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Showing papers on "Rogue wave published in 2018"


Journal ArticleDOI
TL;DR: The results indicate that rogue waves occur when the system hits unlikely pockets of wave configurations that trigger large disturbances of the surface height, and the rogue wave precursors in these pockets are wave patterns of regular height, but with a very specific shape that is identified explicitly, thereby allowing for early detection.
Abstract: The appearance of rogue waves in deep sea is investigated by using the modified nonlinear Schrodinger (MNLS) equation in one spatial dimension with random initial conditions that are assumed to be normally distributed, with a spectrum approximating realistic conditions of a unidirectional sea state. It is shown that one can use the incomplete information contained in this spectrum as prior and supplement this information with the MNLS dynamics to reliably estimate the probability distribution of the sea surface elevation far in the tail at later times. Our results indicate that rogue waves occur when the system hits unlikely pockets of wave configurations that trigger large disturbances of the surface height. The rogue wave precursors in these pockets are wave patterns of regular height, but with a very specific shape that is identified explicitly, thereby allowing for early detection. The method proposed here combines Monte Carlo sampling with tools from large deviations theory that reduce the calculation of the most likely rogue wave precursors to an optimization problem that can be solved efficiently. This approach is transferable to other problems in which the system's governing equations contain random initial conditions and/or parameters.

105 citations


Journal ArticleDOI
TL;DR: Based on the Lax pair of the coupled NLS equation, the determinant representation of the N -fold Darboux transformation(DT) was constructed in this paper, which obtained its higher-order soliton, breather and rogue wave solutions.

104 citations


Journal ArticleDOI
TL;DR: Using Bell’s polynomials and the extended homoclinic test theory, a bilinear form of the gBS equation is derived, which is explicitly constructed as a soliton solutions for the (2+1)-dimensional generalized breaking soliton equation.
Abstract: We consider a (2+1)-dimensional generalized breaking soliton (gBS) equation, which describes the interaction of the Riemann wave propagated along the y -axis with a long wave propagated along the x -axis. By using Bell’s polynomials, we derive a bilinear form of the gBS equation. Based on the resulting Hirota’s bilinear equation, we explicitly construct its soliton solutions. Furthermore, by using the extended homoclinic test theory, its homoclinic breather waves and rogue waves are well derived, respectively. It is hoped that our results can enrich the dynamical behavior of the gBS-type equations.

87 citations


Journal ArticleDOI
TL;DR: In this article, Fokas presented a nonlocal nonlinear Schrodinger (NLS) equation with a self-induced parity-time-symmetric potential, which is a two-spatial dimensional analogue of the nonlinear nonlinear NLS equation.
Abstract: 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.

86 citations


Journal ArticleDOI
TL;DR: In this article, a (3 + 1) -dimensional B-type Kadomtsev-Petviashvili equation was investigated for weakly dispersive waves in a fluid and three different phenomena between a lump and one kink were observed.
Abstract: Under investigation is a (3 + 1) -dimensional B-type Kadomtsev-Petviashvili equation, which describes the weakly dispersive waves in a fluid. Via the Hirota method and symbolic computation, we obtain the mixed lump-kink and mixed rogue wave-kink solutions. Through the mixed lump-kink solutions, we observe three different phenomena between a lump and one kink. For the fusion phenomenon, a lump and a kink are merged with the lump’s energy transferring into the kink gradually, until the lump merges into the kink completely. Fission phenomenon displays that a lump separates from a kink. The last phenomenon shows that a lump travels together with a kink with their amplitudes unchanged. In addition, we graphically study the interaction between a rogue wave and a pair of the kinks. It can be observed that the rogue wave arises from one kink and disappears into the other kink. At certain time, the amplitude of the rogue wave reaches the maximum.

