Stability of Solitary Waves for a Generalized Derivative Nonlinear Schrödinger Equation

Abstract

We consider a derivative nonlinear Schrödinger equation with a general nonlinearity. This equation has a two-parameter family of solitary wave solutions. We prove orbital stability/instability results that depend on the strength of the nonlinearity and, in some instances, on the velocity. We illustrate these results with numerical simulations.

Appendix: Auxiliary Calculations

In this section, we present certain integral relations that are helpful in studying the determinant and trace of d″(ω,c). In the following, we denote $\kappa =\sqrt{4\omega-c^{2}} >0$ and rewrite the solitary solution (1.9) as φ(x)^2σ=f(ω,c)h(ω,c;x)⁻¹, with

$$ f(\omega,c)=\frac{(\sigma+1)\kappa^2 }{2\sqrt{\omega}},\qquad h(x;\sigma;\omega,c)=\cosh(\sigma\kappa x)-\frac{c}{2\sqrt{\omega}}. $$

We also rewrite the functionals Q, P defined in (2.6)–(2.7) and their derivatives in terms of h and f:

(A.1)

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

where

The expressions in (A.3)–(A.6) involve various integrals. The following lemmas show that all of them can be expressed in terms of

$$\alpha_n =\int_0^\infty h^{-\frac{1}{\sigma}-n} \,\mathrm{d}x. $$

Lemma A.1

$$ \alpha_2 = \frac{4\omega}{(\sigma+1)\kappa^2} \alpha_0 + \frac{2c\sqrt{\omega}(2+\sigma)}{(\sigma+1) \kappa^2} \alpha_1 . $$

(A.7)

Proof

We first rewrite α ₀, and then integrate by parts:

Regrouping the terms in this last expression, we obtain (A.7). □

Lemma A.2

We have the following relations:

(A.8)

(A.9)

(A.10)

(A.11)

Proof

By integration by parts, and n integer,

$$ \int_0^\infty h^{-\frac{1}{\sigma}-n}h_c \,\mathrm{d}x = \frac{ c}{\kappa^2 (-\frac{1}{\sigma}-n+1)} \int_0^\infty h^{-\frac{1}{\sigma}-n+1} \,\mathrm{d}x -\frac{1}{2\sqrt{\omega}}\int_0^\infty h^{-\frac{1}{\sigma}-n} \,\mathrm{d}x. $$

Choosing n=2,1, we get (A.8) and (A.9). The relations (A.10) and (A.11) are obtained from (A.8), (A.9), and $h_{\omega}=-\frac{2}{c}h_{c} -\frac{\kappa^{2}}{4\omega^{3/2}c}$. □

Using Lemmas A.1 and A.2, we have the following.

Lemma A.3

Denoting $\tilde{\kappa}= 2^{-\frac{1}{\sigma}-2} \sigma^{-1} (1+\sigma)^{\frac{1}{\sigma}}\kappa^{2(\frac{1}{\sigma}-1)} \omega^{-\frac{1}{2\sigma}-\frac{1}{2}}$, we have

This is used in Sect. 4 to obtain (4.1).