Long Time Motion of NLS Solitary Waves in a Confining Potential

Abstract

We study the motion of solitary-wave solutions of a family of focusing generalized nonlinear Schrödinger equations with a confining, slowly varying external potential, V (x).A Lyapunov-Schmidt decomposition of the solution combined with energy estimates allows us to control the motion of the solitary wave over a long, but finite, time interval.We show that the center of mass of the solitary wave follows a trajectory close to that of a Newtonian point particle in the external potential V (x) over a long time interval.

Dark solitons in external potentials

“…In the end, we mention that the rigorous derivation of the Newton's particle equation for slow dynamics of a bright soliton in an external potential has been reported recently in [6,13,16]. Derivation of its counterpart (1.7) for slow dynamics of a dark soliton is an open problem of analysis.…”

“…(iii) It follows from the decay (4.5) and (4.6) for the four fundamental solutions that there exist constants A ± (λ) and B ± (λ), such that 16) where κ ± are defined by the characteristic equation (4.3) with λ = Λ(ǫ). By expansion (4.4), the expansion of slowly decaying solutions e ∓κ − x gives (4.14), while expansion of fast decaying solutions e ∓κ + x is not written.…”

Untitled

“…For example, it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for Hartree equations; see [Fröhlich et al 2002;2004;Jonsson et al 2006;Abou Salem 2007] and also the remark following Theorem 4. Another very recent application of the nondegeneracy result (1-6) is presented in [Krieger et al 2008], where two soliton solutions to the time-dependent version of (1-4) are constructed.…”

“…This coercivity estimate plays a central role in the stability analysis of solitary waves for NLStype equations and their effective motion in an external potential; see, for example, [Weinstein 1985;Bronski and Jerrard 2000;Fröhlich et al 2004;2007b;Jonsson et al 2006;Abou Salem 2007;Holmer and Zworski 2008].…”

Solitary waves for the Hartree equation with a slowly varying potential

“…Later Bronski and Jerrard [2000] proved a similar theorem in the case of a power nonlinearity, and then more general nonlinearities were treated by Fröhlich, Tsai, and Yau [2002] and by Fröhlich, Gustafson, Jonsson, and Sigal [2004]. More recently Jonsson, Fröhlich, Gustafson, and Sigal [2006] have extended the validity of the effective dynamics to longer time in the case of a confining potential V , and Abou Salem [2008] has treated the case of a potential V that is permitted to vary in time. The case of the cubic nonlinear Schrödinger equation in dimension one was also studied by Holmer and Zworski [2007;.…”