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Orbitally stable standing waves for a system of coupled nonlinear Schrödinger equations. (English) Zbl 0964.35147

From the introduction: We consider the existence and the stability of standing waves (and specially the so-called ground state solutions) for the following system of one-dimensional nonlinear Schrödinger equations: \[ \begin{aligned} iu_t+ u_{xx}+ \alpha_1|u|^{p_1-2} u+\alpha_0|v|^{p_0} |u|^{p_0-2}u &= \delta v_{xx},\\ iv_t+ v_{xx}+ \alpha_2|v|^{p_2-2} v+\alpha_0|u|^{p_0- 2} v &=\delta u_{xx}, \end{aligned} \] where \(\alpha_j\geq 0\) \((j= 1,2)\), \(\alpha_0\in \mathbb{R}\), and \(|\delta|< 1\).
Systems of this type appear in several branches of physics, such as in the study of interactions of waves with different polarizations or in the description of nonlinear modulations of two monochromate waves.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
78A60 Lasers, masers, optical bistability, nonlinear optics
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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