Two-dimensional Rogue Waves Research Articles

Rogue waves of the Nizhnik-Novikov-Veselov equation via self-mapping transformation

This paper introduces a new two-dimensional self-mapping transformation applied to the Nizhnik-Novikov-Veselov equation, resulting in the generation of numerous rogue wave solutions. We discover that temporal-localized and spatiotemporal-localized two-dimensional rogue waves respectively. Notably, these rogue waves emerge from a zero background and subsequently exhibit both algebraic and exponential decay patterns. The proposed technique offers a potential tool for constructing rogue-like waves within (2+1)-dimensional nonlinear wave frameworks. The findings presented here serve as a robust mathematical foundation for advancing both theoretical understanding and practical applications of rogue waves.

Two-dimensional rogue wave clusters in self-focusing Kerr-media

In this paper we consider two-dimensional (2D) rogue waves that can be obtained from the approximate solution of the (2 + 1)-dimensional nonlinear Schrödinger (NLS) equation with Kerr nonlinearity. The approximate method proposed in this paper not only reduces complex operations needed for the solution of high-dimensional nonlinear partial differential equations, but also furnishes an effective mechanism for constructing stable high-dimensional rogue wave clusters, and also provides for more abundant local rogue wave excitations. Such an investigation is important in view of the known tendency for instability of localized solutions to the 2D and 3D NLS equations with focusing Kerr nonlinearity. We adopt the (2 + 1)-dimensional NLS equation in cylindrical coordinates as the basic model, and reduce the complexity of its solution by looking for the radially-symmetric ring-like solutions with the specific feature that the radius of the ring L is much greater than the ring thickness w. In this manner, we obtain an approximate analytical solution with the separated radial and angular functions which depend on the two integer parameters, the azimuthal mode number m and the rogue wave solution's order number n. With the two parameters L and w also included in the approximate solution, we show the profiles of the first-order and the second-order rogue waves, and discuss their characteristics. The method for finding approximate rogue wave solutions of the (2 + 1)-dimensional NLS equation proposed in this paper can be used as an effective way to obtain higher-order rogue wave excitations, and can be extended to other (2 + 1)-dimensional nonlinear systems.

Influence of transverse carrier diffusion on two-dimensional optical rogue waves in broad-area semiconductor lasers with a saturable absorber

The statistics and dynamics of two-dimensional rogue waves in a broad-area semiconductor laser with an intracavity saturable absorber are numerically investigated under the effect of transverse carrier diffusion. We show that lateral diffusion of carriers alters the statistics of rogue waves by enhancing their formation in smaller ratios of carrier lifetimes in active and passive materials while suppressing them when the ratio is larger. The temporal dynamics of the emitted rogue waves is also studied and show that the finite nonzero transverse carrier diffusion coefficient gives them a longer duration. To further approach a realistic experimental situation, we also investigate the statistics and dynamics of rogue waves by simulating a circular disk-shape pump which replaces the flat pump profile typically used in numerical simulations of broad-area lasers. We show that a finite pump shape reduces the number of emitted rogue waves per unit area both below and above the laser threshold for every carrier lifetime ratio. The temporal width of the emitted rogue waves is also shown to decrease as a consequence of removing the nonphysical effects of the infinite flat pump on carrier dynamics.

Self-similar transformation and excitation of rogue waves for (2+1)-dimensional Zakharov equation

