sciences.

New Exact Solutions to a Class of Nonlinear Evolution Equations with Symbolic Computations

2013 ◽  
Vol 645 ◽  
pp. 312-315
Author(s):  
Jian Ya Ge ◽  
Tie Cheng Xia
Keyword(s):  
Exact Solutions ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Algebraic Method ◽  
Inverse Scattering Method ◽  
Nonlinear Evolution Equations ◽  
Scattering Method ◽  
Jacobi Elliptic Function ◽  
New Exact Solutions

Searching for exact solutions to nonlinear evolution equations is an important topic in mathematical physics and engineering. Many methods of finding exact solutions have been presented such as the inverse scattering method, algebraic method and so on. In this paper, by using Fan sub-equation method with the help of Maple, several meaningful solutions are obtained including bell shape solutions, trigonometric function solutions, twist shape solutions and Jacobi elliptic function solutions for a class of nonlinear evolution equation. This method can be applied to other nonlinear partial differential equations.

scholarly journals Nonlinear Evolution Equations with Exponentially Decaying Memory: Existence via Time Discretisation, Uniqueness, and Stability

2020 ◽  
Vol 20 (1) ◽  
pp. 89-108 ◽  
Author(s):  
André Eikmeier ◽  
Etienne Emmrich ◽  
Hans-Christian Kreusler
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Inner Product ◽  
Initial Value ◽  
Time Discretisation ◽  
Volterra Integral Operator ◽  
Stability Results

AbstractThe initial value problem for an evolution equation of type {v^{\prime}+Av+BKv=f} is studied, where {A:V_{A}\to V_{A}^{\prime}} is a monotone, coercive operator and where {B:V_{B}\to V_{B}^{\prime}} induces an inner product. The Banach space {V_{A}} is not required to be embedded in {V_{B}} or vice versa. The operator K incorporates a Volterra integral operator in time of convolution type with an exponentially decaying kernel. Existence of a global-in-time solution is shown by proving convergence of a suitable time discretisation. Moreover, uniqueness as well as stability results are proved. Appropriate integration-by-parts formulae are a key ingredient for the analysis.

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