LYAPUNOV-SCHMIDT REDUCTION IN THE STUDY OF PERIODIC TRAVELLING WAVE SOLUTIONS OF NONLINEAR DISPERSIVE LONG WAVE EQUATION

This article studies the bifurcation of periodic travelling wave solutions of nonlinear dispersive long wave equation by using Lyapunov-Schmidt reduction. We de- termined the conditions for the existence of regular solutions for the reduced equation corresponding to the main problem, also we found the linear approximation of the solu- tions of the main problem.

___

  • Berger,M.S., (1977), Nonlinearity and Functional Analysis, Lectures on Nonlinear problems in Math- ematical Analysis, Academic Press, Inc.
  • Boiti,P., Leon,J.P., and Manna,M. et al., (1987), Spectral transform for a two spatial dimension extension of the dispersive long wave equation, Inverse Problems, 3, pp.371-387.
  • Chen,Y. and Yong,X., (2007), Exact Solutions to the Dispersive Long Wave Equation,International Journal of Nonlinear Science,Vol.4, No.2, pp.147-150.
  • Eslami,M., Neyrame,A., and Ebrahimi,M., (2012), Explicit solutions of nonlinear (2 +1)-dimensional dispersive long wave equation,Journal of King Saud University Science, 24, pp.6971.
  • Fan,E.G., (2003), Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics. Chaos, Solitons and Fractals., 16, pp.819-839.
  • Paquin,G. and Winternitz,P., (1990), Group theoretical analysis of dispersive long wave equations in two space dimensions, Phys D., 46, pp.122-138.
  • Rong,J. and Tang,S., (2009), Bifurcation of travelling wave solutions for (2+1)-dimension nonlinear dispersive long wave equation,Appl. Math. J. Chinese Univ., 24, 3, pp.291-297.
  • Sapronov,Yu.I. and Zachepa V.R., (2002), Local Analysis of Fredholm Equation, Voronezh Univ., Russia. (In Russian)
  • Singh,J., Kumar,D., and Kiliman,A., (2014), Numerical solutions of nonlinear fractional partial differ- ential equations arising in spatial diffusion of biological populations, Abstract and Applied Analysis, Vol.2014.
  • Tang,X.Y. and Lou,S.Y., (2002), Abundant coherent structures of the dispersive long-wave equation in (2+1)- dimensional spaces, Chaos Solitons Fractals, 14, pp.1451-1456.
  • Yomba,E., (2004), Construction of new soliton-like solutions of the (2 + 1) dimensional dispersive long wave equation. Chaos, Solitons and Fractals., 20, pp.1135-1139.