Narrow Results

We present here new solutions of the focusing one-dimensional nonlinear Schrödinger (NLS) equation which appear as deformations of the Peregrine breather of order 10 with 18 real parameters. With this method, we obtain new families of quasi-rational solutions of the NLS equation, and we obtain explicit quotients of polynomial of degree 110 in x and t by a product of an exponential depending on t. We construct new patterns of different types of rogue waves and recover the triangular configurations as well as rings and concentric rings as found for the lower-orders.

In this paper, our purpose is to construct multiwave solutions for the system of equations for the ion sound and Langmuir waves (SEISLWs) by utilizing the traveling wave and logarithmic transformation with the help of symbolic computation and the ansatz function method. We apply three distinctive approaches: namely, positive quadratic function approach, three waves approach and double exponential approach. By choosing suitable parametric values, 3D graphics are plotted to get different types of multiwave solutions including lump waves, rogue wave and multipeak soliton solutions. Moreover, we revealed very interesting multiwaves interactional phenomena.

Two-layer fluid models are used to depict some nonlinear phenomena in fluid mechanics, medical science and thermodynamics. In this paper, we investigate a (3$+$1)-dimensional Yu-Toda-Sasa-Fukuyama equation for the interfacial waves in a two-layer liquid or elastic quasiplane waves in a lattice. Via the Kadomtsev-Petviashvili hierarchy reduction, we derive the rational solutions in the determinant forms and semi-rational solutions. The $n$th-order lump waves and multi-lump waves are obtained, where $n$ is a positive integer. We observe the second-order lump waves: Two-lump waves interact with each other and separate into two new lump waves. Two-lump waves are observed: Overtaking interaction takes place between the two-lump waves; After the interaction, the two-lump waves propagate with their original velocities and amplitudes. Studying the semi-rational solutions, we show the fusion between a lump wave and a bell-type soliton and fission of a bell-type soliton. Interaction between a line rogue wave and a bell-type soliton is shown.

On the basis of the application in the Bose–Einstein condensation, we investigate a (3 + 1)-dimensional Gross–Pitaevskii equation with distributed time-dependent coefficients. With the aid of the Kadomtsev–Petviashvili hierarchy reduction method, we construct the $N$th-order rogue-wave solutions in terms of the Gram determinant by introducing appropriate constraints. Using different coefficients for the diffraction $\beta (t)$ and gain/loss $\gamma (t)$, we demonstrate the behaviors of the first- and second-order rogue waves by analytical and graphical means. We find that only if $\gamma (t)=3\beta (t)$, the rogue waves appear on the constant backgrounds; otherwise, the heights of the backgrounds change as time goes on. With the different choices of $\beta (t)$ and $\gamma (t)$, the long-live, rapid-reducing and periodic rogue waves are discussed. The separated and aggregated second-order rogue waves are also shown on the constant and periodical backgrounds.

Studies on rogue waves (RWs) over constant and periodic wave background are of contemporary interest in several areas of physics. These RWs arise not only on such backgrounds but also appear on double-periodic wave background. In this paper, we derive RW solutions over a double-periodic wave background of a fifth-order nonlinear Schrödinger equation. We create the double-periodic wave background with elliptic functions (in combinations of cn, sn and dn) as seed solutions in the first iteration of Darboux transformation. We then calculate the growth rate of instability for these double-periodic solutions under different values of elliptic modulus parameter. On the top of these background waves, we generate RWs with the help of second independent solution to the eigenvalue problem. Further, we analyze the differences that occur in the appearance of RWs with reference to the lower-order and higher-order dispersions terms. In addition to the above, we examine the derived solutions in detail for certain system and elliptic modulus parameter values and highlight some interesting features that we obtain from our studies.

In this paper, an eighth-order nonlinear Schrödinger equation is investigated in an optical fiber, which can be used to describe the propagation of ultrashort nonlinear pulses. Lax pair and infinitely-many conservation laws are derived to verify the integrability of this equation. Via the Darboux transformation and generalized Darboux transformation, the analytic breather and rogue wave solutions are obtained. Influence of the coefficients of operators in this equation, which represent different order nonlinearity, and the spectral parameter on the propagation and interaction of the breathers and rogue waves is also discussed. We find that (i) the periodic of the breathers decreases as the augment of the spectral parameter; (ii) the coefficients of operators change the compressibility and periodic of the breathers, and can affect the interaction range and temporal–spatial distribution of the rogue waves.

