Instability of Soliton Solutions to the Schamel-nonlinear Schrödinger Equation
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:59736", author = "Sarun Phibanchon and Michael A. Allen", title = "Instability of Soliton Solutions to the Schamel-nonlinear Schrödinger Equation", 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.", volume = "6", number = "1", pages = "66-3", }
