Characteristics of rogue waves on a soliton background in a coupled nonlinear Schrödinger equation

In this paper, a coupled nonlinear Schrödinger (CNLS) equation, which can describe evolution of localized waves in a two‐mode nonlinear fiber, is under investigation. By using the Darboux‐dressing transformation, the new localized wave solutions of the equation are well constructed with a detailed derivation. These solutions reveal rogue waves on a soliton background. Moreover, the main characteristics of the solutions are discussed with some graphics. Our results would be of much importance in predicting and enriching rogue wave phenomena in nonlinear wave fields.     

Rogue waves in the massive Thirring model

In this paper, general rogue wave solutions in the massive Thirring (MT) model are derived by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method and these rational solutions are presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index- and two dimension-ones are proved to be consistent by only one constraint relation on parameters of tau-functions of the KP-Toda hierarchy. It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled integrable systems, the MT model only admits the rogue waves of bright-type, and the higher order rogue waves represent the superposition of fundamental ones in which the nonreducible parameters determine the arrangement patterns of fundamental rogue waves. Particularly, the super rogue wave at each order can be achieved simply by setting all internal parameters to be zero, resulting in the amplitude of the sole huge peak of order N being $2N+1$ times the background. Finally, rogue wave patterns are discussed when one of the internal parameters is large. Similar to other integrable equations, the patterns are shown to be associated with the root structures of the Yablonskii–Vorob'ev polynomial hierarchy through a linear transformation.     

Darboux transformation and Rogue waves of the Kundu–nonlinear Schrödinger equation

In this paper, the Darboux transformation of the Kundu–nonlinear Schrödinger equation is derived and generalized to the matrix of n‐fold Darboux transformation. From known solution Q, the determinant representation of n‐th new solutions of Q^[n] are obtained by the n‐fold Darboux transformation. Then soliton solutions and positon solutions are generated from trivial seed solutions, breather solutions and rogue wave solutions that are obtained from periodic seed solutions. After that, the higher order rogue wave solutions of the Kundu–nonlinear Schrödinger equation are given. We show that free parameters in eigenfunctions can adjust the patterns of the higher order rogue waves. Meanwhile, the third‐order rogue waves are given explicitly. Copyright © 2014 John Wiley & Sons, Ltd.     

Novel high-order breathers and rogue waves in the Boussinesq equation via determinants

By virtue of the bilinear method and the Kadomtsev–Petviashvili (KP) hierarchy reduction technique, wider classes of high-order breather and semirational and rogue wave solutions to the Boussinesq equation are derived. These solutions are presented explicitly in terms of Gram determinants, whose matrix elements have simply algebraic expressions. The breather and rogue wave solutions are derived from two different types of tau functions of a bilinear equation in the single-component KP hierarchy. By taking a long wave limit of high-order breather solutions, a range of hybrid solutions consisting of solitons, breathers, and one fundamental rogue wave are generated. For the rational rogue waves, some typical patterns such as Peregrine-type, triple, and sextuple rogue waves are put forward by modifying the input parameters. Besides, a new rogue wave pattern of third-order rogue waves is found, which features a mixture of a triangular pattern of three fundamental rogue waves and a fundamental pattern of second-order rogue wave. These results may help understand the protean rogue wave manifestations in areas ranging from water waves to fluid dynamics.     

Multi-breather and high-order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

We construct the multi-breather solutions of the focusing nonlinear Schrödinger 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 nontrivial 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 multibreather solutions with different velocities in the limit , 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.     

Rogue wave patterns associated with Okamoto polynomial hierarchies

We show that new types of rogue wave patterns exist in integrable systems, and these rogue patterns are described by root structures of Okamoto polynomial hierarchies. These rogue patterns arise when the τ functions of rogue wave solutions are determinants of Schur polynomials with index jumps of three, and an internal free parameter in these rogue waves gets large. We demonstrate these new rogue patterns in the Manakov system and the three-wave resonant interaction system. For each system, we derive asymptotic predictions of its rogue patterns under a large internal parameter through Okamoto polynomial hierarchies. Unlike the previously reported rogue patterns associated with the Yablonskii–Vorob'ev hierarchy, a new feature in the present rogue patterns is that the mapping from the root structure of Okamoto-hierarchy polynomials to the shape of the rogue pattern is linear only to the leading order, but becomes nonlinear to the next order. As a consequence, the current rogue patterns are often deformed, sometimes strongly deformed, from Okamoto-hierarchy root structures, unless the underlying internal parameter is very large. Our analytical predictions of rogue patterns are compared to true solutions, and excellent agreement is observed, even when rogue patterns are strongly deformed from Okamoto-hierarchy root structures.     

