Abstract
In this paper, we consider the fractional Korteweg-de Vries equations with general nonlinearities. By studying constrained minimization problems and applying the method of concentration-compactness, we obtain the existence of solitary waves for the generalized Korteweg-de Vries equations under some assumptions of the nonlinear term. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, in the sense that a solution which starts near the set will remain near it for all time.
Highlights
1 Introduction This paper is devoted to studying the existence and stability of solitary wave solutions of the generalized Korteweg-de Vries equation ut + f (u) x – L(u) x = in R, ( . )
In Section, we study the existence and stability of solitary waves of equation ( . ) with some special nonlinearities f (u) is devoted to studying equation with general nonlinearities f (u) satisfying the assumption (A)
(i) Iq∞ and Iq are finite and continuous on ( , +∞); for any q > , each minimizing sequence for the problem (Iq∞) or (Iq) is bounded in Hα(R); (ii) Iq∞ ≤ for any q >
Summary
Inspired by the methods used in [ , ], by studying the problem (Iq), we obtain the existence of solitary waves for equation with some special nonlinearities f (u) If {un} is a minimizing sequence for the problem (Iq∗ ), there exists a sequence {yn} ⊂ R and g ∈ Gq∗ such that {un(· + yn)} contains a subsequence converging strongly in Hα(R) to g, and lim inf n→+∞ g∈Gq∗ In Section , we study the existence and stability of solitary waves of equation
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