Rational and semi-rational solutions of the nonlocal Davey-Stewartson equations

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 semi-rational 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 semi-rational 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. Semi-rational solutions describe interactions of line rogue waves and periodic line waves.Comment: 23pages, 12 figures.This is the accepted version by Studies in Applied Mathematic

Transformations between nonlocal and local integrable equations

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 analytical solutions from solutions 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 multi-soliton 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.Comment: 15 pages, 4 figure

Dynamics of Rogue Waves in the Partially PT-symmetric Nonlocal Davey-Stewartson Systems

In this work, we study the dynamics of rogue waves in the partially $\cal{PT}$-symmetric nonlocal Davey-Stewartson(DS) systems. Using the Darboux transformation method, general rogue waves in the partially $\cal{PT}$-symmetric nonlocal DS equations are derived. For the partially $\cal{PT}$-symmetric nonlocal DS-I equation, the solutions are obtained and expressed in term of determinants. For the partially $\cal{PT}$-symmetric DS-II equation, the solutions are represented as quasi-Gram determinants. It is shown that the fundamental rogue waves in these two systems are rational solutions which arises from a constant background at $t\rightarrow -\infty$, and develops finite-time singularity on an entire hyperbola in the spatial plane at the critical time. It is also shown that the interaction of several fundamental rogue waves is described by the multi rogue waves. And the interaction of fundamental rogue waves with dark and anti-dark rational travelling waves generates the novel hybrid-pattern waves. However, no high-order rogue waves are found in this partially $\cal{PT}$-symmetric nonlocal DS systems. Instead, it can produce some high-order travelling waves from the high-order rational solutions.Comment: 22 pages, 26 figure

Families of rational and semi-rational solutions of the partial reverse space-time nonlocal Mel'nikov equation

Inspired by the works of Ablowitz, Mussliman and Fokas, a partial reverse space-time nonlocal Mel'nikov equation is introduced. This equation provides two dimensional analogues of the nonlocal Schrodinger-Boussinesq equation. By employing the Hirota's bilinear method, soliton, breathers and mixed solutions consisting of breathers and periodic line waves are obtained. Further, taking a long wave limit of these obtained soliton solutions, rational and semi-rational solutions of the nonlocal Mel'nikov equation are derived. The rational solutions are lumps. The semi-rational solutions are mixed solutions consisting of lumps, breathers and periodic line waves. Under proper parameter constraints, fundamental rogue waves and a semi-rational solutions of the nonlocal Schrodinger-Boussinesq equation are generated from solutions of the nonlocal Mel'nikov equation

General rogue waves and their dynamics in several reverse time integrable nonlocal nonlinear equations

A study of general rogue waves in some integrable reverse time nonlocal nonlinear equations is presented. Specifically, the reverse time nonlocal nonlinear Schr\"odinger (NLS) and nonlocal Davey-Stewartson (DS) equations are investigated, which are nonlocal reductions from the AKNS hierarchy. By using Darboux transformation (DT) method, several types of rogue waves are constructed. Especially, a unified binary DT is found for this nonlocal DS system, thus the solution formulas for nonlocal DSI and DSII equation can be written in an uniform expression. Dynamics of these rogue waves is separately explored. It is shown that the (1+1)-dimensional rogue waves in nonlocal NLS equation can be bounded for both x and t, or develop collapsing singularities. It is also shown that the (1+2)-dimensional line rogue waves in the nonlocal DS equations can be bounded for all space and time, or have finite-time blowing-ups. All these types depend on the values of free parameters introduced in the solution. In addition, the dynamics patterns in the multi- and higher-order rogue waves exhibits more richer structures, most of which have no counterparts in the corresponding local nonlinear equations.Comment: 22 pages, 12 figure

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

Recently, an integrable system of coupled (2+1)-dimensional nonlinear Schrodinger (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,\textit{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 three-dimensional plots. It is also worthy to note that eq. 2 can reduce to a (1+1)-dimensional \textquotedblleft reverse-space" nonlocal NLS equation by means of a certain transformation, Lastly, main differences between solutions of eqs.2 and 3 are summarized.Comment: Accepted by Chaos, Solitons & Fractals, 13 pages including 10 figure

