Representation formulafor traveling waves to a derivative nonlinear Schrödinger equationwith the periodic boundary condition

Minoru Murai; Kunimochi  Sakamoto; Shoji Yotsutani

doi:10.3934/proc.2015.0878

Article Contents

2015, Volume 2015: 878-900. Doi: 10.3934/proc.2015.0878 open access

This issue Previous Article Global bifurcation sheet and diagrams of wave-pinning in a reaction-diffusion model for cell polarization Next Article Remarks on a dispersive equation in de Sitter spacetime

Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition

1.
Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194
2.
Graduate School of Science Department of Mathematical and Life Sciences, Hiroshima University, Kagamiyama, Higashi-Hiroshima, 739-8526
3.
Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194

Received: September 2014

Revised: January 2015

Published: October 2015

Abstract Related Papers Cited by

Abstract

This paper deals with a derivative nonlinear Schrödinger equation under periodic boundary conditions. Taking advantage of the symmetries of the equation, we search for the traveling wave solutions. The problem is reduced to second order nonlinear nonlocal differential equations. By solving the equations, explicit formulas for the traveling waves are obtained. These formulas allow us to visualize the global structure of the traveling waves with various speeds and profiles.

Keywords:

Derivative nonlinear Schrödinger equation,
travelling waves,
periodic boundary condition,
elliptic functions,
complete elliptic integrals,
exact solution.

Mathematics Subject Classification: Primary: 35Q55, 34A05; Secondary: 34B15.

Citation:

Related Papers

Cited by

References

[1]	J. V. Armitage and W. F. Eberlein, "Elliptic Functions ", Cambridge University Press, Cambridge, 2006.
[2]	S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not., (2006), article 96763.
[3]	H.Ikeda, K.Kondo, H.Okamoto and S.Yotsutani, On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen's spiral flows, Commun. Pure Appl. Anal., 2 (2003), no.3, 381-390.
[4]	K. Imamura, Stability and bifurcation of periodic traveling waves in a derivative non-linear Schrödingier equation, Hiroshima Math. J., 40 (2010), no. 2, 185 - 203.
[5]	S.Kosugi, Y.Morita and S.Yotsutani, A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions, Commun. Pure Appl. Anal., 4 (2005), no.3, 665-682.
[6]	Y.Lou, W-M.Ni and S.Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), no.1-2, 435-458.
[7]	M.Murai, W. Matsumoto and S.Yotsutani, Representation formula for the plane closed elastic curves, Discrete Contin. Dyn. Syst. Supplement 2013, AIMS, (2013), 565-585.