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

Abstract

In this paper, we obtain the N th-order rational solutions for the defocusing non-local nonlinear Schrödinger equation by the Darboux transformation and some limit technique. Then, via an improved asymptotic analysis method relying on the balance between different algebraic terms, we derive the explicit expressions of all asymptotic solitons of the rational solutions with the order 1 ≤ N ≤ 4 . It turns out that the asymptotic solitons are localized in the straight lines or algebraic curves, and the exact solutions approach the curved asymptotic solitons with a slower rate than the straight ones. Moreover, we find that all the rational solutions exhibit just five different types of soliton interactions, and the interacting solitons are divided into two halves with each having the same amplitudes. Particularly for the curved asymptotic solitons, there may exist a slight difference for their velocities between at t and − t with certain parametric conditions. In addition, we reveal that the soliton interactions in the rational solutions with N ≥ 2 are stronger than those in the exponential and exponential-and-rational solutions.

Rational solitons for non-local Hirota equations: Robustness and cascading instability

“…where r(x, t) q*(−x, t), and α, β are real numbers. In contrast with works in the literature on non-local, non-linear Schrödinger equation [37,38], Eq. 2 incorporates third order dispersion and a special form of "self-steepening" cubic non-linearity which maintains the appropriate parity and symmetry.…”

“…Recently there have been tremendous interest in non-local evolution equations, especially those from the NLS family [36][37][38]. For example, rational soliton solutions for focusing and defocusing NLS equations have been studied [37,38].…”

“…Recently there have been tremendous interest in non-local evolution equations, especially those from the NLS family [36][37][38]. For example, rational soliton solutions for focusing and defocusing NLS equations have been studied [37,38]. One motivation is the existence of purely real spectra for parity-time-symmetric (PTsymmetric), non-Hermitian systems [39][40][41][42].…”

Elliptic‐ and hyperbolic‐function solutions of the nonlocal reverse‐time and reverse‐space‐time nonlinear Schrödinger equations

“…[7][8][9][10][11][12][13] Meanwhile, wide classes of explicit solutions were obtained for both 𝜎 = 1 and 𝜎 = −1 cases by some analytical methods. [6][7][8][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31] For example, it was shown that Equation (2) with 𝜎 = −1 admits the exponential, rational, exponential-and-rational solutions, which can display a rich variety of elastic soliton interactions on the nonzero background. 8,[15][16][17]19,21,25,31 Apart from Equation (2), Ablowitz and Musslimani 32 also proposed the nonlocal reverse-time NLS (RTNLS) equation iq t (x, t) + q xx (x, t) + 𝜎q(x, t)q(x, −t)q(x, t) = 0,…”

“…Meanwhile, wide classes of explicit solutions were obtained for both $$$ \sigma =1 $$$ and $$$ \sigma =-1 $$$ cases by some analytical methods 6–8,14–31 . For example, it was shown that Equation () with $$$ \sigma =-1 $$$ admits the exponential, rational, exponential‐and‐rational solutions, which can display a rich variety of elastic soliton interactions on the nonzero background 8,15–17,19,21,25,31 …”