Group Analysis of Differential Equations and Integrable Systems − 2018

Georgi Grahovski (Department of Mathematical Sciences, University of Essex, Colchester, UK)

Integrable nonlocal multi-component equations with ${\cal PT}$ and ${\cal CPT}$ symmetries

We will present extensions of $N$-wave and derivative NLS types of equations with ${\cal PT}$ and ${\cal CPT}$-symmetries [1]. The types of (nonlocal) reductions leading to integrable equations invariant with respect to ${\cal C}$- (charge conjugation), ${\cal P}$- (spatial reflection) and ${\cal T}$- (time reversal) symmetries are described. The corresponding constraints on the fundamental analytic solutions and the scattering data are derived. Based on examples of $3$-wave (related to the algebra $sl(3,{\Bbb C})$) and $4$-wave (related to the algebra $so(5,{\Bbb C})$) systems, the properties of different types of $1$- and $2$-soliton solutions are discussed. It is shown that the ${\cal PT}$ symmetric $3$-wave equations may have regular multi-soliton solutions for some specific choices of their parameters [1]. Furthermore, we will present multi-component generalizations of derivative nonlinear Schrödinger (DNLS) type of related to A.III symmetric spaces and having with ${\cal CPT}$-symmetry [2]. This includes equations of Kaup-Newell (KN) and Gerdjikov-Ivanov (GI) types.

Based on a joint work with Vladimir Gerdjikov and Rossen Ivanov.

[1] V.S. Gerdjikov, G.G. Grahovski, R.I. Ivanov, On the $N$-wave Equations with ${\cal PT}$-symmetry, Theor. Math. Phys. 188 (2016), 1305 - 1321 [E-print: arXiv:1601.01929].
[2] V.S. Gerdjikov, G.G. Grahovski, R.I. Ivanov, On integrable wave interactions and Lax pairs on symmetric spaces, Wave Motion 71 (2017), 53-70 [E-print: arXiv:1607.06940].