Abstract
For a class of nonlinear Schrodinger equations we saw that there is a one to one correspondence between orbital, energetic and linear stability of solitary waves provided d″ (ω) is nonzero. The case d″ (ω) = 0 is critical in the sense that this equivalence breaks down. This result will also extend to the abstract theory provided the linearized operator satisfies the assumptions on its spectrum listed in the introduction. In addition, a more detailed analysis of the critical case will show that for linear stability a conditional stability. result can be obtained which corresponds to the (nonlinear) stability results by M. Weinstein [11].
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