86 citations


Journal ArticleDOI
TL;DR: In this article, exact solutions for the rogue periodic wave were constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux transformations.
Abstract: Rogue periodic waves stand for rogue waves on a periodic background. The nonlinear Schrodinger equation in the focusing case admits two families of periodic wave solutions expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are modulationally unstable with respect to long-wave perturbations. Exact solutions for the rogue periodic waves are constructed by using the explicit expressions for the periodic eigenfunctions of the Zakharov-Shabat spectral problem and the Darboux transformations. These exact solutions generalize the classical rogue wave (the so-called Peregrine's breather). The magnification factor of the rogue periodic waves is computed as a function of the elliptic modulus. Rogue periodic waves constructed here are compared with the rogue wave patterns obtained numerically in recent publications.

84 citations


Journal ArticleDOI
TL;DR: In this article, two families of travelling periodic waves of the modified Korteweg-de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn.
Abstract: Rogue periodic waves stand for rogue waves on a periodic background. Two families of travelling periodic waves of the modified Korteweg–de Vries (mKdV) equation in the focusing case are expressed by the Jacobian elliptic functions dn and cn. By using one-fold and two-fold Darboux transformations of the travelling periodic waves, we construct new explicit solutions for the mKdV equation. Since the dn-periodic wave is modulationally stable with respect to long-wave perturbations, the new solution constructed from the dn-periodic wave is a nonlinear superposition of an algebraically decaying soliton and the dn-periodic wave. On the other hand, since the cn-periodic wave is modulationally unstable with respect to long-wave perturbations, the new solution constructed from the cn-periodic wave is a rogue wave on the cn-periodic background, which generalizes the classical rogue wave (the so-called Peregrine's breather) of the nonlinear Schrodinger equation. We compute the magnification factor for the rogue cn-periodic wave of the mKdV equation and show that it remains constant for all amplitudes. As a by-product of our work, we find explicit expressions for the periodic eigenfunctions of the spectral problem associated with the dn and cn periodic waves of the mKdV equation.

83 citations


Journal ArticleDOI
TL;DR: In this article, the effects of the group velocity dispersion coefficient and the fourth-order nonlinear Schrodinger equation on the first-order and second-order rogue wave solutions were analyzed.
Abstract: In this paper, investigation is made on the coupled variable-coefficient fourth-order nonlinear Schrodinger equations, which describe the simultaneous propagation of optical pulses in an inhomogeneous optical fiber. Via the generalized Darboux transformation, the first- and second-order rogue wave solutions are constructed. Based on such solutions, effects of the group velocity dispersion coefficient and the fourth-order dispersion coefficient on the rogue waves are graphically analyzed. The first-order rogue waves with the eye-shaped distribution, the interactions between the first-order rogue waves with solitons, and the second-order rogue waves with one highest peak and with the triangular structure are displayed. When the value of the group velocity dispersion coefficient or the fourth-order dispersion increases, range of the first-order rogue wave increases. Composite rogue waves are obtained, where the temporal separation between two adjacent rogue waves can be changed if we adjust the group velocity dispersion coefficient and fourth-order dispersion coefficient. Periodic rogue waves are presented. Periods of such rogue waves decrease with the periods of the group-velocity dispersion and fourth-order dispersion coefficient decreasing.

80 citations


Journal ArticleDOI
20 Jul 2018
TL;DR: In this article, the authors measured the statistics of these ultrafast rogue wave patterns with a time lens and developed a numerical model proving that the patterns of the ultra fast rogue waves were generated by the non-instantaneous relaxation of the saturable absorber together with the polarization mode dispersion of the cavity.
Abstract: Fiber lasers are convenient for studying extreme and rare events, such as rogue waves, thanks to the lasers’ fast dynamics. Indeed, several types of rogue wave patterns were observed in fiber lasers at different time-scales: single peak, twin peak, and triple peak. We measured the statistics of these ultrafast rogue wave patterns with a time lens and developed a numerical model proving that the patterns of the ultrafast rogue waves were generated by the non-instantaneous relaxation of the saturable absorber together with the polarization mode dispersion of the cavity. Our results indicate that the dynamics of the saturable absorber is directly related to the dynamics of ultrafast extreme events in lasers.

