Second-order Breather Research Articles

A Kundu-nonlinear Schrödinger equation: Rogue waves, breathers, and mixed interaction solutions.

A Kundu-nonlinear Schrödinger equation that can be utilized to simulate the pulse propagation in optical fibers is researched in this paper. First, the Lax integrability of the above equation is proved and its modulational instability (i.e., the main mechanism for producing the rogue wave solutions and the breather solutions) is calculated. Subsequently, using the generalized perturbation (n,N-n)-fold Darboux transformation, the rogue waves, breathers, and mixed interaction solutions are acquired, as well as the impact of various parameters on the solutions is examined. In particular, when we assume that the coefficient of the equation is θ=mx+dt, some new wave structures are found based on parameter variations, such as the rotational separation of first-order rogue waves, scale-like structures generated by second-order breathers, etc., which offer novel ideas for producing different signals via optical fibers. Ultimately, the classification numbers of mixed solutions of rogue waves and breathers are provided, which can better observe how the two types of the local waves are combined, in accordance with the distributions of the increasing numbers of algebraic equations.

Breathers, rogue waves, and interaction solutions for the variable coefficient Kundu-nonlinear Schrödinger equation

With the inhomogeneity of optical fiber media taken into account, under investigation in this paper is the variable coefficient Kundu-nonlinear Schrödinger equation, which describes the pulses propagation in optical fibers. Based on Lax pair, the Nth-order Darboux transformation is constructed. Depending on plane wave solution, the first- and second-order breather solutions are derived and the interactions between breathers are graphically analyzed. The Kuznetsov–Ma breather, Akhmediev breather, and spatial-temporal breather have been obtained. Moreover, the first-, second-, and third-order rogue wave solutions have been constructed. The usual rogue waves and first- and second-order line rogue waves are observed. The weak and strong interactions between the first-, second-order rogue waves, and spatial-temporal period breather are studied. Furthermore, variable coefficient δ(t) causes rogue waves to produce some interesting evolutionary phenomena, which have been systematically analyzed. In addition, the influences of parameters for the properties of solutions are discussed.

Partially nonlocal ring-like spatiotemporal superimposed second-order breathers under a harmonic potential

By means of reduction expression with solutions of constant-coefficient nonlinear Schrödinger system, analytical partially nonlocal ring-like spatiotemporal superimposed second-order Akhmediev and Ma breather solution is derived from the Darboux approach. In xyt or xyz coordinate, the cylindrical multilayer structure of Akhmediev and Ma breathers is embedded in the center, and rings of Akhmediev breather and plane of Ma breather extend out. Moreover, the influence of the radius and thickness parameters is studied. The Hermite parameter has impact on the layer structure of ring-like superimposed second-order breather in the z direction.

Solitonic rogue waves induced by the modulation instability in a split-ring-resonator-based left-handed coplanar waveguide

Modulation instability and nonlinear excitation modes in lower and upper frequency bands are investigated in the nonlinear left-handed coplanar waveguide, where nonlinear capacitance (voltage-dependent) is used. The multi-scale method is employed to derive a system of two coupled nonlinear Schödinger equations governing the evolution of the kink and bright solitons of these modes. A modulation instability is investigated to show the effects of the left-handed and right-handed influences on the modulation instability gain and the intensity of the plane wave. In the lower and upper frequency modes, the breathing structures have strong values for the left-handed component. At the end, the numerical simulation of the continuous wave is used to show the development of the modulated waves and localized objects. Another relevant localized structure has been obtained to display the second-order breathers and Akhmediev breathers of types A and B under the variation of the perturbed wave number. Both linear inductances and perturbed wave numbers are tools for controlling the amplitude of the localized structures, and modulation instability is sensitive to these parameters.

