Abstract: We present a rational solution for a mixed nonlinear Schrodinger (MNLS) equation. This solution has two free parameters, a and b, representing the contributions of self-steepening and self phase-modulation (SPM) of an associated physical system, respectively. It describes five soliton states: a paired bright-bright soliton, a single soliton, a paired bright-grey soliton, a paired bright-black soliton, and a rogue wave state. We show that the transition among these five states is induced by self-steepening and SPM through tuning the values of a and b. This is a unique and potentially fundamentally important phenomenon in a physical system described by the MNLS equation.
Abstract: 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...
TL;DR: A first-order rogue wave (RW) solution of the KMN equation is presented by the one-fold Darboux transformation from a non-zero “seed” solution and it satisfies the complex modified Korteweg–de Vries (mKdV) and the NLS equation by two different transformations of variables.
Abstract: Recently, Anjan Kundu, Abhik Mukherjee and Tapan Naskar (KMN) have introduced a (2+1)-dimensional equation as a new extension of the well-known nonlinear Schrodinger (NLS) equation, which is called KMN equation in this paper. We provide a triplet Lax pair of the KMN equation. Basing on triplet Lax pair, we present the first-order rogue wave (RW) solution of the KMN equation by the one-fold Darboux transformation from a non-zero “seed” solution. This RW solution also satisfies the complex modified Korteweg–de Vries (mKdV) and the NLS equation by two different transformations of variables. Moreover, we discuss the localization of rogue wave of KMN equation, observing that the area of rogue wave is a constant which is just related to the amplitude of seed solution.
Abstract: We consider a next-higher-order extension of the Chen–Lee–Liu equation, i.e., a higher-order Chen–Lee–Liu (HOCLL) equation with third-order dispersion and quintic nonlinearity terms. We construct the n-fold Darboux transformation (DT) of the HOCLL equation in terms of the n × n determinants. Comparing this with the nonlinear Schrodinger equation, the determinant representation Tn of this equation is involved with the complicated integrals, although we eliminate these integrals in the final form of the DT, so that the DT of the HOCLL equation is unusual. We provide explicit expressions of multi-rogue wave (RW) solutions for the HOCLL equation. It is concluded that the rogue wave solutions are likely to be crucial when considering higher-order nonlinear effects.
Abstract: The breather solutions of the Maxwell–Bloch equations in a two-level resonant system associated with the self-induced transparency phenomenon are constructed by the Darboux transformation. After constructing the formulas of the second-order breather solutions, the double degeneration and hybrid solutions are studied by the analytical form as well as figures. Our results might be helpful in such application or prevention of the rogue waves from breather solution interactions and degeneration in the nonlinear optical systems associated with the Maxwell–Bloch equations.
Abstract: We study the AB system describing marginally unstable baroclinic wave packets in geophysical fluids and also ultrashort pulses in nonlinear optics. We show that the breather can be converted into different types of stationary nonlinear waves on constant backgrounds, including the multi-peak soliton, M-shaped soliton, W-shaped soliton and periodic wave. We also investigate the nonlinear interactions between these waves, which display some novel patterns due to the nonpropagating characteristics of the solitons: (1) Two antidark solitons can produce a W-shaped soliton instead of a higher-order antidark one; (2) the interaction between an antidark soliton and a W-shaped soliton can not only generate a higher-order antidark soliton, but also form a W-shaped soliton pair; and (3) the interactions between an oscillation W-shaped soliton and an oscillation M-shaped soliton show the multi-peak structures. We find that the transition occurs at a modulational stability region in a low perturbation frequency region.
Abstract: Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson  and Dysthe . A number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's  soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al..In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.
Abstract: IF a conducting liquid is placed in a constant magnetic field, every motion of the liquid gives rise to an E. M. F. which produces electric currents. Owing to the magnetic field, these currents give mechanical forces which change the state of motion of the liquid. Thus a kind of combined electromagnetic-hydro-dynamic wave is produced which, so far as I know, has as yet attracted no attention.
TL;DR: Novel soliton solutions for the nonautonomous nonlinear Schrödinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonaut autonomous solitONS that interact elastically and generally move with varying amplitudes, speeds, and spectra.
Abstract: Novel soliton solutions for the nonautonomous nonlinear Schr\"odinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies.
TL;DR: The detection of oscillatory phenomena associated with a large bright-point group that is 430,000 square kilometers in area and located near the solar disk center suggests that these torsional Alfvén oscillations are induced globally throughout the entire brightening.
Abstract: The flow of energy through the solar atmosphere and the heating of the Sun's outer regions are still not understood. Here, we report the detection of oscillatory phenomena associated with a large bright-point group that is 430,000 square kilometers in area and located near the solar disk center. Wavelet analysis reveals full-width half-maximum oscillations with periodicities ranging from 126 to 700 seconds originating above the bright point and significance levels exceeding 99%. These oscillations, 2.6 kilometers per second in amplitude, are coupled with chromospheric line-of-sight Doppler velocities with an average blue shift of 23 kilometers per second. A lack of cospatial intensity oscillations and transversal displacements rules out the presence of magneto-acoustic wave modes. The oscillations are a signature of Alfven waves produced by a torsional twist of ±22 degrees. A phase shift of 180 degrees across the diameter of the bright point suggests that these torsional Alfven oscillations are induced globally throughout the entire brightening. The energy flux associated with this wave mode is sufficient to heat the solar corona.
Abstract: For nonlinear short-pulse propagation in long optical fibers, the conventional static approximation of the nonlinear terms in the wave equation must be extended to include the derivative of the pulse envelope. As a result, an initially symmetric pulse will develop an asymmetric self-phase modulation and a self-steepening, which ultimately lead to shock formation unless balanced by dispersion. This effect may be responsible for the pulse asymmetries observed in recent experiments.