77 citations


Journal ArticleDOI
TL;DR: In this article, the authors investigated the rogue wave dynamics governed by coupled nonlinear Schrodinger equation (NLSE) models and reported the evidence of a group of three dark rogue waves, the so-called dark three-sister rogue waves.
Abstract: Over the past decade, the rogue wave debate has stimulated the comparison of predictions and observations among different branches of wave physics, particularly between hydrodynamics and optics, in situations where analogous dynamical behaviors can be identified, thanks to the use of common universal models. Although the scalar nonlinear Schr\"odinger equation (NLSE) has constantly played a central role for rogue wave investigations, moving beyond the standard NLSE model is relevant and needful for describing more general classes of physical systems and applications. In this direction, the coupled NLSEs are known to play a pivotal role for the understanding of the complex wave dynamics in hydrodynamics and optics. Benefiting from the advanced technology of high-speed telecommunication-grade components, and relying on a careful design of the nonlinear propagation of orthogonally polarized optical pump waves in a randomly birefringent telecom fiber, this work explores, both theoretically and experimentally, the rogue wave dynamics governed by such coupled NLSEs. We report, for the first time, the evidence of a group of three dark rogue waves, the so-called dark three-sister rogue waves, where experiments, numerics, and analytics show a very good consistency.

76 citations


Journal ArticleDOI
TL;DR: It is shown that the rogue wave possess a growing and decaying line profile that arises from a nonconstant background and then retreat back to the same nonconstants background again, which can be used to illustrate the interactions of water waves in shallow water.
Abstract: Based on Hirota bilinear method, N -solitons, breathers, lumps and rogue waves as exact solutions of the (3+1)-dimensional nonlinear evolution equation are obtained. The impacts of the parameters on these solutions are analyzed. The parameters can influence and control the phase shifts, propagation directions, shapes and energies for these solutions. The single-kink soliton solution and interactions of two and three-kink soliton overtaking collisions of the Hirota bilinear equation are investigated in different planes. The breathers in three dimensions possess different dynamics in different planes. Via a long wave limit of breathers with indefinitely large periods, rogue waves are obtained and localized in time. It is shown that the rogue wave possess a growing and decaying line profile that arises from a nonconstant background and then retreat back to the same nonconstant background again. The results can be used to illustrate the interactions of water waves in shallow water. Moreover, figures are given out to show the properties of the explicit analytic solutions.

Journal ArticleDOI
TL;DR: It is observed that the rogue wave, possessing a peak wave profile, arises from one of the resonance stripe solitons, moves to the other, and then disappears, therefore, a rogue wave can be generated by the interaction between the lump soliton and the pair of resonance stripesolitons.
Abstract: The rogue wave and a pair of resonance stripe solitons to KP equation are discovered. First, based on the bilinear method, some lump solutions are obtained containing six parameters, four of which must cater to the non-zero conditions so as to insure the solution analytic and rationally localized. Second, a one-stripe-soliton-lump solution is presented and the interaction shows that the lump soliton can be drowned or swallowed by the stripe soliton, conversely, the lump soliton is spit out from the stripe soliton. Finally, a new ansatz of combination of positive quadratic functions and hyperbolic functions is introduced, and thus a novel nonlinear phenomenon is explored. It is interesting that a rogue wave can be excited. It is observed that the rogue wave, possessing a peak wave profile, arises from one of the resonance stripe solitons, moves to the other, and then disappears. Therefore, a rogue wave can be generated by the interaction between the lump soliton and the pair of resonance stripe solitons. However, compared with classic rouge wave, the dynamics of above nonlinear waves are quite different, which are graphically demonstrated.

Journal ArticleDOI
TL;DR: The Kadomtsev–Petviashvili equation is investigated, which describes the dynamics of nonlinear waves in plasma physics and fluid dynamics and a new family of two wave solutions, rational breather wave and rogue wave solutions of the equation is constructed.
Abstract: In this paper, a ( 3 + 1 ) -dimensional generalized Kadomtsev–Petviashvili (gKP) equation is investigated, which describes the dynamics of nonlinear waves in plasma physics and fluid dynamics. By employing the extended homoclinic test method, we construct a new family of two wave solutions, rational breather wave and rogue wave solutions of the equation. Moreover, by virtue of some ansatz functions and the Riccati equation method, its analytical bright soliton, dark soliton and traveling wave solutions are derived. Finally, we obtain its exact power series solution with the convergence analysis. In order to further understand the dynamics, we provide some graphical analysis of these solutions.

