
Group Analysis of Differential Equations and Integrable Systems − 2018
Georgi Grahovski
(Department of Mathematical Sciences, University of Essex, Colchester, UK)
Integrable nonlocal multicomponent equations with ${\cal PT}$ and ${\cal CPT}$ symmetries
Abstract:
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 multisoliton solutions for some specific choices of their parameters [1].
Furthermore, we will present multicomponent generalizations of derivative nonlinear SchrÃ¶dinger (DNLS) type of related to A.III symmetric spaces and having with ${\cal CPT}$symmetry [2]. This includes equations of KaupNewell (KN) and GerdjikovIvanov (GI) types.
Based on a joint work with Vladimir Gerdjikov and Rossen Ivanov.
References:
[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 [Eprint: 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), 5370 [Eprint: arXiv:1607.06940].