Resonance Y-type soliton and hybrid solutions for a (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Konopelchenko-Schiff system in a fluid or plasma

In this paper, a (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Konopelchenko-Schiff system in a fluid or plasma is investigated. Via the Hirota method and symbolic computation, we work out some two-resonance Y-type soliton solutions as well as some hybrid solutions composed of the two-resonance Y-type solitons and solitons/breathers. Graphically, we display some two-resonance Y-type solitons. We present the interactions between the two-resonance Y-type soliton and one soliton, among the two-resonance Y-type soliton and two solitons, between the two-resonance Y-type soliton and first-order breather as well as among the two-resonance Y-type soliton and second-order breathers.

Pfaffian, breather, and hybrid solutions for a (2 + 1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics

Fluid mechanics is seen as the study on the underlying mechanisms of liquids, gases and plasmas, and the forces on them. In this paper, we investigate a (2 + 1)-dimensional generalized nonlinear system in fluid mechanics and plasma physics. By virtue of the Pfaffian technique, the Nth-order Pfaffian solutions are derived and proved, where N is a positive integer. Based on the Nth-order Pfaffian solutions, the first- and second-order breather solutions are obtained. In addition, Y-type and X-type breather solutions are constructed. Furthermore, we investigate the influence of the coefficients in the system on those breathers as follows: The locations and periods of those breathers are related to δ1, δ2, δ3, δ4, and δ5, where δc's (c=1,2,3,4,5) are the constant coefficients in the system. Moreover, hybrid solutions composed of the breathers and solitons are derived. Interactions between the Y/X-type breather and Y-type soliton are illustrated graphically, respectively. Then, we show the influence of the coefficients in the system on the interactions between the Y/X-type breather and Y-type soliton.

Bilinear form and nonlinear waves of a (1+1)-dimensional generalized Boussinesq equation for the gravity waves over water surface

Gravity waves are studied in, e.g., the oceanic and atmospheric sciences. In this paper, for the head-on interaction of oblique gravity wave profiles over the water surface, we investigate a (1+1)-dimensional generalized Boussinesq equation. Bilinear form is derived, and the higher-order rogue waves, breather and hybrid solutions in the determinant form are constructed via the Hirota bilinear method and Kadomtsev–Petviashvili hierarchy reduction. Influence of α, β and γ on the solutions is discussed, where α, β and γ are the nonzero coefficients in that equation. For the first-order rogue-wave solutions, we find that the wave widths and amplitudes are related to α and β, and the locations of the extreme points depend on α, β and γ. Structures of the higher-order rogue waves are also given. We construct three types of the first-order breathers, and the second-order breathers are obtained through the combinations of the first-order ones. α, β and γ are found to influence the amplitudes and locations of the first-order breathers. Hybrid solutions comprising the one rogue wave, the higher-order breathers and solitons are discussed graphically. Those results reveal the nonlinear properties of oblique gravity waves over the water surface, which can help the study on nonlinear waves in fluid mechanics, plasma physics and fiber optics.

Hybrid solutions for the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics

In fluid mechanics, the higher-dimensional and higher-order equations are constructed to describe the propagations of nonlinear waves. In this paper, we investigate the (2+1)-dimensional variable-coefficient Caudrey-Dodd-Gibbon-Kotera-Sawada equation in fluid mechanics. The Nth-order Pfaffian solutions are constructed and proved via the modified Pfaffian technique. The higher-order soliton, first- and second-order breather solutions are constructed based on the Nth-order Pfaffian solutions. We graphically demonstrate that the amplitudes and velocities of the solitons are affected by some variable coefficients. Hybrid solutions composed of breathers, lumps and solitons are illustrated graphically. It can be found that when certain parameters are chosen, the breathers, lumps and solitons included in the hybrid solutions possess the same properties as those of the breather and lump solutions.

