Instability of Soliton Solutions to the Schamel-nonlinear Schrödinger Equation

Authors: Sarun Phibanchon, Michael A. Allen

Abstract:

A variational method is used to obtain the growth rate of a transverse long-wavelength perturbation applied to the soliton solution of a nonlinear Schr¨odinger equation with a three-half order potential. We demonstrate numerically that this unstable perturbed soliton will eventually transform into a cylindrical soliton.

Keywords: Soliton, instability, variational method, spectral method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1075701

References:

[1] E. Infeld and G. Rowlands, Nonlinear Waves, Solitons and Chaos, 2nd ed. Cambridge: Cambridge University Press, 2000.
[2] H. Washimi and T. Taniuti, "Propagation of ion-acoustic solitary waves of small amplitude," Phys. Rev. Lett., vol. 17, pp. 996-8, 1966.
[3] H. Schamel, "Stationary solitary, snoidal and sinusoidal ion acoustic waves," Plasma Phys., vol. 14, no. 10, pp. 905-24, 1972.
[4] G. P. Agrawal, Nonlinear Fiber Optics. New York: Academic Press, 2001.
[5] R. K. Bullough, P. M. Jack, P. W. Kitchenside, and R. Saunders, "Solitons in laser physics," Phys. Scr., vol. 20, pp. 364-381, 1979.
[6] R. Fedele, H. Schamel, and P. K. Shukla, "Solitons in the Madelung-s fluid," Phys. Scr., vol. T98, pp. 18-23, 2002.
[7] S. Phibanchon and M. A. Allen, "Numerical solutions of the nonlinear Schr¨odinger equation with a square root nonlinearity," in The 2010 International Conference on Computational Science and its Applications (ICCSA 2010). Los Alamitos, CA, USA: IEEE Computer Society, 2010, pp. 293-5.
[8] G. Rowlands, "Stability of nonlinear plasma waves," J. Plasma Phys., vol. 3, pp. 567-76, 1969.
[9] D. Anderson, M. Lisak, and A. Berntson, "A variational approach to nonlinear evolution equations in optics," Pramana J. Phys., vol. 57, pp. 917-36, 2001.
[10] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, 2nd ed. Cambridge: Cambridge University Press, 1992.
[11] P. Frycz and E. Infeld, "Spontaneous transition from flat to cylindrical solitons," Phys. Rev. Lett., vol. 63, no. 4, pp. 384-5, 1989.
[12] S. Phibanchon and M. A. Allen, "Time evolution of perturbed solitons of modified Kadomtsev-Petviashvili equations," in The 2007 International Conference on Computational Science and its Applications (ICCSA 2007). Los Alamitos, CA, USA: IEEE Computer Society, 2007, pp. 20-3.