Families of exact solutions of a new extended (2+1)-dimensional Boussinesq equation

A new variant of the $(2+1)$-dimensional [$(2+1)d$] Boussinesq equation was recently introduced by J. Y. Zhu, arxiv:1704.02779v2, 2017; see eq. (3). First, we derive in this paper the one-soliton solutions of both bright and dark types for the extended $(2+1)d$ Boussinesq equation by using the traveling wave method. Second, $N$-soliton, breather, and rational solutions are obtained by using the Hirota bilinear method and the long wave limit. Nonsingular rational solutions of two types were obtained analytically, namely: (i) rogue-wave solutions having the form of W-shaped lines waves and (ii) lump-type solutions. Two generic types of semi-rational solutions were also put forward. The obtained semi-rational solutions are as follows: (iii) a hybrid of a first-order lump and a bright one-soliton solution and (iv) a hybrid of a first-order lump and a first-order breather.Comment: Accepted by Nonlinear Dynamics, 19 pages including 10 figure

Local and nonlocal (2+1)(2+1)-dimensional Maccari systems and their soliton solutions

In this work, by using the Hirota bilinear method, we obtain one- and two-soliton solutions of integrable $(2+1)$-dimensional $3$-component Maccari system which is used as a model describing isolated waves localized in a very small part of space and related to very well-known systems like nonlinear Schr\"{o}dinger, Fokas, and long wave resonance systems. We represent all local and Ablowitz-Musslimani type nonlocal reductions of this system and obtain new integrable systems. By the help of reduction formulas and soliton solutions of the $3$-component Maccari system, we obtain one- and two-soliton solutions of these new integrable local and nonlocal reduced $2$-component Maccari systems. We also illustrate our solutions by plotting their graphs for particular values of the parameters.Comment: 26 pages, 28 figure

Reductions of the (4 + 1)-dimensional Fokas equation and their solutions

An integrable extension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations is investigated in this paper.We will refer to this integrable extension as the (4+1)-dimensional Fokas equation. The determinant expressions of soliton, breather, rational, and semi-rational solutions of the (4 + 1)-dimensional Fokas equation are constructed based on the Hirota's bilinear method and the KP hierarchy reduction method. The complex dynamics of these new exact solutions are shown in both three-dimensional plots and two-dimensional contour plots. Interestingly, the patterns of obtained high-order lumps are similar to those of rogue waves in the (1 + 1)-dimensions by choosing different values of the free parameters of the model. Furthermore, three kinds of new semi-rational solutions are presented and the classification of lump fission and fusion processes is also discussed. Additionally, we give a new way to obtain rational and semi-rational solutions of (3 + 1)-dimensional KP equation by reducing the solutions of the (4 + 1)-dimensional Fokas equation. All these results show that the (4 + 1)-dimensional Fokas equation is a meaningful multidimensional extension of the KP and DS equations. The obtained results might be useful in diverse fields such as hydrodynamics, non-linear optics and photonics, ion-acoustic waves in plasmas, matter waves in Bose-Einstein condensates, and sound waves in ferromagnetic media.Comment: Nonlinear Dynamics, 99, (2020) 3013-302

Semi-rational solutions of the third-type Davey-Stewartson equation

General dark solitons and mixed solutions consisting of dark solitons and breathers for the third-type Davey-Stewartson (DS-III) equation are derived by employing the bilinear method. By introducing the two differential operators, semi-rational solutions consisting of rogue waves, breathers and solitons are generated. These semi-rational solutions are given in terms of determinants whose matrix elements have simple algebraic expressions. Under suitable parametric conditions, we derive general rogue wave solutions expressed in terms of rational functions. It is shown that the fundamental (simplest) rogue waves are line rogue waves. It is also shown that the multi-rogue waves describe interactions of several fundamental rogue waves, which would generate interesting curvy wave patterns. The higher order rogue waves originate from a localized lump and retreat back to it. Several types of hybrid solutions composed of rogue waves, breathers and solitons have also been illustrated. Specifically, these semi-rational solutions have a new phenomenon: lumps form on dark solitons and gradual separation from the dark solitons is observed.Comment: 15 pages including 16 figure