Remarks on solitary waves and Cauchy problem for Half-wave-Schrödinger equations

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  • Yakine Bahri
    Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada
  • Slim Ibrahim
    Department of Mathematics and Statistics, University of Victoria, 3800 Finnerty Road, Victoria, BC V8P 5C2, Canada
  • Hiroaki Kikuchi
    Department of Mathematics, Tsuda University, 2-1-1 Tsuda-machi, Kodaira-shi, Tokyo 187-8577, Japan

抄録

<jats:p> In this paper, we study solitary wave solutions of the Cauchy problem for Half-wave-Schrödinger equation in the plane. First, we show the existence and the orbital stability of the ground states. Second, we prove that given any speed [Formula: see text], traveling wave solutions exist and converge to the zero wave as the velocity tends to [Formula: see text]. Finally, we solve the Cauchy problem for initial data in [Formula: see text], with [Formula: see text]. The critical case [Formula: see text] still stands as an interesting open problem. </jats:p>

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