General stationary solutions of the nonlocal nonlinear Schrödinger equation and their relevance to the PT-symmetric system

Abstract

With the stationary solution assumption, we establish the connection between the nonlocal nonlinear Schrödinger (NNLS) equation and an elliptic equation. Then, we obtain the general stationary solutions and discuss the relevance of their smoothness and boundedness to some integral constants. Those solutions, which cover the known results in the literature, include the unbounded elliptic-function and hyperbolic-function solutions, the bounded sn-, cn-and dn-function solutions, as well as the bright and dark soliton solutions. By the imaginary translation invariance of the NNLS equation, we also derive the complex-amplitude stationary solutions, in which all the bounded cases obey either the PT -or anti-PT -symmetric relation. In particular, the complex tanh-function solution can exhibit no spatial localization in addition to the dark and anti-dark soliton profiles, where is sharp contrast with the common dark soliton. Considering the physical relevance to PT -symmetric systems, we show that the complex-amplitude stationary solutions can yield a wide class of complex and time-independent PT -symmetric potentials, and the symmetry breaking does not occur in the PT -symmetric linear systems with the associated potentials.

Discrete solitons dynamics in $$\mathscr {PT}$$-symmetric oligomers with complex-valued couplings

“…This includes models with saturable and higher-order nonlinearity [9][10][11][12][13][14][15][16], as well as fractional-order diffraction effect [17][18][19][20][21][22][23]. There are also works on nonlocal models that involve PT -symmetry, such as the nonlocal NLS equation [24][25][26][27][28] and nonlocal Davey-Stewartson equations [29][30][31]. Other models outside the NLS realm where PT -symmetry plays a role include a nonlocal nonlinear Hirota equation [32], a nonlocal long-wave-short wave interaction equation [33], double-ring optical power resonator [34], quantum master Lindblad equation [35], and dampled harmonic Liénard oscillator equations [36].…”

Bose–Einstein Condensation Processes with Nontrivial Geometric Multiplicities Realized via 𝒫𝒯−Symmetric and Exactly Solvable Linear-Bose–Hubbard Building Blocks

“…In our present paper, we paid attention to the less widespread simulation of the condensation processes in which the underlying dynamical equations are required P T −symmetric, i.e., invariant with respect to the simultaneous action of parity P and of the antilinear time reversal T [13]. As we already indicated above, the scope and impact of such a methodical innovation is not yet fully explored, with the existing results ranging from an amendment of our understanding of the topological phase transitions (for example, in a non-Hermitian Aubry-Andre-Harper model [47] or in quasicrystals [48]) up to the new approaches to the perception of the conservation laws [49], and from the theoretical studies of the interference between channels [50,51] and of the mechanisms of squeezing [52] up to the detailed, experiment-oriented simulations of the properties of the specific BEC-type condensates [41]. In this framework, promising results are also being obtained in the area of related mathematics.…”

Rational solutions of the defocusing non-local nonlinear Schrödinger equation: asymptotic analysis and soliton interactions

“…As a new simple integrable model, equation (1.1) has been actively studied from different mathematical aspects, including the inverse scattering transform schemes of initial value problem with zero and non-zero boundary conditions [4,6,17], Hamiltonian structures for the NNLS hierarchy [18], long-time asymptotic behaviour with decaying boundary conditions [19], equivalent transformation between the NLS and NNLS equations [20], local well-posedness and blow-up instability of arbitrarily small initial data [21], etc. Meanwhile, various analytical methods were used to derive wide classes of explicit solutions for both ε = 1 and ε = −1 cases of equation (1.1) [4,6,17,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. It has been shown that the focusing case possesses the bright-soliton, dark-soliton, rogue-wave and breather solutions, which may develop the blow-up behaviour in finite time [4,[25][26][27][28][29][30], whereas the defocusing case admits the non-singular soliton solutions, which in general exhibit the pairwise soliton interactions [17,[33][34][35][36]…”

“…Meanwhile, various analytical methods were used to derive wide classes of explicit solutions for both ε = 1 and ε = −1 cases of equation (1.1) [4,6,17,[22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. It has been shown that the focusing case possesses the bright-soliton, dark-soliton, rogue-wave and breather solutions, which may develop the blow-up behaviour in finite time [4,[25][26][27][28][29][30], whereas the defocusing case admits the non-singular soliton solutions, which in general exhibit the pairwise soliton interactions [17,[33][34][35][36][37][38].…”

“…Thus, the NNLS equation may have potential applications in PT -symmetric physics [42]. In this regard, Zhang et al [39] discussed the similarity between the time-dependent complex potential generated from an exact solution of equation (1.1) and the gain/loss distribution in a PT symmetric optical system; Moreover, Xu et al [27] showed that the general stationary solutions of equation (1.1) can be used to yield a wide class of complex and time-independent PT symmetric potentials whose associated Hamiltonians keep the PT symmetry unbroken.…”