79 pags., 30 figs., 3 apps. -- Open Access funded by Creative Commons Atribution Licence 3.0

Versatile rogue waves in scalar, vector, and multidimensional nonlinear systems

In this paper, the partially party-time (PT) symmetric nonlocal Davey–Stewartson (DS) equations with respect to x is called x-nonlocal DS equations, while a fully PT symmetric nonlocal DSII equation is called nonlocal DSII equation. Three kinds of solutions, namely, breather, rational, and semirational solutions for these nonlocal DS equations are derived by employing the bilinear method. For the x-nonlocal DS equations, the usual (2 + 1)-dimensional breathers are periodic in x direction and localized in y direction. Nonsingular rational solutions are lumps, and semirational solutions are composed of lumps, breathers, and periodic line waves. For the nonlocal DSII equation, line breathers are periodic in both x and y directions with parallels in profile, but localized in time. Nonsingular rational solutions are (2 + 1)-dimensional line rogue waves, which arise from a constant background and disappear into the same constant background, and this process only lasts for a short period of time. Semirational solutions describe interactions of line rogue waves and periodic line waves.

Rational and Semirational Solutions of the Nonlocal Davey–Stewartson Equations

Optical solitons in (2+1)–Dimensions with Kundu–Mukherjee–Naskar equation by extended trial function scheme

Optical solitons in presence of higher order dispersions and absence of self-phase modulation

Recently, a number of nonlocal integrable equations, such as the PT-symmetric nonlinear Schrodinger (NLS) equation and PT-symmetric Davey–Stewartson equations, were proposed and studied. Here, we show that many of such nonlocal integrable equations can be converted to local integrable equations through simple variable transformations. Examples include these nonlocal NLS and Davey–Stewartson equations, a nonlocal derivative NLS equation, the reverse space-time complex-modified Korteweg–de Vries (CMKdV) equation, and many others. These transformations not only establish immediately the integrability of these nonlocal equations, but also allow us to construct their Lax pairs and analytical solutions from those of the local equations. These transformations can also be used to derive new nonlocal integrable equations. As applications of these transformations, we use them to derive rogue wave solutions for the partially PT-symmetric Davey–Stewartson equations and the nonlocal derivative NLS equation. In addition, we use them to derive multisoliton and quasi-periodic solutions in the reverse space-time CMKdV equation. Furthermore, we use them to construct many new nonlocal integrable equations such as nonlocal short pulse equations, nonlocal nonlinear diffusion equations, and nonlocal Sasa–Satsuma equations.

Transformations between Nonlocal and Local Integrable Equations

We construct an analytical and explicit representation of the Darboux transformation (DT) for the KunduEckhaus (KE) equation. Such solution and n-fold DT Tn are given in terms of determinants whose...

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2015.0236

The Darboux transformation of the Kundu–Eckhaus equation

Recently, Fokas presented a nonlocal Davey–Stewartson I (DSI) equation (Fokas 2016 Nonlinearity 29 319–24), which is a two-spatial dimensional analogue of the nonlocal nonlinear Schrodinger (NLS) equation (Ablowitz and Musslimani 2013 Phys. Rev. Lett. 110 064105), involving a self-induced parity-time-symmetric potential. For this equation, high-order periodic line waves and line breathers are derived by employing the bilinear method. The long wave limit of these periodic solutions yields two kinds of fundamental rogue waves, namely, kink-shaped and W-shaped line rogue waves. The interaction of fundamental line rogue waves generate higher-order rogue waves, which have several interesting patterns with different curvy profiles. Furthermore, semi-rational solutions are constructed, which are line rogue waves on a background of periodic line waves. Finally, two particular solutions of the nonlocal NLS equation, namely, a first-order rogue wave and a semi-rational solution are obtained as reductions of the corresponding solutions of the nonlocal DSI equation.

https://iopscience.iop.org/article/10.1088/1361-6544/aac761/pdf

Rogue waves of the nonlocal Davey–Stewartson I equation

We construct higher order rogue wave solutions for the Gerdjikov–Ivanov equation explicitly in terms of a determinant expression. The dynamics of both soliton and non-soliton solutions is discussed. A family of solutions with distinct structures, which are new to the Gerdjikov–Ivanov equation, are presented.

https://iopscience.iop.org/article/10.1088/0031-8949/89/03/035501/pdf

The higher order rogue wave solutions of the Gerdjikov–Ivanov equation

We consider the Riemann–Hilbert method for initial problem of the vector Gerdjikov–Ivanov equation, and obtain the formula for its N-soliton solution, which is expressed as a ratio of (N + 1) × (N + 1) determinant and N × N determinant. Furthermore, by applying asymptotic analysis, the simple elastic interactions of N-soliton are confirmed, and the shifts of phase and position are also explicitly displayed.

https://www.tandfonline.com/doi/pdf/10.1080/14029251.2017.1313475

Yongshuai Zhang

Papers

The Darboux transformation of the Kundu–Eckhaus equation

Rogue waves of the nonlocal Davey–Stewartson I equation

The higher order rogue wave solutions of the Gerdjikov–Ivanov equation

Riemann-Hilbert method and N-soliton for two-component Gerdjikov-Ivanov equation

Riemann–Hilbert method for the Wadati–Konno–Ichikawa equation: N simple poles and one higher-order pole