Stability theory of solitary waves in the presence of symmetry, II

Orbital Stability: Analysis Meets Geometry

“…This has the advantage of eliminating the introduction of the domain D (and therefore of condition (ii)) and is precisely the definition of "solution" to (6.5) used in [GSS87,GSS90]. In [Stu08], E is a Hilbert space and still a different formulation is adopted.…”

“…The argument in the proof of Theorem 8.5 is extracted from the proof of Theorem 5.3 in [GSS87] and is used in [GSS90] as well. We point out however, that conditions (i) and (iii) are not made explicit there.…”

The Stability of Solitary Waves of Depression

“…In fact, the proof of stability of positive solitary waves appears in the appendix of [5]. The method of Benjamin has subsequently been refined and extended, and a general theory has been developed [2,4,8,10,11,16]. It appears however that almost all previous work has exclusively focused on positive solitary waves.…”

“…It appears however that almost all previous work has exclusively focused on positive solitary waves. In order to treat negative solitary waves, the general theory developed in [10,16] cannot be applied straightforwardly, and it is our purpose here to indicate a complete proof of stability of negative solitary waves. Thus the main contribution of the present article is the proof of the following theorem.…”

“…Proof. The proof is based on the techniques of of Bona, Grillakis, Souganidis Shatah, and Strauss in [8,10]. In particular, the theorem will be proved by contradiction as follows.…”