The search for the excitation of two-dimensional rogue wave in a (2+1)-dimensional nonlinear evolution model is a research hotspot. In this paper, the self-similar transformation of the (2+1)-dimensional Zakharov equation is established, and this equation is transformed into the (1+1)-dimensional nonlinear Schrödinger equation. Based on the similarity transformation and the rational formal solution of the (1+1)-dimensional nonlinear Schrödinger equation, the rogue wave excitation of the (2+1)-dimensional Zakharov equation is obtained by selecting appropriate parameters. We can see that the shape and amplitude of the rogue waves can be effectively controlled. Finally, the propagation characteristics of line rogue waves are diagrammed visually. We also find that the line-type characteristics of two-dimensional rogue wave are present in the <i>x-y</i> plane when the parameter <inline-formula><tex-math id="M5">\begin{document}$ \gamma = 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M5.png"/></alternatives></inline-formula>. The line rogue wave is converted into discrete localized rogue wave in the <i>x-y</i> plane when the parameter <inline-formula><tex-math id="M6">\begin{document}$ \gamma \ne 1 $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="8-20211181_M6.png"/></alternatives></inline-formula>. The spatial localized rogue waves with short-life can be obtained in the required <i>x-y</i> plane region. This is similar to the Peregrine soliton (PS) first discovered by Peregrine in the (1+1)-dimensional NLS equation, which is the limit case of the “Kuznetsov-Ma soliton” (KMS) or “Akhmediev breather” (AB). The proposed approach to constructing the line rogue waves of the (2+1) dimensional Zakharov equation can serve as a potential physical mechanism to excite two-dimensional rogue waves, and can be extended to other (2+1)-dimensional nonlinear systems.

Rogue breathers and rogue lumps on a background of dark line solitons for the Maccari system

The Maccari system is a nonlinear dynamical model for the description of two-dimensional rogue waves occuring in diverse physical settings. For this two-dimensional nonlinear system, the time localized breathers and doubly localized lumps termed as rogue breathers and rogue lumps on a background of dark line solitons are investigated. The rogue breather behaves as a breather appearing and disappearing on a background of two dark line solitons, and is generated from the resonant collision between a breather and two dark line solitons. The limit case of a rogue breather is a rogue lump, which is truly localized in time and in the two-dimensional space. The Nth-order rogue lumps on a background of (N+1) dark line solitons are constructed by employing the Kadomtsev-Petviashvili hierarchy reduction method combined with the Hirota bilinear method. These higher-order lumps arise from a background of dark line solitons and then disappear into the background again after living for a very short period of time. The rogue lumps possess the key features of two-dimensional rogue waves occurring in a plethora of physical settings.

Doubly Localized Two-Dimensional Rogue Waves in the Davey–Stewartson I Equation

Doubly Localized Two-Dimensional Rogue Waves in the Davey–Stewartson I Equation

Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg-de Vries equation* *Project supported by the National Natural Scinece Foundation of China (Grant Nos. 11671219, 11871446, 12071304, and 12071451).

Within the (2 + 1)-dimensional Korteweg–de Vries equation framework, new bilinear Bäcklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation. By introducing an arbitrary function ϕ(y), a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method. By choosing the appropriate parameters, their interesting dynamic behaviors are shown in three-dimensional plots. Furthermore, novel rational solutions are generated by taking the limit of the obtained solitons. Additionally, two-dimensional (2D) rogue waves (localized in both space and time) on the soliton plane are presented, we refer to them as deformed 2D rogue waves. The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane, and its evolution process is analyzed in detail. The deformed 2D rogue wave solutions are constructed successfully, which are closely related to the arbitrary function ϕ(y). This new idea is also applicable to other nonlinear systems.

Self-similarity transformation and two-dimensional rogue wave construction of non-autonomous Kadomtsev-Petviashvili equation

