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2015, 2015(special): 878-900. doi: 10.3934/proc.2015.0878

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

Minoru Murai 1, , Kunimochi Sakamoto 2, and  Shoji Yotsutani 3,

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, Japan
3. 

Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194

Received  September 2014 Revised  January 2015 Published  November 2015

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: periodic boundary condition, travelling waves, Derivative nonlinear Schrödinger equation, exact solution., complete elliptic integrals, elliptic functions.
Mathematics Subject Classification: Primary: 35Q55, 34A05; Secondary: 34B1.
Citation: Minoru Murai, Kunimochi Sakamoto, Shoji Yotsutani. Representation formula for traveling waves to a derivative nonlinear Schrödinger equation with the periodic boundary condition. Conference Publications, 2015, 2015 (special) : 878-900. doi: 10.3934/proc.2015.0878
References:
[1]

J. V. Armitage and W. F. Eberlein, "Elliptic Functions ", Cambridge University Press, Cambridge, 2006. Google Scholar

[2]

S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not., (2006), article 96763.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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. Google Scholar

show all references

References:
[1]

J. V. Armitage and W. F. Eberlein, "Elliptic Functions ", Cambridge University Press, Cambridge, 2006. Google Scholar

[2]

S. Herr, On the Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition, Int. Math. Res. Not., (2006), article 96763.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

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