Under investigation in this letter is a (3+1)-dimensional generalized B-type Kadomtsev-Petviashvili equation, which describes the weakly dispersive waves propagating in a fluid. Employing the Hirota method and symbolic computation, we obtain the lump, breather-wave and rogue-wave solutions under certain constraints. We graphically study the lump waves with the influence of the parameters $h_1$, $h_3$ and $h_5$ which are all the real constants: When $h_1$ increases, amplitude of the lump wave increases, and location of the peak moves; when $h_3$ increases, lump wave’s amplitude decreases, but location of the peak keeps unchanged; when $h_5$ changes, lump wave’s peak location moves, but amplitude keeps unchanged. Breather waves and rogue waves are displayed: Rogue waves emerge when the periods of the breather waves go to the infinity.

High-order rogue wave solutions of the Benjamin–Ono equation and the nonlocal nonlinear Schrödinger equation are derived by employing the bilinear method, which are expressed by simple polynomials. Typical dynamics of these high-order rogue waves are studied by analytical and graphical ways. For the Benjamin–Ono equation, there are two types of rogue waves, namely, bright rogue waves and dark rogue waves. In particular, the fundamental rogue wave pattern is different from the usual fundamental rogue wave patterns in other soliton equations. For the nonlocal nonlinear Schrödinger equation, the exact explicit rogue wave solutions up to the second order are presented. Typical rogue wave patterns such as Peregrine-type, triple and fundamental rogue waves are put forward. These high-order rogue wave patterns have not been shown before in the nonlocal Schrödinger equation.

We study a (2 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which characterizes the formation of patterns in liquid drops. By using Bell’s polynomials, an effective way is employed to succinctly construct the bilinear form of the gKP equation. Based on the resulting bilinear equation, we derive its solitary waves, rogue waves and homoclinic breather waves, respectively. Our results can help enrich the dynamical behavior of the KP-type equations.

Rational solutions and hybrid solutions from N-solitons are obtained by using the bilinear method and a long wave limit method. Line rogue waves and lumps in the (2+1)-dimensional nonlinear Schrödinger (NLS) equation are derived from two-solitons. Then from three-solitons, hybrid solutions between kink soliton with breathers, periodic line waves and lumps are derived. Interestingly, after the collision, the breathers are kept invariant, but the amplitudes of the periodic line waves and lumps change greatly. For the four-solitons, the solutions describe as breathers with breathers, line rogue waves or lumps. After the collision, breathers and lumps are kept invariant, but the line rogue wave has a great change.

In this paper, the generalized variable-coefficient forced Kadomtsev–Petviashvili (gvcfKP) equation is investigated, which can be used to characterize the water waves of long wavelength relating to nonlinear restoring forces. Using a dependent variable transformation and combining the Bell’s polynomials, we accurately derive the bilinear expression for the gvcfKP equation. By virtue of bilinear expression, its solitary waves are computed in a very direct method. By using the Riemann theta function, we derive the quasiperiodic solutions for the equation under some limitation factors. Besides, an effective way can be used to calculate its homoclinic breather waves and rogue waves, respectively, by using an extended homoclinic test function. We hope that our results can help enrich the dynamical behavior of the nonlinear wave equations with variable-coefficient.

In this paper, the (3+1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation is discussed, which can be used to describe certain characteristics of soliton in a nonlinear media with weak dispersion. By using the virtue of Bell polynomial, we construct the exact bilinear formalism and soliton wave of the equation, respectively. We also analyze its stability analysis. Moreover, based on the resulting bilinear formalism, we obtain its rouge wave solutions with a direct method. Finally, we also discuss the interaction phenomena between solitary wave solutions and rogue wave solutions. It is hoped that our results can be used to enrich the dynamics of the (3+1)-dimensional nonlinear wave fields.

Lattices are used in such fields as electricity, optics and magnetism. Under investigation in this paper is an inhomogeneous discrete nonlinear Schrödinger equation, which models the wave propagation in a lattice. Employing the Kadomtsev–Petviashvili (KP) hierarchy reduction, we obtain the rogue-wave solutions, and see that the rogue waves are affected by the coefficient of the on-site external potential. We see (1) the first-order rogue wave with one peak and two hollows; (2) the second-order rogue waves, each of which is with one peak or three humps; (3) the third-order rogue waves, each of which is with one peak or six humps, and those humps exhibit the triangular pattern, anti-triangular pattern and circular pattern. When the coefficient of the on-site external potential is a constant, the rogue waves periodically appear. When the coefficient of the on-site external potential monotonously changes, oscillations emerge on the constant background.