<sec>Rogue wave is a kind of natural phenomenon that is fascinating, rare, and extreme. It has become a frontier of academic research. The rogue wave is considered as a spatiotemporal local rational function solution of nonlinear wave model. There are still very few (2 + 1)-dimensional nonlinear wave models which have rogue wave solutions, in comparison with soliton and Lump waves that are found in almost all (2 + 1)-dimensional nonlinear wave models and can be solved by different methods, such as inverse scattering method, Hirota bilinear method, Darboux transform method, Riemann-Hilbert method, and homoclinic test method. The structure and evolution characteristics of the obtained (2 + 1)-dimensional rogue waves are quite different from the prototypes of the (1 + 1)-dimensional nonlinear Schrödinger equation. Therefore, it is of great value to study two-dimensional rogue waves.</sec><sec>In this paper, the non-autonomous Kadomtsev-Petviashvili equation is first converted into the Kadomtsev-Petviashvili equation with the aid of a similar transformation, then two-dimensional rogue wave solutions represented by the rational functions of the non-autonomous Kadomtsev-Petviashvili equation are constructed based on the Lump solution of the first kind of Kadomtsev-Petviashvili equation, and their evolutionary characteristics are illustrated by images through appropriately selecting the variable parameters and the dynamic stability of two-dimensional single rogue waves is numerically simulated by the fast Fourier transform algorithm. The obtained two-dimensional rogue waves, which are localized in both space and time, can be viewed as a two-dimensional analogue to the Peregrine soliton and thus are a natural candidate for describing the rogue wave phenomena. The method presented here provides enlightenment for searching for rogue wave excitation of (2 + 1)-dimensional nonlinear wave models.</sec><sec>We show that two-dimensional rogue waves are localized in both space and time which arise from the zero background and then disappear into the zero background again. These rogue-wave solutions to the non-autonomous Kadomtsev-Petviashvili equation generalize the rogue waves of the nonlinear Schrödinger equation into two spatial dimensions, and they could play a role in physically understanding the rogue water waves in the ocean.</sec>

Optical turbulence and transverse rogue waves in a cavity with triple-quantum-dot molecules

We show that optical turbulence extreme events can exist in the transverse dynamics of a cavity containing molecules of triple quantum dots under conditions close to tunneling-induced transparency. These nanostructures, when coupled via tunneling, form a four-level configuration with tunable energy-level separations. We show that such a system exhibits multistability and bistability of Turing structures in instability domains with different critical wave vectors. By numerical simulation of the mean-field equation that describes the transverse dynamics of the system, we show that the simultaneous presence of two transverse solutions with opposite nonlinearities gives rise to a series of turbulent structures with the capability of generating two-dimensional rogue waves.

New Patterns of the Two-Dimensional Rogue Waves: (2+1)-Dimensional Maccari System* *Supported by the National Natural Science Foundation of China under Grant No. 11671219, the K. C. Wong Magna Fund in Ningbo University, Gai-Hua Wang is also supported by the Scientific Research Foundation of Graduate School of Ningbo University

The ocean rogue wave is one kind of puzzled destructive phenomenon that has not been understood thoroughly so far. The two-dimensional nature of this wave has inspired the vast endeavors on the recognizing new patterns of the rogue waves based on the dynamical equations with two-spatial variables and one-temporal variable, which is a very crucial step to prevent this disaster event at the earliest stage. Along this issue, we present twelve new patterns of the two-dimensional rogue waves, which are reduced from a rational and explicit formula of the solutions for a (2+1)-dimensional Maccari system. The extreme points (lines) of the first-order lumps (rogue waves) are discussed according to their analytical formulas. For the lower-order rogue waves, we show clearly in formula that parameter b 2 plays a significant role to control these patterns.

Two-dimensional dissipative rogue waves due to time-delayed feedback in cavity nonlinear optics

We demonstrate a way to generate two-dimensional rogue waves in two types of broad area nonlinear optical systems subject to time-delayed feedback: in the generic Lugiato-Lefever model and in the model of a broad-area surface-emitting laser with saturable absorber. The delayed feedback is found to induce a spontaneous formation of rogue waves. In the absence of delayed feedback, spatial pulses are stationary. The rogue waves are exited and controlled by the delay feedback. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical contribution, which is often considered as a signature of the presence of rogue waves, appears for sufficiently strong feedback. The generality of our analysis suggests that the feedback induced instability leading to the spontaneous formation of two-dimensional rogue waves is a universal phenomenon.