Order Rogue Wave Solutions Research Articles

Rogue wave solutions for the [formula omitted]-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili equation

In this paper, the 3+1-dimensional generalized Camassa–Holm–Kadomtsev–Petviashvili equation is investigated by Hirota bilinear method, from which the first, second and third order rogue wave solutions are derived. The relationships between the parameters and the amplitude, width and center of rogue wave are discussed. At the same time, several evolution and density plots are present to exhibit the dynamical structure of the rogue wave solutions. The results may be useful in nonlinear optics and other physically relevant integrable systems.

Dynamics of general higher-order rogue waves in the two-component long wave–short wave model of Newell type

An integrable two component long-wave–short-wave (2-LS) model of Newell type governing the resonant interaction between two capillary waves with a common gravity wave is considered. The general higher order rogue wave (RW) solutions of this system are obtained by employing the bilinear KP hierarchy reduction method. Our study explores three different families of RWs depending upon the root structure of an algebraic equation characterizing them. Those are the Nth-order bounded rogue waves, mixed bounded (N1,N2)th-order rogue waves and degradable bounded (N¯1,N¯2)th-order rogue waves, respectively, where N,N1,N2 are arbitrary positive integer and N¯1,N¯2 are arbitrary non-negative integers. It is interesting to note that the fundamental rogue wave in one short wave (SW) component can attain a peak amplitude as high as more than six times the background amplitude while it is three times the background amplitude in the other SW component. We identify that the Nth-order bounded rogue waves are composed of N(N+1)2 fundamental rogue waves, the mixed bounded (N1,N2)th-order rogue waves comprise N1th-order bounded rogue waves mixing with another type N2th-order bounded rogue waves existing in different state, and the degradable bounded (N¯1,N¯2)th-order rogue waves consist of N¯12+N¯22−N¯1(N¯2−1) fundamental rogue waves. Non-trivial super rouge waves are reported.

Rogue waves dynamics of cubic–quintic nonlinear Schrödinger equation with an external linear potential through a modified Noguchi electrical transmission network

In this paper, we have considered a modified Noguchi nonlinear transmission network with a dispersive element. Using the reductive perturbation method in the semi-discrete limit, we show how wave dynamics in the network can be governed by a cubic–quintic nonlinear Schrödinger equation (CQ-NLSE) with an external linear potential. Through the modified Darboux transformation, we constructed the first, second and third order rogue wave solutions. Although the linear potential function Γ(τ) does not influence the instability criterion of waves propagating in the network, it has a significant effect on the dynamics of the amplitude, phase and shirp of rogue waves. Our results also reveal that it is possible both to predict the position where the amplitude of rogue waves reaches its maximum value and to reduce their spatial width.

General higher—order breathers and rogue waves in the two‐component long‐wave–short‐wave resonance interaction model

AbstractGeneral higher order breather and rogue wave (RW) solutions to the two‐component long wave–short wave resonance interaction (2‐LSRI) model are derived via the bilinear Kadomtsev–Petviashvili hierarchy reduction method and are given in terms of determinants. Under particular parametric conditions, the breather solutions can reduce to homoclinic orbits, or a mixture of breathers and homoclinic orbits. There are three families of RW solutions, which correspond to a simple root, two simple roots, and a double root of an algebraic equation related to the dimension reduction procedure. The first family of RW solutions consists of bounded fundamental RWs, the second family is composed of bounded fundamental RWs coexisting with another fundamental RWs of different a bounded state ( being positive integers), while the third one has fundamental bounded RWs ( being nonnegative integers). The second family can be regarded as the superpositions of the first family, while the third family can be the degenerate case of the first family under particular parameter choices. These diverse RW patterns are illustrated graphically.

General rogue wave solution to the discrete nonlinear Schrödinger equation

In the present paper, we are concerned with the rogue wave solution to an integrable discrete nonlinear Schrödinger (NLS) equation. First, through the KP–Toda reduction method, the general breather solution of the discrete NLS equation is deduced from two bilinear equations in discrete two-dimensional Toda-lattice hierarchy. Then, by taking an ingenious and successive limit to the breather solution, we derive the higher order rogue wave solution to the discrete NLS equation.

General breather and rogue wave solutions to the complex short pulse equation

In the present paper, we attempt to construct both the breather and rogue wave solutions to the focusing complex short pulse (fCSP) equation via the KP–Toda reduction method. Following a procedure of reducing the bilinear equations satisfied by tau functions of Kadomtsev–Petviashvili (KP)–Toda hierarchy to the ones of the fCSP equation with nonzero boundary condition, we first deduce the general breather solution of the fCSP equation starting from a specially arranged tau-function of the KP–Toda hierarchy, then we construct and prove the Nth order rogue wave solutions of the fCSP equation and express them in two different but equivalent forms of determinants. The dynamical behaviors of both the breather and rogue wave solutions are illustrated and analyzed.