Under investigation in this work is a generalized nonlinear equation, which can be widely applied to describe various phenomena in nonlinear physical science field. The equation can be reduced to Kadomtsev–Petviashvili-type equations and Jimbo–Miwa-type equations as its special cases. By using Bell’s polynomial, its bilinear representation is well constructed. Based on the obtained bilinear formalism, we derive the solitary wave solutions of the equation. We also consider its kinky breather wave, rational breather wave and rogue wave solutions by employing the extended homoclinic test method, respectively. Moreover, the dynamic behaviors of these solutions are given by graphical analysis. It is hoped that our works can be helpful to understand dynamical behavior of the nonlinear equation.

In this paper, we construct the breathers of the (3+1)-dimensional Jimbo–Miwa (JM) equation by means of the Hirota bilinear method, then based on the Hirota bilinear method with a new ansatz form, the multiple rogue wave solutions are constructed. Here, we discuss the general breathers, first-order rogue waves, the second-order rogue waves and the third-order rogue waves. Then we draw the 3- and 2-dimensional plots to illustrate the dynamic characteristics of breathers and multiple rogue waves. These interesting results will help us better reveal (3+1)-dimensional JM equation evolution mechanism.

By considering a simple one-dimensional Gross–Pitaevskii equation with variable coefficient, we study various rogue waves by the choice of different trapping potential coefficient. The trapping potential coefficient is used as an independent parameter function; a simple procedure is established to obtain different chasses of the scattering length and rogue wave solutions by using similarity transformation. A few properties of rogue wave solutions are also discussed. Our results demonstrate that the rogue waves can be controlled by selecting appropriate trapping potential coefficients.

We illustrate the analytical control of the Akhmediev breather frequency in Bose–Einstein condensate in a time varying parabolic trap. This is achieved by varying Feshbach managed scattering length R(t) within the integrability framework. Recognizing the mathematical relation between R(t) and parabolic trap with the linear Schrödinger equation, we use isospectral Hamiltonian approach, which leads to a one parameter family of nonlinearity control functions, directly affecting the characteristic scales of the solution space, for the same trapping potential. The chirp equation and the trap equation being same, the phase of the wave for the obtained class of nonlinearity function remains identical. For explicating the general applications of the present approach, we exhibit breather frequency control in a number of cases, with asymptotically vanishing nonlinearity, for expulsive parabolic trap. Interestingly, the same procedure is found to provide a free parameter, which controls the Rogue wave amplitude.

We have numerically generated the rogue waves supported by discrete nonlinear Schrödinger equation (DNLSE). The investigation has been made on pure cubic DNLSE as well as on cubic-quintic discrete nonlinear Schrödinger equation (CQDNLSE). Runge–Kutta fourth (RK4) order method has been employed to obtain these results. For the cubic DNLSE breathers are found to be localized in one of the dimensions and periodic in the other dimensions, whereas rogue waves are found to be appearing from nowhere and disappearing without a trace. For CQDNLSE it is reported that the sign of quintic term possesses a significant effect on the intensity of the rogue waves. Further these waves are appearing from nowhere but instead of disappearing these show periodic reappearance. For larger value of quintic coefficient this phenomenon is visible at shorter length scale. Breathers are reported for the first time for the cubic DNLSE. The obtained results for CQDNLSE are unobserved previously in any constant coefficient discrete nonlinear equation.

Giant anomalous events may occur in systems characterized by many waves, with extremely high amplitude waves appearing with a probability much higher than expected for a random dynamics. Such rare, and giant, events are known to occur on the ocean surface and have been named, in that context, "rogue waves". We present here some recent experimental results on the statistical properties of giant optical waves, or "optical rogue waves", observed in spatially extended optical systems. We will present the main statistical features of such extreme events, then, based on both a linear and a nonlinear experiment, we will outline a way enabling us to identify two key ingredients at the origin of optical rogue waves, namely, granularity, that is, a minimal size of the individual light spots, and inhomogeneity, that is, clustering of the light spots into separate domains with different average intensities. In the linear experiment we measure also the statistics of the waiting times between successive rogue waves and we show that it follows a log-Poisson distribution, which is characteristics of event separation observed in cooperative, complex systems and in such different fields as geophysics and biology.

In this paper, a generalized Darboux transformation is obtained for Fordy–Kulish NLS (nonlinear Schrödinger) systems on general Hermitian symmetric spaces in order to rigorously obtain rogue wave solutions for these systems. In particular, we express the generalized algebraic relations in a simple and elegant compact form. As an illustration, we derive multi-soliton, breather-type and mainly rogue wave solutions of triangular patterns for single- and multi-component NLS systems on $C{P}^1$ and $SP(2)/U(2),$ respectively. We also analyze the modulation instability of proper plane wave solutions. In order to get visual intuition for the dynamics of the result and solutions for the running examples, the associated simulations of profiles are furnished as well.