Journal ArticleDOI
TL;DR: In this paper, a (2 + 1 ) -dimensional Breaking Soliton equation, which can describe various nonlinear scenarios in fluid dynamics, is described using the Bell polynomials.

Journal ArticleDOI
TL;DR: In this article, an effective and straightforward method is presented to succinctly construct the bilinear representation of the (4+1)-dimensional nonlinear Fokas equation, which is an important physics model.
Abstract: Under investigation in this paper is the (4+1)-dimensional nonlinear Fokas equation, which is an important physics model. With the aid of Bell’s polynomials, an effective and straightforward method is presented to succinctly construct the bilinear representation of the equation. By using the resulting bilinear formalism, the soliton solutions and Riemann theta function periodic wave solutions of the equation are well constructed. Furthermore, the extended homoclinic test method is employed to construct the breather wave solutions and rogue wave solutions of the equation. Finally, a connection between periodic wave solutions and soliton solutions is systematically established. The results show that the periodic waves tend to solitary waves under a limiting procedure.

Journal ArticleDOI
TL;DR: In this paper, the authors investigate the (2 + 1 )-dimensional B-type Kadomtsev-Petviashvili (BKP) equation, which can be used to describe weakly dispersive waves propagating in the quasi media and fluid mechanics.

Journal ArticleDOI
TL;DR: A symbolic computation approach to constructing higher order rogue waves with a controllable center of the nonlinear systems is presented, making use of their Hirota bilinear forms.
Abstract: A symbolic computation approach to constructing higher order rogue waves with a controllable center of the nonlinear systems is presented, making use of their Hirota bilinear forms. As some examples, it turns out that some higher order rogue wave solutions of the Kadomtsev–Petviashvili (KP) type equations in ( 3 + 1 ) and ( 2 + 1 ) -dimensions are obtained. Some features of controllable center of rogue waves are graphically discussed.

Journal ArticleDOI
TL;DR: In this article, the multi-breather solutions of the focusing nonlinear Schrodinger equation (NLSE) on the background of elliptic functions were constructed by the Darboux transformation, and the dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of background.
Abstract: We construct the multi-breather solutions of the focusing nonlinear Schrodinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to\pm\infty$, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher-order rogue wave solution.

Journal ArticleDOI
TL;DR: The results show that the extreme behavior of the breather wave yields the rogue wave for the higher-order nonlinear Schrodinger equation.

Journal ArticleDOI
TL;DR: Four kinds of localized waves, namely, solitons, lumps, breathers, and rogue waves are constructed for the(3+1)-dimensional generalized KP equation based on the Hirota bilinear method and long wave limit.
Abstract: Based on the Hirota bilinear method and long wave limit, four kinds of localized waves, namely, solitons, lumps, breathers, and rogue waves are constructed for the(3+1)-dimensional generalized KP equation. N-soliton solutions are obtained by employing bilinear method, then breathers, two breathers and interaction breather solutions are obtained by selecting appropriate parameters on two-soliton solution and four-soliton solution. These breathers own different dynamic behaviors in the different planes. Taking a long wave limit on the two and four soliton solutions under special parameter constraints, one-order lumps and rogue waves, two-order lumps and rogue waves, and interaction solutions between lumps and rogue waves are derived. Applying the same method on the three soliton solution, interaction solutions between kink solitons with periodic solutions, lumps and rogue waves are constructed, respectively. The influence of parameters on the solution is analyzed. The propagation directions, phase shifts, energies and shapes for these solutions can be affected and controlled by the parameters. Moreover, graphics are presented to demonstrate the properties of the explicit analytical localized wave solutions.

Journal ArticleDOI
TL;DR: The results show that rogue waves can come from the extreme behavior of the breather solitary waves for the (2+1)-dimensional gCHKP equation.