Multiple-order breathers for a generalized (3+1)-dimensional Kadomtsev–Petviashvili Benjamin–Bona–Mahony equation near the offshore structure

In this paper, a generalized (3+1)-dimensional Kadomtsev–Petviashvili Benjamin–Bona–Mahony equation which describes the fluid flow in the case of an offshore structure, is investigated. Here, making use of the bilinear form and symbolic computation, we construct four kinds of rogue wave solutions consisting of independent breathers. Among these solutions, the fourth order rogue wave solution is rarely considered in nonlinear system. Exact locations of the highest and lowest peaks as well as the extreme values of the wave heights are systematically analyzed. The obtained rogue waves observe certain “circularity structure”, the highest or lowest peaks both sit at the same circular. Moreover, we show that the rogue waves are stable during the propagation.

The robust inverse scattering method for focusing Ablowitz–Ladik equation on the non-vanishing background

In this paper, we consider the robust inverse scattering method for the Ablowitz–Ladik (AL) equation on the non-vanishing background, which can be used to deal with arbitrary-order poles on the branch points and spectral singularities in a unified way. The Darboux matrix is constructed with the aid of loop group method and considered within the framework of robust inverse scattering transform. Various soliton solutions are constructed without using the limit technique. These solutions include general soliton, breathers, as well as high order rogue wave solutions.

Rogue Wave Solutions of the (3+1)-Dimensional Generalized Kadomtsev-Petviashvili Equation

The rogue wave solutions of the (3+1)-dimensional generalized Kadomtsev-Petviashvili (gKP) equation are constructed by Hirota bilinear method and symbolic computation approach. By choosing proper polynomial function, the first, the second and the third order rogue wave solutions are systematically obtained. The maximum amplitude and the minimum amplitude of the first order rogue wave solution are given. Moreover, the first three order generalised rogue wave solutions are also explicitly presented. Finally, some features of rogue wave solutions are graphically discussed.

Intricate dynamics of rogue waves governed by the Sasa–Satsuma equation

We present the second order rogue wave solutions of the Sasa–Satsuma evolution equation that have significantly more complicated structure than in the case of the nonlinear Schrödinger equation. This equation takes into account the third order dispersion, self-frequency shift and self-steepening effects, thus making it closer to realistic physical applications. These effects are important both for water waves and pulses in optical fibres. Although the Sasa–Satsuma equation is integrable, the degree of complexity of these new solutions has prevented them being found previously. Here, we present in explicit form the second-order rogue wave solutions of the Sasa–Satsuma equation for three free parameter values of the family of rogue wave solutions. Special choices of the free parameters allow us to present these solutions in the simplest possible way and to illustrate them graphically.

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

AbstractWe 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.

Modulational instability and higher order-rogue wave solutions for the generalized discrete Hirota equation

This paper investigates the modulational instability and higher order-rogue waves in the generalized discrete Hirota system. The Lax pair and conservation laws for this system are constructed. Then the existent conditions for its modulational instability to form the rogue waves are given starting from the plane wave solution. Furthermore, the higher-order discrete rogue waves of this system are reported using the novel discrete version of generalized perturbation (n,N−n)-fold Darboux transformation. Finally, the dynamical behaviors of the strong and weak interactions of these higher-order discrete rogue waves are discussed analytically and numerically, which exhibits abundant nonlinear wave structures. These results may be useful for understanding some physical phenomena in optical fibers and relevant fields.

A symbolic computation approach to constructing rogue waves with a controllable center in the nonlinear systems

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.

Rogue waves in exciton–polariton condensates under homogeneous pumping

We predict the emergence of rogue wave solutions in one-dimensional exciton–polariton condensates under homogeneous pumping. We model the condensate dynamics in a microwire using the dissipative Gross–Pitaevskii equation for the polariton field, with considers attractive nonlinearity, coupled to the rate equation of the excitonic reservoir density. With the help of the direct ansatz method and similarity transformation, deformed first order rogue wave solutions are constructed and its dynamics analyzed. We show that the deformed rogue wave has a curved background controlled by the pump power and the strength of the nonlinear interaction of polaritons. Moreover, the maximal population of the polaritons appears where high energy of rogue wave is concentrated.