Rogue waves in baroclinic flows

We investigate an AB system, which can be used to describe marginally unstable baroclinic wave packets in a geophysical fluid. Using the generalized Darboux transformation, we obtain higher-order rogue wave solutions and analyze rogue wave propagation and interaction. We obtain bright rogue waves with one and two peaks. For the wave packet amplitude and the mean-flow correction resulting from the self-rectification of the nonlinear wave, the positions and values of the wave crests and troughs are expressed in terms of a parameter describing the state of the basic flow, in terms of a parameter responsible for the interaction of the wave packet and the mean flow, and in terms of the group velocity. We show that the interaction of the wave packet and mean flow and also the group velocity affect the propagation and interaction of the amplitude of the wave packet and the self-rectification of the nonlinear wave.     

On rogue wave in the Kundu-DNLS equation

In this paper, the determinant representation of the n-fold Darboux transformation (DT) of the Kundu-DNLS equation is given. Based on our analysis, the soliton solutions, positon solutions and breather solutions of the Kundu-DNLS equation are given explicitly. Further, we also construct the rogue wave solutions which are given by using the Taylor expansion of the breather solution. Particularly, these rogue wave solutions possess several free parameters. With the help of these parameters, these rogue waves constitute several patterns, such as fundamental pattern, triangular pattern, circular pattern.     

Novel rogue waves in an inhomogenous nonlinear medium with external potentials

Employing the similarity transformation connected with the standard constant coefficient nonlinear Schrödinger equation, we obtain the analytical rogue wave solutions to a generalized variable coefficient nonlinear Schrödinger equation with external potentials describing the pulse propagation in nonlinear media with transverse and longitudinal directions nonuniformly distributed. Based on the obtained solutions, abundant structures of rogue waves are constructed by selecting some special parameters. The main properties as well as the dynamic behaviors of these rogue waves are discussed by direct computer simulations.     

Doubly localized two-dimensional rogue waves generated by resonant collision in Maccari system

In this paper, the Maccari system is investigated, which is viewed as a two-dimensional extension of nonlinear Schrödinger equation. We derive doubly localized two-dimensional rogue waves on the dark solitons of the Maccari system with Kadomtsev–Petviashvili hierarchy reduction method. The two-dimensional rogue waves include line segment rogue waves and rogue-lump waves, which are localized in two-dimensional space and time. These rogue waves are generated by the resonant collision of rational solitary waves and dark solitons, the whole process of transforming elastic collision into resonant collision is analytically studied. Furthermore, we also discuss the local characteristics and asymptotic properties of these rogue waves. Simultaneously, the generating conditions of the line segment rogue wave and rogue-lump wave are also given, which provides the possibility to predict rogue wave. Finally, a new way to obtain the high-order rogue waves of the nonlinear Schrödinger equation are given by proper reduction from the semi-rational solutions of the Maccari system.     

The realization of controllable three dimensional rogue waves in nonlinear inhomogeneous system

We obtain self-similar first-order and second-order rogue wave solutions for the (3+1)-dimensional inhomogeneous nonlinear Schrödinger equation. Based on these solutions, we investigate the control and manipulation of rogue waves in the dispersion decreasing fibers with Logarithmic profile and Gaussian profile. Our results indicate that the propagation behaviors of rogue waves, such as fast excitation, sustainment and restraint, can be manipulated by modulating the relation between the maximum value of the effective propagation distance Z_m and the parameter Z₀ relating to the excited types of rogue wave. The comparison of the propagation behavior of rogue wave in the dispersion decreasing waveguides with Logarithmic profile, Gaussian profile and hyperbolic profile is also given.     

Multiple rogue wave solutions for a (3+1)-dimensional Hirota bilinear equation

In this paper, we considered the multiple rogue wave solutions of a (3+1)-dimensional Hirota bilinear equation by using a symbolic computation approach. Based on the bilinear form of this equation, the first-order rogue waves, the second-order rogue waves and the third-order rogue waves are presented. Moreover, some basic properties of multiple rogue waves and their collision structures are explained by drawing the three dimensional plot.     

Analytic rogue wave solutions for a generalized fourth‐order Boussinesq equation in fluid mechanics

A bilinear transformation method is proposed to find the rogue wave solutions for a generalized fourth‐order Boussinesq equation, which describes the wave motion in fluid mechanics. The one‐ and two‐order rogue wave solutions are explicitly constructed via choosing polynomial functions in the bilinear form of the equation. The existence conditions for these solutions are also derived. Furthermore, the system parameter controls on the rogue waves are discussed. The three parameters involved in the equation can strongly impact the wave shapes, amplitudes, and distances between the wave peaks. The results can be used to deeply understand the nonlinear dynamical behaviors of the rogue waves in fluid mechanics.     

