Discrete symmetry’s chains and links between integrable equations

Abstract

The discrete symmetry's dressing chains of the nonlinear Schrödinger equation (NLS) and Davey-Stewartson equations (DS) are consider. The modified NLS (mNLS) equation and the modified DS (mDS) equations are obtained. The explicitly reversible Bäcklund auto-transformations for the mNLS and mDS equations are constructed. We demonstrate discrete symmetry's conjugate chains of the KP and DS models. The two-dimensional generalization of the P 4 equation are obtained.

The Cauchy Problem for the Generalized Hyperbolic Novikov–Veselov Equation via the Moutard Symmetries

“…The system (36) and (37) depends on four constants: φ 0 , Φ 0 , φ 0 andΦ 0 . Three of them can be chosen arbitrarily, whereas the fourth one would have to satisfy the Equation 35, namely:…”

“…The answer lies in repeating the Moutard transformation process we described in Section 2! Indeed, since the unknown functions φ(x) and Φ(y) satisfy the relations (36) and (37), all we really have to do is substitute them into the system (32), derive A(x, t) and B(y, t), and substitute them in Equation (21) to find out the sought after u[1](x, y, t), which will conclude the problem.…”

Closed dressing chains of 1D and 2D Toda lattice

“…5. This work is continuation of our research in [6], [7] which, for one's turn, was inspired by the article [8].…”

“…Formulas (26) were obtained from the discrete symmetries chains of NLS in our work [7]. There is another (more spontaneous) way to obtain them.…”

“…In turn the Goursat equation produce 2D mKdV hierarchy so transformations (45) (which don't conflict with (46)) can be used to construct exact solutions of equations from this hierarchy. The DS equations (36) admits ST ( [7], [16]):…”