Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. (English) Zbl 1336.35326
Summary: 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.
MSC:
35Q55 | NLS equations (nonlinear Schrödinger equations) |
34A05 | Explicit solutions, first integrals of ordinary differential equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
35C07 | Traveling wave solutions |
33E05 | Elliptic functions and integrals |
Keywords:
derivative nonlinear Schrödinger equation; travelling waves; periodic boundary condition; elliptic functions; complete elliptic integrals; exact solutionReferences:
[1] | J. V. Armitage, “Elliptic Functions ”,, Cambridge University Press (2006) · Zbl 1105.14001 |
[2] | S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition,, Int. Math. Res. Not. (2006) · Zbl 1149.35074 |
[3] | H. Ikeda, On the global branches of the solutions to a nonlocal boundary-value problem arising in Oseen’s spiral flows,, Commun. Pure Appl. Anal., 2, 381 (2003) · Zbl 1056.34021 |
[4] | K. Imamura, Stability and bifurcation of periodic traveling waves in a derivative non-linear Schrödingier equation,, Hiroshima Math. J., 40 (2010) · Zbl 1227.35121 |
[5] | S. Kosugi, A complete bifurcation diagram of the Ginzburg-Landau equation with periodic boundary conditions,, Commun. Pure Appl. Anal., 4, 665 (2005) · Zbl 1117.35304 |
[6] | Y. Lou, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Partial differential equations and applications,, Discrete Contin. Dyn. Syst., 10, 1 (2004) |
[7] | M. Murai, Representation formula for the plane closed elastic curves,, Discrete Contin. Dyn. Syst. Supplement 2013, 565 (2013) · Zbl 1310.53004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.