Bilinear form, soliton, breather, hybrid and periodic-wave solutions for a (3+1)-dimensional Korteweg–de Vries equation in a fluid

Fluids are studied in such disciplines as atmospheric science, oceanography and astrophysics. In this paper, we investigate a (3+1)-dimensional Korteweg–de Vries equation in a fluid. Bilinear form and N-soliton solutions are obtained, where N is a positive integer. Via the N-soliton solutions, we derive the higher-order breather solutions. We observe the interaction between the two perpendicular first-order breathers on the $$x-y$$ and $$x-z$$ planes and the interaction between the periodic line wave and the first-order breather on the $$y-z$$ plane, where x, y and z are the independent variables in the equation. We discuss the effects of $$\alpha $$ , $$\beta $$ , $$\gamma $$ and $$\delta $$ on the amplitude of the second-order breather, where $$\alpha $$ , $$\beta $$ , $$\gamma $$ and $$\delta $$ are the constant coefficients in the equation: Amplitude of the second-order breather decreases as $$\alpha $$ increases; amplitude of the second-order breather increases as $$\beta $$ increases; amplitude of the second-order breather keeps invariant as $$\gamma $$ or $$\delta $$ increases. Via the N-soliton solutions, hybrid solutions comprising the breathers and solitons are derived. Based on the Riemann theta function, we obtain the periodic-wave solutions, and find that the periodic-wave solutions approach to the one-soliton solutions under a limiting condition.

Darboux dressing transformation and superregular breathers for a coupled nonlinear Schrödinger system with the negative coherent coupling in a weakly birefringent fibre

For a coupled nonlinear Schrödinger system with the negative coherent coupling, which describes two orthogonally polarized pulses in a weakly birefringent fibre, we construct the Darboux dressing transformation and the N-th-order breather solutions with N as a positive integer. When the retarded time tends to be infinite, limits of the ratios between the N-th-order breather solutions and seed solutions are obtained. For the first-order breathers, we present the condition to distinguish the degenerate and nondegenerate cases. For the nondegenerate breathers, we analyse whether the breathers could be kink-type based on the above limits. For the second-order breathers, superregular breathers (SRBs) are derived, where each SRB could consist of two (1) kink-type, (2) single-hump or (3) double-hump quasi-Akhmediev breathers (quasi-ABs). Before and after the interaction, profiles of two quasi-ABs change for Case (1) but keep unchange for Case (2) or (3).

Breather solutions of the nonlocal nonlinear self-focusing Schrödinger equation

The first- and second-order breather solutions of the self-focusing nonlocal nonlinear Schrödinger (NNLS) equation are obtained by employing Hirota's bilinear method. The NNSE also happens to be an example of Schrödinger equation with parity-time (PT) symmetry. With the help of recurrence relations in the Hirota bilinear form, the nth-order breather solutions on the nonzero background of the NNLS equation are obtained, and the collision, superposition and separation of transmission modes is studied respectively. When the parameters describing these breathers are selected as some special values, they display plentiful spatial structures which provide effective methods for controlling the localized optical waves in nonlocal nonlinear media.

Multi-soliton solutions for a higher-order coupled nonlinear Schrödinger system in an optical fiber via Riemann–Hilbert approach

The main objective of this work is to study the multi-soliton solutions for the higher-order coupled nonlinear Schrodinger system in an optical fiber. Firstly, by using the Riemann–Hilbert (RH) approach as well as analyzing the related spectral problem of the Lax pair, a matrix RH problem for the system is strictly formulated. Secondly, a series of multi-soliton solutions including breathers and interaction solutions can be computed from the RH problem with the reflectionless case. Thirdly, the propagation and collision dynamic behaviors as well as localized wave characteristics of these solutions are presented by selecting appropriate parameters with some graphics. The innovation and highlights of this article are shown through obtained interesting results. The one is that the higher-order linear and nonlinear term $$\varepsilon $$ has important impact on the velocity, phase, period, and wavewidth of wave dynamics. The other is that collisions for the second-order breathers and soliton solutions are elastic interaction which imply they remain bounded all the time. Nevertheless, third-order breathers and soliton solutions are inelastic interaction and the amplitude decreases rapidly with time when collisions occur.

