Galilean-invariant Nonlinear PDEs and their Exact Solutions

Abstract

All systems of (n+1)-dimensional quasilinear evolutional second-order equations invariant under the chain of algebras AG(1.n) ⊂ AG 1 (1.n) ⊂ AG 2 (1.n) are described. The obtained results are illustrated by examples of nonlinear Schrödinger equations.

Symmetries, ansätze and exact solutions of nonlinear second-order evolution equations with convection terms

“…The Galilei algebra of the linear heat equation contains the unit (1.3)) and is essentially different from that of Burger's equation. Nonlinear equations and systems of equations, preserving the Galilei algebra of the linear heat equation, have been described by Cherniha [21], Fushchych & Cherniha [11,22] and Fushchych et al [6].…”

The Darboux Transformation and N-Soliton Solutions of Gerdjikov–Ivanov Equation on a Time–Space Scale

“…Furthermore, there is another well-known equation, the Eckhaus equation, which possesses a very similar structure. The Eckhaus equation is also integrable and has soliton-like solutions expressed in terms of the hyperbolic functions [37,38]. Therefore, we will find the most effective way to reduce structures of solutions ( 68) and (71) on C and Kp, and study the Eckhaus equation on a time-space scale in our future work.…”

New Spherically Symmetric Solutions of Nonlinear Schrödinger Equations

“…Moreover, using the transformations (26), (27) and the translations in variable t (for details see [3,4]), we can construct the following formula for obtaining new solutions of the NSchE (29)…”

“…In many papers (see, for example, [1][2][3][4]) there are constructed and investigated solitonlike ones of the nonlinear Schrödiner equation (NSchE) iΨ t + ∆Ψ + λ|Ψ| 2d Ψ = 0,…”