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Symmetries and soliton solutions of the Galilean complex Sine-Gordon equation. (English) Zbl 1364.35347

Summary: We discuss a new equation, the Galilean version of the complex Sine-Gordon equation in \(1+1\) dimensions, \(\Psi_{x x}(1-\Psi^\ast\Psi) + 2 \mathrm{i}m\Psi_t +\Psi^\ast \Psi_x^2 -\Psi(1-\Psi^\ast\Psi)^2 = 0\), derived from its relativistic counterpart via Galilean covariance. We determine its Lie point symmetries, discuss some group-invariant solutions, and examine some soliton solutions. The reduction under Galilean symmetry leads to an equation similar to the stationary Gross-Pitaevskii equation. This work is motivated in part by recent applications of the relativistic complex Sine-Gordon equation to the dynamics of Q-balls.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35Q53 KdV equations (Korteweg-de Vries equations)
35C08 Soliton solutions

Software:

SADE
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Full Text: DOI

References:

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