Invariant Manifolds of Traveling Waves of the 3D Gross–Pitaevskii Equation in the Energy Space

Abstract

We study the local dynamics near general unstable traveling waves of the 3D Gross-Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We also prove that (i) the center-unstable manifold attracts nearby orbits exponentially before they get away from the traveling waves along the center directions and (ii) if an initial data is not on the center-stable manifolds, then the forward flow will be ejected away from traveling waves exponentially fast. Furthermore, under a non-degenerate assumption, we show the orbital stability of the traveling waves on the center manifolds, which also implies the local uniqueness of the local invariant manifolds. Our approach based on a geometric bundle coordinates should work for a general class of Hamiltonian PDEs.

Dynamics near the solitary waves of the supercritical gKDV equations

“…In this section, we set up the bundle coordinates near M precisely and discuss its smoothness. This subsection is in the same spirit as Section 2.2 in [10]. Fixing c > 0, define a vector bundle X e c over R with fibers X e c,y as (2.14)…”

“…Instead, the above bundle coordinate system (2.20), where V e ∈ X e c,y is not directly parametrized by a translation in y and a rescaling in c, represents a somewhat different framework based on the observation that, while the parametrization by the spatial translation of y and rescaling of c are not smooth in H 1 with respect to y and c respectively, the vector bundles X T,e,+,− c,y over M are smooth in c and y as given in Lemma 2.2. This geometric bundle coordinate system has been used in [2,10], in the latter of which we construct local invariant manifolds near unstable traveling waves of the Gross-Pitaevskii equation.…”

“…where X e c (δ) defined in (2.15). Our construction follows the procedure in [10]. Though it has been carried out in details in [10], for the sake of completeness we briefly describe the procedure here.…”

“…They constructed a clever nonlinear "optimal mobile distance" to overcome this difficulty. In this paper, we follow the approach as in [2,10] to utilize a smooth bundle coordinate system. Namely, any U in a tubular neighborhood of M is written as…”

“…To the best of our knowledge, this current work is the first one constructing invariant manifolds of a global soliton manifold for a dispersive PDE with derivative nonlinearities. Our approach, involving using the bundle coordinates and deriving space-time estimates with small exponential growth, seems to be rather general and, with minimal essential modifications, applicable to unstable relative equilibria (including ground and excited states) of a class of Hamiltonian PDEs with natural symmetries (see also [10]). This paper is organized as following.…”