New families of exact solitary patterns solutions for the nonlinearly dispersive R(m,n) equations

Yan, Zhenya

doi:10.1016/s0960-0779(02)00204-7

Cited by 14 publications

(4 citation statements)

References 12 publications

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“…Generally speaking, it has been shown that many nonlinear wave equations with fully dispersive terms possess these two kinds of solutions, such as K(m, n) equation, [2−4] [2−11] More recently we showed that B(1, n) equation with linear dispersive term and R(1, n) equation with linearly dispersive term also possesses these two solutions. [8,12] In 1999, Kumar and Panigrahi [13] presented that some nonlinear evolution equations with linear dispersive terms admitted the compacton-like solution in the form…”

Section: U(x T) =mentioning

confidence: 99%

New Compacton-Like and Solitary Pattern-Like Solutions of (2+1)-Dimensional Generalization of Modified KdV Equation

Chen

Yan

2005

Commun. Theor. Phys.

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Recently some (1+1)-dimensional nonlinear wave equations with linearly dispersive terms were shown to possess compacton-like and solitary pattern-like solutions. In this paper, with the aid of Maple, new solutions of (2+1)-dimensional generalization of mKdV equation, which is of only linearly dispersive terms, are investigated using three new transformations. As a consequence, it is shown that this (2+1)-dimensional equation also possesses new compacton-like solutions and solitary pattern-like solutions.

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Section: U(x T) =mentioning

confidence: 99%

New Compacton-Like and Solitary Pattern-Like Solutions of (2+1)-Dimensional Generalization of Modified KdV Equation

Chen

Yan

2005

Commun. Theor. Phys.

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“…Parker [11,12] investigated another regularized long-wave Boussinseq (RLW-Boussinseq) equation u tt + 1/2(u 2 ) xx − u xxtt = 0, (1.1) and obtained several smooth solitary waves and periodic waves by using the bilinear transformation method. Motivated by the rich mathematical and physical properties of the RLW-Boussinseq equation (1.1) in (1+1)-dimensional space, Yan [15,16] introduced a family of regularized long-wave Boussinseq equations (R(m, n) equations in short)…”

Section: Introductionmentioning

confidence: 99%

“…When m > 1, (1.2) becomes a family of the R(m, n) equations with nonlinear dispersion, and when m = 1, (1.2) becomes a family of the R(m, n) equations with linear dispersion. Some exact solitary waves and periodic waves of (1.2) for certain special values of m and n were obtained by Yan [15,16] and by Inc [4,5]. Since the analysis of general solutions is much too difficult and just a few exact solutions can been found, it is very important to do the qualitative analysis of the solutions.…”

Section: Introductionmentioning

confidence: 99%

Dynamics and bifurcations of travelling wave solutions of R(m, n) equations

Feng

2007

Proc Math Sci

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By using the bifurcation theory and methods of planar dynamical systems to R(m, n) equations, the dynamical behavior of different physical structures like smooth and non-smooth solitary wave, kink wave, smooth and non-smooth periodic wave, and breaking wave is obtained. The qualitative change in the physical structures of these waves is shown to depend on the systemic parameters. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of the above waves are given. Moreover, some explicit exact parametric representations of travelling wave solutions are listed.

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“…In recent years one can observe a splash of papers where authors presented a lot of different approaches to look for exact solutions of nonlinear differential equations [1,2,3,4,5,6,7,8,9]. There are two reasons to make the study in this direction.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

Kudryashov

2004

Zeitschrift Für Naturforschung A

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New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption that nonlinear differential equations have exact solution which is general solution of the simplest integrable equation. The Riccati equation is shown to be a building block to find a lot of nonlinear differential equations with exact solutions. Nonlinear differential equations of the second, third and fourth order with special solutions are given. Most of these equations are used at the description of processes in physics and in theory of nonlinear waves.

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New families of exact solitary patterns solutions for the nonlinearly dispersive R(m,n) equations

Cited by 14 publications

References 12 publications

New Compacton-Like and Solitary Pattern-Like Solutions of (2+1)-Dimensional Generalization of Modified KdV Equation

New Compacton-Like and Solitary Pattern-Like Solutions of (2+1)-Dimensional Generalization of Modified KdV Equation

Dynamics and bifurcations of travelling wave solutions of R(m, n) equations

Nonlinear Differential Equations With Exact Solutions Expressed Via The Weierstrass Function

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