Fabrice Béthuel, Philippe Gravejat, Jean-Claude Saut
We provide a rigorous mathematical derivation of the convergence in the long-wave transonic limit of the minimizing travelling waves for the two-dimensional Gross-Pitaevskii equation towards ground states for the Kadomtsev-Petviashvili equation (KP I).
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We investigate the properties of finite energy travelling waves to the nonlinear Schrodinger equation with nonzero conditions at infinity for a wide class of nonlinearities. In space dimension two and three we prove that travelling waves converge in the transonic limit (up to rescaling) to ground states of the Kadomtsev-Petviashvili equation. Our results generalize an earlier result of F. Bethuel, P. Gravejat and J-C. Saut for the two-dimensional Gross-Pitaevskii equation, and provide a rigorous proof to a conjecture by C. Jones and P. H. Roberts about the existence of an upper branch of trave…
We construct a smooth branch of travelling wave solutions for the 2 dimensional Gross-Pitaevskii equations for small speed. These travelling waves exhibit two vortices far away from each other. We also compute the leading order term of the derivatives with respect to the speed. We construct these solutions by an implicit function type argument.
In the previous paper, we constructed a smooth branch of travelling waves for the 2 dimensional Gross-Pitaevskii equation. Here, we continue the study of this branch. We show some coercivity results, and we deduce from them the kernel of the linearized operator, a spectral stability result, as well as a uniqueness result in the energy space. In particular, our result proves the non degeneracy of these travelling waves, which is a key step in the classification of these waves and for the construction of multi-travelling waves.
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 study the Gross-Pitaevskii equation in dimension two with periodic conditions in one direction, or equivalently on the product space $\mathbb{R} \times \mathbb{T}_L$ where $L > 0$ and $\mathbb{T}_L = \mathbb{R} / L \mathbb{Z}.$ We focus on the variational problem consisting in minimizing the Ginzburg-Landau energy under a fixed momentum constraint. We prove that there exists a threshold value for $L$ below which minimizers are the one-dimensional dark solitons, and above which no minimizer can be one-dimensional.
This paper presents recent results concerning the existence and qualitative properties of travelling wave solutions to the Gross-Pitaevskii equation posed on the whole space R^N. Unlike the defocusing nonlinear Schrödinger equations with null condition at infinity, the presence of non-zero conditions at infinity yields a rather rich and delicate dynamics. We focus on the case N = 2 and N = 3, and also briefly review some classical results on the one-dimensional case. The works we survey provide rigorous justifications to the impressive series of results which Jones, Putterman and Roberts estab…
Construction of minimizing travelling waves for the Gross-Pitaevskii equation on $\mathbb{R} \times \mathbb{T}$
本研究针对$\mathbb{R} \times \mathbb{T}$上的Gross-Pitaevskii方程，通过适用于周期情形的新型对称化论证得到极小化序列紧性，构造出固定动量下金兹堡-朗道能量极小的有限能量行波分支，明确了不同横向长度下极小解的形态。
Logarithmic Gross-Pitaevskii equation
本文研究带对数非线性及无穷远非平凡边界条件的薛定谔方程，证明其柯西问题在能量空间中全局适定，并刻画了一维孤立波与行波。
Let $q(x,y)$ be an nondegenerate lump solution to KP-I (Kadomtsev-Petviashvili-I) equation $$\partial_x^4q-2\sqrt{2}\partial_x^2q-3\sqrt{2}\partial_x((\partial_xq) ^2)-2\partial_y^2q=0. $$ We prove the existence of a traveling wave solution $ u_{\e} (x-ct, y)$ to GP (Gross-Pitaevskii) equation $$ i\partial_{t}Ψ+ΔΨ+(1-|Ψ|^{2})Ψ=0,\ \ \ \mbox{in} \ {\mathbb R}^2 $$ in the transonic limit $$ c=\sqrt{2}-ε^2 $$ with $$ u_ε=1 + i εq(x,y) + {\mathcal O} (ε^2). $$ This proves the existence of finite energy solutions in the so-called Jones-Roberts program in the transonic range $ c \in (\sqrt{2}-ε^2, \…
Minimizing travelling waves for the one-dimensional nonlinear Schrödinger equation with non-zero condition at infinity
针对无穷远处非零的一维非线性 Schrödinger 方程，论文用固定动量约束下的能量极小化和集中紧性方法构造行波解，并证明极小化子的稳定性。
In this paper, we proceed along our analysis of the Korteweg-de Vries approximation of the Gross-Pitaevskii equation initiated in a previous paper. At the long-wave limit, we establish that solutions of small amplitude to the one-dimensional Gross-Pitaevskii equation split into two waves with opposite constant speeds $\pm \sqrt{2}$, each of which are solutions to a Korteweg-de Vries equation. We also compute an estimate of the error term which is somewhat optimal as long as travelling waves are considered. At the cost of higher regularity of the initial data, this improves our previous estimat…
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