The higher-order lump, breather and hybrid solutions for the generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation in fluid mechanics

Under investigation in this paper is the (2 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation, which can be utilized to describe certain nonlinear phenomena in fluid mechanics. We obtain the higher-order lump, breather and hybrid solutions, and analyze the effects of the constant coefficients $$h_1$$ , $$h_2$$ , $$h_4$$ and $$h_5$$ in that equation on those solutions, since the higher-order lump solutions are generalized via the long-wave limit method, and since the higher-order breather solutions and hybrid solutions composed of the solitons, breathers and lumps are derived. With the help of the analytic and graphic analysis, we get the following: (1) amplitudes of the humps and valleys of the first-order lumps are related to $$h_1$$ , $$h_2$$ , $$h_4$$ and $$h_5$$ , proportional to $$h_4$$ while inversely proportional to $$h_2$$ . Velocities of the first-order lumps are proportional to $$h_4$$ . The second-order lumps describe the interaction between the two first-order lumps, which is elastic since those lumps keep their shapes, velocities and amplitudes unchanged after the interaction. Effects of $$h_2$$ and $$h_4$$ on the second-order lumps are graphically illustrated. (2) Amplitudes of the first-order breathers are proportional to $$h_2$$ . Interaction between the breather waves is graphically presented. Effects of $$h_2$$ and $$h_1$$ on the amplitudes and shapes of the second-order breathers are graphically discussed. (3) Elastic interactions are graphically illustrated, between the first-order breathers and one solitons, the first-order lumps and one solitons, as well as the first-order breathers and first-order lumps. Also graphically illustrated, amplitudes of all those three kinds of hybrid solutions are inversely proportional to $$h_2$$ , and velocity of the one soliton is positively correlated to $$h_4$$ .

A generalized complex mKdV equation: Darboux transformations and explicit solutions

A new generalized complex modified Korteweg–de Vries equation associated with a 3 × 3 matrix spectral problem is proposed by resorting to the zero-curvature equation. Based on the gauge transformations between the Lax pairs, a Darboux transformation for the generalized complex modified Korteweg–de Vries equation is constructed, from which the corresponding N-fold Darboux transformations are derived in terms of both iterative technique and determinants. As an application of the resulting Darboux transformations, explicit solutions, like one-soliton, two-soliton, and three-soliton solutions, first-order breather and second-order breather solutions, for the generalized complex modified Korteweg–de Vries equation are obtained.

Extended generalized Darboux transformation to hybrid rogue wave and breather solutions for a nonlinear Schrödinger equation

Abstract An extended generalized Darboux transformation method is proposed to construct the hybrid rogue wave and breather solutions for a classical nonlinear Schrodinger equation. Three types of hybrid wave solutions are obtained: (i) the hybrid first-order rogue wave and breather; (ii) the hybrid second-order rogue wave and first-order breather; (iii) the hybrid first-order rogue wave and second-order breather. These solutions are novel and can be used to investigate the dynamical characteristic of the hybrid rogue waves and breathers. The control and interaction based on the parameters of the hybrid wave solution are graphically demonstrated. An exact link is established between the hybrid solutions and the rogue wave solutions via setting the parameter at special value.

Dark–bright semi-rational solitons and breathers for a higher-order coupled nonlinear Schrödinger system in an optical fiber

Optical-fiber communication is an indispensable part of the modern communication. In this letter, we study a higher-order coupled nonlinear Schrödinger system, which describes the simultaneous propagation of two electric fields in an optical fiber. We construct the Nth-order dark–bright semi-rational soliton solutions for the mixed interactions, and the Nth-order breather solutions for the focusing interactions, where N is a positive integer. For the mixed interactions, we show the first- and second-order dark–bright semi-rational solitons. For the focusing interactions, we present the first- and second-order Y-shaped breathers, and the line-shaped breather with one component possessing an anti-eye-shaped breather and the other component containing an eye-shaped breather, as well as the second-order breather with each component consisting of two curved breathers.

