Solitons and periodic solutions to a couple of fractional nonlinear evolution equations

This paper studies a couple of fractional nonlinear evolution equations using first integral method. These evolution equations are foam drainage equation and Klein–Gordon equation (KGE), the latter of which is considered in (2 + 1) dimensions. For the fractional evolution, the Jumarie’s modified Riemann–Liouville derivative is considered. Exact solutions to these equations are obtained.

[1]  Anjan Biswas,et al.  1-soliton solution of the K(m,n) equation with generalized evolution , 2008 .

[2]  Wen-Xiu Ma,et al.  Travelling wave solutions to a seventh order generalized KdV equation , 1993 .

[3]  Ahmet Bekir,et al.  Travelling wave solutions of nonlinear evolution equations by using the first integral method , 2009 .

[4]  N. Taghizadeh,et al.  Exact solutions of the nonlinear Schrödinger equation by the first integral method , 2011 .

[5]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[6]  The fluid dynamics of foams , 2003 .

[7]  Mostafa Eslami,et al.  Exact solutions of the Kudryashov–Sinelshchikov equation and nonlinear telegraph equation via the first integral method , 2012 .

[8]  P. Hartman.  Ordinary Differential Equations , 1965 .

[9]  P. C. Ray,et al.  SOLUTION OF NON-LINEAR KLEIN-GORDON EQUATION WITH A QUADRATIC NON-LINEAR TERM BY ADOMIAN DECOMPOSITION METHOD , 2009 .

[10]  R. Hirota.  Exact solution of the Korteweg-deVries equation for multiple collision of solitons , 1971 .

[11]  A. Bekir,et al.  Analytic treatment of nonlinear evolution equations using first integral method , 2012 .

[12]  Nikolay K. Vitanov,et al.  Application of the method of simplest equation for obtaining exact traveling-wave solutions for two classes of model PDEs from ecology and population dynamics , 2010 .

[13]  Ismail Aslan.  Travelling wave solutions to nonlinear physical models by means of the first integral method , 2011 .

[14]  I. Podlubny.  Fractional differential equations , 1998 .

[15]  Wenxiu Ma,et al.  A multiple exp-function method for nonlinear differential equations and its application , 2010, 1010.3324.

[16]  Yongxiang Li,et al.  Existence of mild solutions for fractional evolution equations with mixed monotone nonlocal conditions , 2013, Zeitschrift für angewandte Mathematik und Physik.

[17]  Yi Zhang,et al.  Hirota bilinear equations with linear subspaces of solutions , 2012, Appl. Math. Comput..

[18]  Wenxiu Ma,et al.  A transformed rational function method and exact solutions to the 3+1 dimensional Jimbo–Miwa equation , 2009, 0903.5337.

[19]  Anjan Biswas,et al.  Solitary wave solution for KdV equation with power-law nonlinearity and time-dependent coefficients , 2009 .

[20]  İsmail Aslan,et al.  The first integral method for constructing exact and explicit solutions to nonlinear evolution equations , 2012 .

[21]  Denis Weaire,et al.  The foam drainage equation , 1996 .

[22]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[23]  Andrew G. Glen,et al.  APPL , 2001 .

[24]  B. Lu.  The first integral method for some time fractional differential equations , 2012 .

[25]  İsmail Aslan.  Some exact solutions for Toda type lattice differential equations using the improved (G′/G)-expansion method , 2012 .

[26]  Ismail Aslan,et al.  Exact and explicit solutions to nonlinear evolution equations using the division theorem , 2011, Appl. Math. Comput..

[27]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[28]  Zhaosheng Feng,et al.  The first-integral method to study the Burgers–Korteweg–de Vries equation , 2002 .

[29]  Wen-Xiu Ma,et al.  Solving the (3 + 1)-dimensional generalized KP and BKP equations by the multiple exp-function algorithm , 2012, Appl. Math. Comput..

[30]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[31]  B. M. Fulk.  MATH , 1992 .

[32]  W. Malfliet.  Solitary wave solutions of nonlinear wave equations , 1992 .

[33]  Wenxiu Ma,et al.  Explicit and exact solutions to a Kolmogorov-Petrovskii-Piskunov equation , 1995, solv-int/9511005.

[34]  Gang Chen,et al.  Solution of the Klein–Gordon for exponential scalar and vector potentials , 2005 .

[35]  Optical Solitons with Time-Dependent Dispersion, Nonlinearity and Attenuation in a Kerr-Law Media , 2009 .

[36]  A. Biswas,et al.  Topological and non-topological solitons of the Klein–Gordon equations in 1+2 dimensions , 2010 .

[37]  Anjan Biswas,et al.  Soliton perturbation theory for the quadratic nonlinear Klein-Gordon equation , 2008, Appl. Math. Comput..

[38]  Anjan Biswas,et al.  Solitary wave solution for the generalized Kawahara equation , 2009, Appl. Math. Lett..

[39]  Holger Kantz,et al.  Modified method of simplest equation and its application to nonlinear PDEs , 2010, Appl. Math. Comput..