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Register a new userInvariant Manifolds of traveling waves of the 3D Gross-Pitaevskii equation in the energy space
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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.
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We consider the 3D Gross-Pitaevskii equation begin{equation} onumber ipartial_t psi +Delta psi+(1-|psi|^2)psi=0 text{ for } psi:mathbb{R}times mathbb{R}^3 rightarrow mathbb{C} end{equation} and construct traveling waves solutions to this equation. These are solutions of the form $psi(t,x)=u(x_1,x_2,x_3-Ct)$ with a velocity $C$ of order $varepsilon|logvarepsilon|$ for a small parameter $varepsilon>0$. We build two different types of solutions. For the first type, the functions $u$ have a zero-set (vortex set) close to an union of $n$ helices for $ngeq 2$ and near these helices $u$ has degree 1. For the second type, the functions $u$ have a vortex filament of degree $-1$ near the vertical axis $e_3$ and $ngeq 4$ vortex filaments of degree $+1$ near helices whose axis is $e_3$. In both cases the helices are at a distance of order $1/(varepsilonsqrt{|log varepsilon|)}$ from the axis and are solutions to the Klein-Majda-Damodaran system, supposed to describe the evolution of nearly parallel vortex filaments in ideal fluids. Analogous solutions have been constructed recently by the authors for the stationary Gross-Pitaevskii equation, namely the Ginzburg-Landau equation. To prove the existence of these solutions we use the Lyapunov-Schmidt method and a subtle separation between even and odd Fourier modes of the error of a suitable approximation.
In this paper we study the existence of finite energy traveling waves for the Gross-Pitaevskii equation. This problem has deserved a lot of attention in the literature, but the existence of solutions in the whole subsonic range was a standing open problem till the work of Maris in 2013. However, such result is valid only in dimension 3 and higher. In this paper we first prove the existence of finite energy traveling waves for almost every value of the speed in the subsonic range. Our argument works identically well in dimensions 2 and 3. With this result in hand, a compactness argument could fill the range of admissible speeds. We are able to do so in dimension 3, recovering the aforementioned result by Maris. The planar case turns out to be more difficult and the compactness argument works only under an additional assumption on the vortex set of the approximating solutions.
We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by Gerard cite{Gerard}. In this paper we prove scattering for small data in the same space with some additional angular regularity, and in particular in the radial case we obtain small energy scattering.
The purpose of this paper is to provide a rigorous mathematical proof of the existence of travelling wave solutions to the Gross-Pitaevskii equation in dimensions two and three. Our arguments, based on minimization under constraints, yield a full branch of solutions, and extend earlier results, where only a part of the branch was built. In dimension three, we also show that there are no travelling wave solutions of small energy.
We establish the orbital stability of the black soliton, or kink solution, $v_0(x) = th big(frac{x}{sqrt{2}} big)$, to the one-dimensional Gross-Pitaevskii equation, with respect to perturbations in the energy space.
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