Symmetries and soliton solutions of the Galilean complex Sine-Gordon equation

• We discuss a new non-linear Galilean complex Sine-Gordon equation.
• We find the Lie point symmetries and perform a symmetry reduction of this equation.
• We find one-soliton solutions of this new Galilean equation.

We discuss a new equation, the Galilean version of the complex Sine-Gordon equation in 1+11+1 dimensions, Ψxx(1−Ψ⁎Ψ)+2imΨt+Ψ⁎Ψx2−Ψ(1−Ψ⁎Ψ)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.