Bounded traveling waves of the oceanic currents motion equations

The bifurcation methods of differential equations are employed to investigate traveling waves of the oceanic currents motion equations. The sufficient conditions to guarantee the existence of different kinds of bounded traveling wave solutions are rigorously determined. Further, due to the existence of a singular line in the corresponding traveling wave system, the smooth periodic traveling wave solutions gradually lose their smoothness and evolve to periodic cusp waves. The results of numerical simulation accord with theoretical analysis.     

Bi-solitons, breather solution family and rogue waves for the (2+1)-dimensional nonlinear Schr\"{o}dinger equation

In this paper, bi-solitons, breather solution family and rogue waves for the (2+1)-Dimensional nonlinear Schr\"{o}dinger equations are obtained by using Exp-function method. These solutions derived from one unified formula which is solution of the standard (1+1) dimension nonlinear Schr\"{o}dinger equation. Further, based on the solution obtained by other authors, higher-order rational rogue wave solution are obtained by using the similarity transformation. These results greatly enriched the diversity of wave structures for the (2+1)-dimensional nonlinear Schr\"{o}dinger equations     

扩展的(G'/G)展开法求量子Zakharov-Kuznetsov方程的精确解

为得到量子Zakharov-Kuznetsov方程的一些新精确解,借助行波解的思想,结合齐次平衡原理和一类非线性常微分方程解的结构,利用扩展的(G'/G)展开方法,研究了其相应的更加丰富的精确解表达形式.新精确解的表达式主要由双曲函数、三角函数和有理数函数构成,出现了某些怪波解的情形.通过对比不同情况下解的形式,利用M...     

Stokes waves with constant vorticity: I. Numerical computation

Periodic traveling waves are numerically computed in a constant vorticity flow subject to the force of gravity. The Stokes wave problem is formulated via a conformal mapping as a nonlinear pseudodifferential equation, involving a periodic Hilbert transform for a strip, and solved by the Newton‐GMRES method. For strong positive vorticity, in the finite or infinite depth, overhanging profiles are found as the amplitude increases and tend to a touching wave, whose surface contacts itself at the trough line, enclosing an air bubble; numerical solutions become unphysical as the amplitude increases further and make a gap in the wave speed versus amplitude plane; another touching wave takes over and physical solutions follow along the fold in the wave speed versus amplitude plane until they ultimately tend to an extreme wave, which exhibits a corner at the crest. Touching waves connected to zero amplitude are found to approach the limiting Crapper wave as the strength of positive vorticity increases unboundedly, while touching waves connected to the extreme waves approach the rigid body rotation of a fluid disk.     

Instability of nonlinear dispersive solitary waves

We consider linear instability of solitary waves of several classes of dispersive long wave models. They include generalizations of KDV, BBM, regularized Boussinesq equations, with general dispersive operators and nonlinear terms. We obtain criteria for the existence of exponentially growing solutions to the linearized problem. The novelty is that we dealt with models with nonlocal dispersive terms, for which the spectra problem is out of reach by the Evans function technique. For the proof, we reduce the linearized problem to study a family of nonlocal operators, which are closely related to properties of solitary waves. A continuation argument with a moving kernel formula is used to find the instability criteria. These techniques have also been extended to study instability of periodic waves and of the full water wave problem.     

Solitary wave solutions to the Isobe-Kakinuma model for water waves

We consider the Isobe-Kakinuma model for two-dimensional water waves in the case of a flat bottom. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for a Lagrangian approximating Luke's Lagrangian for water waves. We show theoretically the existence of a family of small amplitude solitary wave solutions to the Isobe-Kakinuma model in the long wave regime. Numerical analysis for large amplitude solitary wave solutions is also provided and suggests the existence of a solitary wave of extreme form with a sharp crest.     

Bound-state soliton and rogue wave solutions for the sixth-order nonlinear Schrödinger equation via inverse scattering transform method

In this work, inverse scattering transform for the sixth-order nonlinear Schrödinger equation with both zero and nonzero boundary conditions at infinity is given, respectively. For the case of zero boundary conditions, in terms of the Laurent's series and generalization of the residue theorem, the bound-state soliton is derived. For nonzero boundary conditions, using the robust inverse scattering transform, we present a matrix Riemann–Hilbert problem of the sixth-order nonlinear Schrödinger equation. Then, based on the obtained Riemann–Hilbert problem, the rogue wave solutions are derived through a modified Darboux transformation. Besides, according to some appropriate parameters choices, several graphical analysis are provided to discuss the dynamical behaviors of the rogue wave solutions and analyze how the higher-order terms affect the rogue wave.