## Asymptotic stability of the black soliton for the Gross-Pitaevskii equation.(English)Zbl 1326.35346

This paper considers finite-energy solutions of the Gross-Pitaevskii (GP) equation $i\partial_t \Psi+\partial_{xx}\Psi+\Psi(1-|\Psi|^2)=0,$ where $$\Psi$$ is a complex-valued function of $$(x,t)\in \mathbb{R}^2$$ satisfying the boundary condition $$|\Psi(x,t)|\to 1$$ as $$|x|\to +\infty.$$ Solitons in this paper are travelling wave solutions of the form $$\Psi(x,t)=U_c(x-ct).$$ There exist finite-energy non-constant solitons of speed $$c$$ if and only if $$|c|<\sqrt{2},$$ which are uniquely given by the formula $U_c(x)=\sqrt{\dfrac{2-c^2}{2}}\text{tanh}\left(\dfrac{\sqrt{2-c^2}}{2}x \right)+i\dfrac{c}{\sqrt{2}},$ up to the invariances of the problem, that is, multiplication by a constant of modulus one and translation. The particular solution when the speed $$c=0$$ is called the black soliton of the GP equation.
This paper studies the nonlinear stability of the black soliton. The first result of this paper concerns the orbital stability of the black soliton in the energy space. The proof relies on a variational approach, in the spirit of the work by M. I. Weinstein [Commun. Pure Appl. Math. 39, 51–67 (1986; Zbl 0594.35005)] and M. Grillakis et al. [J. Funct. Anal. 74, 160–197 (1987; Zbl 0656.35122)]. The main ingredient in the proof is to establish the coercivity of the energy functional. The second result concerns the asymptotic stability of the black soliton. The proof of the asymptotic stability uses ideas and techniques developed by Y. Martel and F. Merle [Math. Ann. 341, No. 2, 391–427 (2008; Zbl 1153.35068)] for the generalized Korteweg-de Vries equation.

### MSC:

 35Q55 NLS equations (nonlinear Schrödinger equations) 35B35 Stability in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs 35C08 Soliton solutions 35C07 Traveling wave solutions

### Citations:

Zbl 0594.35005; Zbl 0656.35122; Zbl 1153.35068
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### References:

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