Journal ArticleDOI
TL;DR: In this article, a set of coherently-coupled nonlinear Schrodinger equations with the positive coherent coupling terms, which are related to the optical fiber communication, are studied through the binary Darboux transformation with the dimensional reduction.
Abstract: A set of the coherently-coupled nonlinear Schrodinger equations with the positive coherent coupling terms, which are related to the optical fiber communication, are studied through the binary Darboux transformation with the dimensional reduction. Formalisms of the solutions appear as the mixtures of the polynomial functions with exponential functions. When the spectral parameter is real, we obtain different kinds of the solutions, such as the soliton, degenerate-soliton, periodic, and soliton-like rational solutions. When the spectral parameter is complex with a non-zero imaginary part, we obtain the rogue waves and twisted rogue-wave pairs, and show that an eye-shaped rogue wave splits into a twisted rogue-wave pair.

Journal ArticleDOI
TL;DR: In this article, the authors derived a bilinear equation for the BKP-Boussinesq equation via using Bell's polynomials and derived the Backlund transformation.
Abstract: Under investigation in this paper is the $$(3+1)$$ -dimensional B-type Kadomtsev–Petviashvili–Boussinesq (BKP–Boussinesq) equation, which can display the nonlinear dynamics in fluid. By using Bell’s polynomials, we explicitly derive a bilinear equation for the equation via a very natural and effective way. Then, three types of exchange identities of Hirota’s bilinear operators are presented to derive its Backlund transformation. Based on that, we construct the traveling wave solutions, kink solitary wave solutions, rational breathers and rogue waves of the equation. Finally, some properties of interaction phenomena are also provided, which can be used to study the domain of lump solutions. It is hoped that our results can be used to enrich the dynamical behavior of the $$(3+1)$$ -dimensional nonlinear evolution equations.

Journal ArticleDOI
05 Oct 2018-EPL
TL;DR: In this paper, a coupled nonlinear Schrodinger (NLS) equation can be reduced to the generalized NLS equation by constituting a certain constraint and a generalized Darboux transformation (DT) is constructed.
Abstract: We consider a coupled nonlinear Schrodinger (NLS) equation, which can be reduced to the generalized NLS equation by constituting a certain constraint. We first construct a generalized Darboux transformation (DT) for the coupled NLS equation. Then, by using the resulting DT, we analyse the solutions with vanishing boundary condition and non-vanishing boundary condition, respectively, including positon wave, breather wave and higher-order rogue wave solutions for the coupled NLS equation. Moreover, in order to better understand the dynamic behavior, the characteristics of these solutions are discussed through some diverting graphics under different parameters choices.

Journal ArticleDOI
TL;DR: In this article, it is shown that the fundamental first-order rogue wave can be classified into three patterns: four-petal state, dark state, and bright state by choosing different values of parameter values.
Abstract: General high-order rogue waves of the nonlinear Schrodinger–Boussinesq equation are obtained by the KP-hierarchy reduction theory, and the N-order rogue waves are expressed with the determinants, whose entries are all algebraic forms, which is shown in the theorem. It is found that the fundamental first-order rogue waves can be classified into three patterns: four-petal state, dark state, bright state by choosing different values of parameter $$\alpha $$ . An interesting phenomenon is discovered as the evolution of the parameter $$\alpha $$ : the rogue wave changes from four-petal state to dark state, whereafter bright state, which are consistent with the change in the corresponding critical points to the function of two variables. Furthermore, the dynamical property of second-order and third-order rogue waves is plotted, which can be regarded as the nonlinear superposition of the fundamental first-order rogue waves.

Journal ArticleDOI
TL;DR: The ( 3 + 1 ) -dimensional KP–Boussinesq equation, which can be used to describe the nonlinear dynamic behavior in scientific and engineering applications, is investigated and general high–order soliton solutions are derived by using the Hirota’s bilinear method combined with the perturbation expansion technique.

