Journal ArticleDOI
The tanh method: I. Exact solutions of nonlinear evolution and wave equations
Willy Malfliet,Willy Hereman +1 more
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TLDR
In this article, a systemized version of the tanh method is used to solve particular evolution and wave equations, where the boundary conditions are implemented in this expansion, and the associated velocity can then be determined a priori, provided the solution vanishes at infinity.Abstract:
A systemized version of the tanh method is used to solve particular evolution and wave equations. If one deals with conservative systems, one seeks travelling wave solutions in the form of a finite series in tanh. If present, boundary conditions are implemented in this expansion. The associated velocity can then be determined a priori, provided the solution vanishes at infinity. Hence, exact closed form solutions can be obtained easily in various cases.read more
Citations
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Linear and nonlinear waves
TL;DR: The study of waves can be traced back to antiquity where philosophers, such as Pythagoras, studied the relation of pitch and length of string in musical instruments and the subject of classical acoustics was laid down and presented as a coherent whole by John William Strutt in his treatise Theory of Sound.
Journal ArticleDOI
One method for finding exact solutions of nonlinear differential equations
TL;DR: In this article, a method for finding exact solutions of nonlinear differential equations is considered and modifications of the method are discussed, showing that the method is one of the most effective approaches for solving nonlinear problems.
Journal ArticleDOI
A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation
Wen-Xiu Ma,Jyh-Hao Lee +1 more
TL;DR: In this article, a direct approach to exact solutions of nonlinear partial differential equations is proposed, by using rational function transformations, which provides a more systematical and convenient handling of the solution process of non-linear equations, unifying the tanh-function type methods, the homogeneous balance method, the exp-function method, and the mapping method.
Journal ArticleDOI
The tanh method for traveling wave solutions of nonlinear equations
TL;DR: The tanh method is employed for traveling wave solutions of nonlinear equations and the study is extended to equations that do not have tanh polynomial solutions.
Journal ArticleDOI
Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm
Wen-Xiu Ma,Zuo-Nong Zhu +1 more
TL;DR: The multiple exp-function algorithm, as a generalization of Hirota’s perturbation scheme, is used to construct multiple wave solutions to the (3 + 1)-dimensional generalized KP and BKP equations.
References
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Journal ArticleDOI
Dust -acoustic waves in dusty plasmas
TL;DR: In this paper, a balance of dust particle inertia and plasma pressure is investigated and it is shown that these waves can propagate linearly as a normal mode in a dusty plasma, and non-linearly as supersonic solitons of either positive or negative electrostatic potential.
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Solitary wave solutions of nonlinear wave equations
TL;DR: In this article, a method for obtaining traveling-wave solutions of nonlinear wave equations that are essentially of a localized nature is proposed based on the fact that most solutions are functions of a hyperbolic tangent.
Journal ArticleDOI
The tanh method: II. Perturbation technique for conservative systems
Willy Malfliet,Willy Hereman +1 more
TL;DR: In this paper, a general wave profile, with a perturbative solitary-wave contribution superposed, was obtained for a particular choice of the parameters, and a comparison with the exact solution was made.
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On types of nonlinear nonintegrable equations with exact solutions
TL;DR: In this article, the Riccati equation and the equation for the anharmonic oscillator are expressed in terms of the solutions of a non-integrable nonlinear equation.
Journal ArticleDOI
Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA
Willy Hereman,M Takaoka +1 more
TL;DR: The direct algebraic method for constructing travelling wave solutions on nonlinear evolution and wave equations has been generalized and systematized in this paper, where the class of solitary wave solutions is extended to analytic functions of the real exponential solutions of the linearized equation Expanding the solutions in an infinite series in these real exponentials, an exact solution of the nonlinear PDE is obtained, whenever the series can be summed.