Rogue wave solutions for the higher-order nonlinear Schrödinger equation with variable coefficients by generalized Darboux transformation

In this paper, we study a higher-order variable coefficient nonlinear Schrödinger (NLS) equation, which plays an important role in the control of the ultrashort optical pulse propagation in nonlinear optical systems. Then, we construct a generalized Darboux transformation (GDT) for the higher-order variable coefficient NLS equation. The [Formula: see text]th order rogue wave solution is obtained by the iterative rule and it can be expressed by the determinant form. As application, we calculate rogue waves (RWs) from first- to fourth-order in accordance with different kinds of parameters. In particular, the dynamical properties and spatial-temporal structures of RWs are discussed and compared with Hirota equation through some figures.

Multi-soliton, multi-breather and higher order rogue wave solutions to the complex short pulse equation

In the present paper, we are concerned with the general analytic solutions to the complex short pulse (CSP) equation including soliton, breather and rogue wave solutions. With the aid of a generalized Darboux transformation, we construct the N-bright soliton solution in a compact determinant form, the N-breather solution including the Akhmediev breather and a general higher order rogue wave solution. The first and second order rogue wave solutions are given explicitly and analyzed. The asymptotic analysis is performed rigorously for both the N-soliton and the N-breather solutions. All three forms of the analytical solutions admit either smoothed-, cusped- or looped-type ones for the CSP equation depending on the parameters. It is noted that, due to the reciprocal (hodograph) transformation, the rogue wave solution to the CSP equation can be a smoothed, cusponed or a looped one, which is different from the rogue wave solution found so far.

The Darboux transformation of the Kundu–Eckhaus equation

We construct an analytical and explicit representation of the Darboux transformation (DT) for the Kundu–Eckhaus (KE) equation. Such solution andn-fold DTTnare given in terms of determinants whose entries are expressed by the initial eigenfunctions and ‘seed’ solutions. Furthermore, the formulae for the higher order rogue wave (RW) solutions of the KE equation are also obtained by using the Taylor expansion with the use of degenerate eigenvaluesλ2k−1→λ1=−12a+βc2+ic,k=1,2,3,…, all these parameters will be defined latter. These solutions have a parameterβ, which denotes the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effects. We apply the contour line method to obtain analytical formulae of the length and width for the first-order RW solution of the KE equation, and then use it to study the impact of theβon the RW solution. We observe two interesting results on localization characters ofβ, such that ifβis increasing froma/2: (i) the length of the RW solution is increasing as well, but the width is decreasing; (ii) there exist a significant rotation of the RW along the clockwise direction. We also observe the oppositely varying trend ifβis increasing toa/2. We define an area of the RW solution and find that this area associated withc=1 is invariant whenaandβare changing.

Localized Properties of Rogue Wave for a Higher-Order Nonlinear Schrödinger Equation* *Supported by the National Natural Science Foundation of China under Grant No. 11271210, the K.C. Wong Magna Fund in Ningbo University

In this paper, we provide determinant representation of the n-th order rogue wave solutions for a higher-order nonlinear Schrödinger equation (HONLS) by the Darboux transformation and confirm the decomposition rule of the rogue wave solutions up to fourth-order. These solutions have two parameters α and β which denote the contribution of the higher-order terms (dispersions and nonlinear effects) included in the HONLS equation. Two localized properties, i.e., length and width of the first-order rogue wave solution are expressed by above two parameters, which show analytically a remarkable influence of higher-order terms on the rogue wave. Moreover, profiles of the higher-order rogue wave solutions demonstrate graphically a strong compression effect along t-direction given by higher-order terms.

Higher order rogue wave solutions of general coupled nonlinear Schrödinger equations

We present first, second, and third order rogue wave (RW) solutions of general two coupled nonlinear Schrödinger equations. We construct higher order RW solutions through a modified Darboux transformation method. In the first iteration we construct a first order RW solution. The solution obtained in the first iteration (first order RW) has been considered as the seed solution to generate the second iterated solution (second order RW). The latter has been given as the seed solution to generate the third iterated solution (third order RW). We also briefly discuss the characteristics of these RWs.

Rogue Wave Solutions for Nonlinear Schrödinger Equation with Variable Coefficients in Nonlinear Optical Systems

In this paper, the generalized Darboux transformation is constructed to variable coefficient nonlinear Schrödinger (NLS) equation. The N-th order rogue wave solution of this variable coefficient NLS equation is obtained by determinant expression form. In particular, we present rogue waves from first to third-order through some figures and analyze their dynamics.