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From Ginzburg-Landau to Gross-Pitaevskii. (English) Zbl 1126.35063
Summary: In this note we consider the Gross-Pitaevskii equation \(i\varphi_t +\Delta \varphi +\varphi(1-|\varphi|^2) = 0\), where \(\varphi\) is a complex-valued function defined on \(\mathbb R^N \times \mathbb R\), and study the following 2-parameters family of solitary waves: \(\varphi(x,t)=e^{i\omega t} v(x_1-ct,x')\), where \((\omega,c)\in \mathbb R^2\), \(\upsilon \in L_{\text{loc}}^3 (\mathbb R^N,\mathbb C)\) and \(x'\) denotes the vector of the last \(N-1\) variables in \(\mathbb R^N\). We prove that every distribution solution \(\varphi \), of the considered form, satisfies the following universal (and sharp) \(L ^\infty\)-bound: \[ \|\varphi\|_{L^{\infty(\mathbb R^N\times\mathbb R)}}^2 \leq \max\left\{0,1-\omega+\frac{c^2}{4}\right\}. \] This bound has two consequences. The first one is that \(\varphi\) is smooth and the second one is that a solution \(\phi \not\equiv 0\) exists, if and only if \(1- \omega+ \frac{c^2}{4} > 0\). We also prove a non-existence result for some solitary waves having finite energy. Some more general nonlinear Schrödinger equations are considered in the third and last section. The proof of our theorems is based on previous results of the author [Differ. Integral Equ. 11, No. 6, 875–893 (1998; Zbl 1074.35504)] concerning the Ginzburg-Landau system of equations in \(\mathbb R^N\).

35Q55 NLS equations (nonlinear Schrödinger equations)
35B45 A priori estimates in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
35J45 Systems of elliptic equations, general (MSC2000)
35J60 Nonlinear elliptic equations
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