Integrable nonlocal vector nonlinear Schrödinger equation with self-induced parity-time-symmetric potential

Abstract

A two component nonlocal vector nonlinear Schrödinger equation (VNLSE) is considered with a self-induced PT symmetric potential. It is shown that the system possess a Lax pair and an infinite number of conserved quantities and hence integrable. Some of the conserved quantities like number operator, Hamiltonian etc. are found to be real-valued, in spite of these charges being non-hermitian. The soliton solution for the same equation is obtained through the method of inverse scattering transformation and the condition of reduction from nonlocal to local case is also mentioned. An inhomogeneous version of this VNLSE with space -time modulated nonlinear interaction term is also considered and a mapping of this Eq. with standard VNLSE through similarity transformation is used to generate its solutions.

Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions

“…To explore general soliton solutions, we adopt the non-standard bilinearization procedure developed for the scalar NNLS equation [14]. Using this procedure, we bilinearize both the nonlocal Manakov equation and the following a pair of coupled equations that arise in the zero curvature condition [43], that is…”

Discrete symmetries and nonlocal reductions

“…This means that the reduced systems admit recursion operators and Lax pairs. We can obtain N -soliton solutions of the reduced systems by the inverse scattering method [1]- [3], [10], [11], [14], [17], [19], [27], by the Darboux transformation [9], [16], [18], [22], [23], and by the Hirota bilinear method [7], [13], [15], [21], [31]- [33].…”

“…After Ablowitz and Musslimani's works there is a huge interest in obtaining nonlocal reductions of systems of integrable equations and finding interesting wave solutions of these systems. Specific examples are nonlocal NLS equation [1]- [14], nonlocal mKdV equation [2]- [4], [13], [15]- [18], nonlocal SG equation [2]- [4], [19], nonlocal DS equation [3], [20]- [24], nonlocal Fordy-Kulish equations [13], [25], nonlocal N -wave systems [3], [26], nonlocal vector NLS equations [27]- [30], nonlocal (2 + 1)dimensional negative AKNS systems [31], nonlocal coupled Hirota-Iwao mKdV systems [32]. See [33] for the discussion of superposition of nonlocal integrable equations, and [34] for the nonlocal reductions of the integrable equations of hydrodynamic type.…”

Nonlocal hydrodynamic type of equations

“…Example 1. We take the algebra A with k = 0 and consider the Lax equation (12) with a Lax operator (15). For n = 2, we obtain the shallow water waves system…”