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Exact solutions of semilinear radial Schrödinger equations by separation of group foliation variables. (English) Zbl 1315.35194

Summary: Explicit solutions are obtained for a class of semilinear radial Schrödinger equations with power nonlinearities in multi-dimensions. These solutions include new similarity solutions and other new group-invariant solutions, as well as new solutions that are not invariant under any symmetries of this class of equations. Many of the solutions have interesting analytical behavior connected with blow-up and dispersion. Several interesting nonlinearity powers arise in these solutions, including the case of the critical (pseudo-conformal) power. In contrast, standard symmetry reduction methods lead to nonlinear ordinary differential equations for which few if any explicit solutions can be derived by standard integration methods.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C05 Solutions to PDEs in closed form
35B44 Blow-up in context of PDEs

Software:

CRACK
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Full Text: DOI arXiv

References:

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