Two (2+1)-dimensional integrable nonlocal nonlinear Schrödinger equations: Breather, rational and semi-rational solutions

Abstract

Recently, an integrable system of coupled (2 + 1)-dimensional nonlinear Schrödinger (NLS) equations was introduced by Fokas (eq. (18) in Nonlinearity 29, 319324 (2016)). Following this pattern, two integrable equations [eqs.(2) and (3)] with specific parity-time symmetry are introduced here, under different reduction conditions. For eq. (2), two kinds of periodic solutions are obtained analytically by means of the Hirota's bilinear method. In the long-wave limit, the two periodic solutions go over into rogue waves (RWs) and semi-rational solutions, respectively. The RWs have a line shape, while the semi-rational states represent RWs built on top of the background of periodic line waves. Similarly, semi-rational solutions consisting of a line RW and line breather are derived. For eq. (3), three kinds of analytical solutions,viz., breathers, lumps and semi-rational solutions, representing lumps, periodic line waves and breathers are obtained, using the Hirota method. Their dynamics are analyzed and demonstrated by means of threedimensional plots. It is also worthy to note that eq. (2) can reduce to a (1 + 1)-dimensional "reverse-space" nonlocal NLS equation by means of a certain transformation, Lastly, main differences between solutions of eqs. (2) and (3) are summarized.

Rogue waves and lumps on the nonzero background in the ‐symmetric nonlocal Maccari system

“…The nonlocal DS‐I equation gives rise to periodic solitons 73 and line breather solutions 31,70 . The nonlocal (2+1)‐dimensional NLS equation produces line breather solution 90 . Soliton solutions. The nonlocal Maccari system () generates three kinds of two‐soliton solutions, namely, dark‐dark solitons, dark‐antidark solitons, and antidark‐antidark solitons.…”

“…The simplest periodic wave solutions and are generated as It is a series of periodic line (effectively one‐dimensional) waves. As can be seen in Figure 1, a series of periodic line waves appear from the constant plane and annihilate rapidly, and its dynamic behaviors are similar to the line breather 48,70,87,90 . However, the periodic line wave solution possesses one maximum amplitude and one minimum amplitude, which is different from the line breather with two minimum amplitudes and one maximum amplitude.…”

“…The first‐order rational solution is a line RW in the ‐plane, its dynamical behavior being similar to that of the first‐order RW solution introduced in Refs. 48, 50, 70, 90. Additionally, higher order RWs composed of more line RWs can also be generated for lager .…”

“…In comparison to the results reported in Refs. 31, 70, 90, the corresponding solution always describes the evolution of the cross‐shaped RW on top of the background of periodic line waves.…”

Numerical Solution of High-Dimensional Shockwave Equations by Bivariate Multi-Quadric Quasi-Interpolation

“…One reason is that it is a fundamental shockwave PDE from fluid mechanics which often occurs in applied and computational mathematics, such as modeling of dynamics, heat conduction, and acoustic wave [1,2]. More importantly, it often plays the role of a testing equation to check the feasibility of the scheme, i.e., once the scheme is efficient for Burgers' equation, there is great possibility that it is applicable to other equations with shock or soliton waves, such as non-linear Schrodinger equation [3], Sine-Gordon equation [4], and Klein-Gordon equation [5].…”