Nonlinear evolution equations and ordinary differential equations of painlevè type

1978 ◽  
Vol 23 (9) ◽  
pp. 333-338 ◽  
Author(s):  
M. J. Ablowitz ◽  
A. Ramani ◽  
H. Segur
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations

A Direct Transformation Method and its Application to Variable Coefficient Nonlinear Equations of Schrödinger Type

2009 ◽  
Vol 64 (11) ◽  
pp. 697-708 ◽  
Author(s):  
Li-Hua Zhang ◽  
Xi-Qiang Liu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Variable Coefficient ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Transformation Method ◽  
Nonlinear Evolution Equations ◽  
Nls Equation ◽  
Direct Transformation ◽  
Nonlinear Schrödinger

In this paper, the generalized variable coefficient nonlinear Schrödinger (NLS) equation and the cubic-quintic nonlinear Schrödinger (CQNLS) equation with variable coefficients are directly reduced to simple and solvable ordinary differential equations by means of a direct transformation method. Taking advantage of the known solutions of the obtained ordinary differential equations, families of exact nontravelling wave solutions for the two equations have been constructed. The characteristic feature of the direct transformation method is, that without much extra effort, we circumvent the integration by directly reducing the variable coefficient nonlinear evolution equations to the known ordinary differential equations. Another advantage of the method is that it is independent of the integrability of the given nonlinear equation. The method used here can be applied to reduce other variable coefficient nonlinear evolution equations to ordinary differential equations.

FROBENIUS INTEGRABLE DECOMPOSITIONS FOR TWO CLASSES OF NONLINEAR EVOLUTION EQUATIONS WITH VARIABLE COEFFICIENTS

2009 ◽  
Vol 23 (12) ◽  
pp. 1519-1524 ◽  
Author(s):  
FUCAI YOU ◽  
TIECHENG XIA ◽  
JIAO ZHANG
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Evolution Equations ◽  
Boussinesq Equation ◽  
Nonlinear Evolution ◽  
Variable Coefficients ◽  
Nonlinear Evolution Equations ◽  
Partial Differential ◽  
Kdv Equations

Frobenius integrable decompositions are introduced for partial differential equations with variable coefficients. Two classes of partial differential equations with variable coefficients are transformed into Frobenius integrable ordinary differential equations. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations, the Boussinesq equation and the Camassa–Holm equation with variable coefficients.

scholarly journals On the relations between some well-known methods and the projective Riccati equations

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 613-618
Author(s):  
Şamil Akçağıl
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Riccati Equations ◽  
Nonlinear Evolution Equations ◽  
Alternative Methods ◽  
Computational Techniques ◽  
Partial Differential ◽  
Traditional Methods

AbstractSolving nonlinear evolution equations is an important issue in the mathematical and physical sciences. Therefore, traditional methods, such as the method of characteristics, are used to solve nonlinear partial differential equations. A general method for determining analytical solutions for partial differential equations has not been found among traditional methods. Due to the development of symbolic computational techniques many alternative methods, such as hyperbolic tangent function methods, have been introduced in the last 50 years. Although all of them were introduced as a new method, some of them are similar to each other. In this study, we examine the following four important methods intensively used in the literature: the tanh–coth method, the modified Kudryashov method, the F-expansion method and the generalized Riccati equation mapping method. The similarities of these methods attracted our attention, and we give a link between the methods and a system of projective Riccati equations. It is possible to derive new solution methods for nonlinear evolution equations by using this connection.

Application of the Subequation Method to Some Differential Equations of Time-Fractional Order

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Ahmet Bekir ◽  
Esin Aksoy
Keyword(s):  
Differential Equations ◽  
Exact Solutions ◽  
Fractional Order ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
New Exact Solutions

The main goal of this paper is to develop subequation method for solving nonlinear evolution equations of time-fractional order. We use the subequation method to calculate the exact solutions of the time-fractional Burgers, Sharma–Tasso–Olver, and Fisher's equations. Consequently, we establish some new exact solutions for these equations.

scholarly journals Nonpoint Symmetry and Reduction of Nonlinear Evolution and Wave Type Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Ivan Tsyfra ◽  
Tomasz Czyżycki
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Hyperbolic Equations ◽  
Nonlinear Partial Differential Equations ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Heat Equation ◽  
Nonlinear Hyperbolic Equations ◽  
Arbitrary Smooth Function ◽  
Independent Variables

We study the symmetry reduction of nonlinear partial differential equations with two independent variables. We propose new ansätze reducing nonlinear evolution equations to system of ordinary differential equations. The ansätze are constructed by using operators of nonpoint classical and conditional symmetry. Then we find solution to nonlinear heat equation which cannot be obtained in the framework of the classical Lie approach. By using operators of Lie-Bäcklund symmetries we construct the solutions of nonlinear hyperbolic equations depending on arbitrary smooth function of one variable too.

scholarly journals HYPERBOLIC TYPE SOLUTIONS FOR THE COUPLE BOITI-LEON-PEMPINELLI SYSTEM

2020 ◽  
pp. 523
Author(s):  
Asif Yokus ◽  
Hülya Durur ◽  
Hijaz Ahmad
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Symbolic Computation ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Partial Differential Equations ◽  
Expansion Method ◽  
Hyperbolic Type ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool

In this paper, the (1/G')-expansion method is used to solve the coupled Boiti-Leon-Pempinelli (CBLP) system. The proposed method was used to construct hyperbolic type solutions of the nonlinear evolution equations. To asses the applicability and effectiveness of this method, some nonlinear evolution equations have been investigated in this study. It is shown that with the help of symbolic computation, the (1/G')-expansion method provides a powerful and straightforward mathematical tool for solving nonlinear partial differential equations.

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