The higher-order nonlinear Schrödinger equation with non-zero boundary conditions: Robust inverse scattering transform, breathers, and rogons

An integrable three-parameter fifth-order nonlinear Schrödinger equation with non-zero boundary conditions was studied staring from its Lax pair and a robust inverse scattering transform such that one-, and n-fold modified Darboux transforms (DTs) are presented. For distinct parameters, the one-fold DT is used to simultaneously investigate its first- and second-order breathers, rational W-shaped soliton-like solutions, and rogons. Moreover, we also analyze their wave structures, and relations between these solutions and special parameters. The n-fold DT is also established for the fifth-order NLS equation such that its (2n−1,2n)th-order rogue wave solutions can be found simultaneously. Particularly, we exhibit the third- and fourth-order rogon structures. All these results can also reduce to ones for the special cases of the fifth-order NLS equation such as the NLS equation, Hirota equation, and Lakshmanan-Porsezian-Daniel equation.

General Higher-Order Breather and Hybrid Solutions of the Fokas System**Supported by the National Natural Science Foundation of China under Grant No. 11671219, the Natural Science Foundation of Zhejiang Province under Grant No. LZ19A010001, and the K. C. Wong Magna Fund in Ningbo University

Fokas system is the simplest (2+1)-dimensional extension of the nonlinear Schrödinger (NLS) equation (Eq. (), Inverse Problems 10 (1994) L19-L22). By appropriately limiting on soliton solutions generated by the Hirota bilinear method, the explicit forms of n-th breathers and semi-rational solutions for the Fokas system are derived. The obtained first-order breather exhibits a range of interesting dynamics. For high-order breather, it has more rich dynamical behaviors. The first-order and the second-order breather solutions are given graphically. Using the long wave limit in soliton solutions, rational solutions are obtained, which are used to analyze the mechanism of the rogue wave and lump respectively. By taking a long waves limit of a part of exponential functions in f and g appeared in the bilinear form of the Fokas system, many interesting hybrid solutions are constructed. The hybrid solutions illustrate various superposed wave structures involving rogue waves, lumps, solitons, and periodic line waves. Their rather complicated dynamics are revealed.

Lax pair, breather-to-soliton conversions, localized and periodic waves for a coupled higher-order nonlinear Schrödinger system in a birefringent optical fiber

A coupled higher-order nonlinear Schrodinger system, which describes the ultrashort pulses in a birefringent optical fiber, is analytically investigated. For the complex envelopes of the electric field in the fiber, we construct a Lax pair which is different from that in the existing literature, and derive out the corresponding first- and second-order breather solutions. We present the first- and second-order breather-to-soliton conversion conditions, related to the strength of the higher-order linear and nonlinear effects. We find that the strength affects the peak numbers of solitons, and see the multi-peak soliton, W-shaped soliton, M-shaped soliton, anti-dark soliton and two kinds of periodic waves. From the second-order breather solutions under the first-order breather-to-soliton conversion condition, interactions between the breather and a W-shaped soliton, an M-shaped soliton or periodic waves are given. From the second-order breather solutions under the second-order breather-to-soliton conversion conditions, we present the interactions between the two M-shaped solitons, the two anti-dark solitons, a W-shaped soliton and an M-shaped soliton, or a W-shaped soliton and two kinds of the periodic waves. Those results might provide certain assistance for the studies on the birefringent optical fibers.

Localized-wave interactions for the discrete nonlinear Schrödinger equation under the nonvanishing background

In this paper, localized-wave interactions under the nonvanishing background are studied through the Darboux transformation (DT) for the discrete nonlinear Schrödinger equation. First of all, via the elementary DT, we obtain the first-order breather solution and give the parameter conditions for generating Kuznetsov–Ma breathers, Akhmediev breathers, breathers with a number of bunches and spatio-temporal breathers. Moreover, we analyze the effects of parameters on the velocity and period of the breathers. Secondly, we derive the second-order solution and discuss the dynamic behaviors of breather–breather and breather-rogue wave interactions on the nonvanishing background. Finally, via the generalized DT, we construct the equal-eigenvalue degenerate second-order breather solution, and analyze the characteristics of interactions between two breathers, which is found to be different from the ones given by the elementary DT.