Journal ArticleDOI
TL;DR: In this paper, the properties of homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional nonlinear Schrodinger (NLS) equation, were studied.
Abstract: We study numerically the properties of (statistically) homogeneous soliton gas depending on soliton density (proportional to number of solitons per unit length) and soliton velocities, in the framework of the focusing one-dimensional nonlinear Schr\"odinger (NLS) equation. To model such gas we use $N$-soliton solutions ($N$-SS) with $N\ensuremath{\sim}100$, which we generate with specific implementation of the dressing method combined with 100-digits arithmetics. We examine the major statistical characteristics, in particular the kinetic and potential energies, the kurtosis, the wave-action spectrum and the probability density function (PDF) of wavefield intensity. We show that in the case of small soliton density the kinetic and potential energies, as well as the kurtosis, are very well described by the analytical relations derived without taking into account soliton interactions. With increasing soliton density and velocities, soliton interactions enhance, and we observe increasing deviations from these relations leading to increased absolute values for all of these three characteristics. The wave-action spectrum is smooth, decays close to exponentially at large wavenumbers and widens with increasing soliton density and velocities. The PDF of wave intensity deviates from the exponential (Rayleigh) PDF drastically for rarefied soliton gas, transforming much closer to it at densities corresponding to essential interaction between the solitons. Rogue waves emerging in soliton gas are multisoliton collisions, and yet some of them have spatial profiles very similar to those of the Peregrine solutions of different orders. We present example of three-soliton collision, for which even the temporal behavior of the maximal amplitude is very well approximated by the Peregrine solution of the second order.

Journal ArticleDOI
TL;DR: Within the coupled Fokas-Lenells equations framework, it is numerically confirmed that the Peregrine soliton beyond the threefold limit can be reproduced from either a deterministic initial profile or a chaotic background field, hence anticipating the feasibility of its experimental observation.
Abstract: Within the coupled Fokas-Lenells equations framework, we show explicitly that, in contrast to the expected threefold-amplitude magnification, Peregrine solitons can reach a peak amplitude as high as 5 times the background level. Besides, the interaction of two such anomalous Peregrine solitons can generate a spikelike rogue wave of extremely high peak amplitude, depending on the parameters used. We numerically confirm that the Peregrine soliton beyond the threefold limit can be reproduced from either a deterministic initial profile or a chaotic background field, hence anticipating the feasibility of its experimental observation.

Journal ArticleDOI
TL;DR: In this article, Kuznetsov-Ma and superregular Schrodinger equation (NLSE) breathers play key roles in the dynamics of a wide class of localized condensate perturbations.
Abstract: The one-dimensional focusing nonlinear Schrodinger equation (NLSE) on an unstable condensate background is the fundamental physical model that can be applied to study the development of modulation instability (MI) and formation of rogue waves. The complete integrability of the NLSE via inverse scattering transform enables the decomposition of the initial conditions into elementary nonlinear modes: breathers and continuous spectrum waves. The small localized condensate perturbations (SLCP) that grow as a result of MI have been of fundamental interest in nonlinear physics for many years. Here, we demonstrate that Kuznetsov-Ma and superregular NLSE breathers play the key role in the dynamics of a wide class of SLCP. During the nonlinear stage of MI development, collisions of these breathers lead to the formation of rogue waves. We present new scenarios of rogue wave formation for randomly distributed breathers as well as for artificially prepared initial conditions. For the latter case, we present an analytical description based on the exact expressions found for the space-phase shifts that breathers acquire after collisions with each other. Finally, the presence of Kuznetsov-Ma and superregular breathers in arbitrary-type condensate perturbations is demonstrated by solving the Zakharov-Shabat eigenvalue problem with high numerical accuracy.

Journal ArticleDOI
TL;DR: In this paper, the first exact rational solution of the set of rational solutions of the modified Korteweg-de Vries equation was presented, which can be considered as rogue waves of the corresponding equation.
Abstract: We present the first four exact rational solutions of the set of rational solutions of the modified Korteweg–de Vries equation. These solutions can be considered as rogue waves of the corresponding equation. Comparison with rogue wave solutions of the nonlinear Schrodinger equation shows a strong analogy between their characteristics, especially for amplitude-to-background ratio. The new solutions may be useful in the theory of rogue waves in shallow water and for light propagation in cubic nonlinear media involving only a few optical cycles.