First-order Rogue Wave Research Articles

Rogue waves for a variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics

Under investigation in this paper is a variable-coefficient Kadomtsev–Petviashvili equation in fluid mechanics, which describes the shallow-water waves with the weak nonlinearity and dispersion. Employing the Kadomtsev–Petviashvili hierarchy reduction, we obtain the rogue-wave solutions in terms of the Gramian. Periodic, cubic- and s-shaped line rogue waves are presented with different forms of the dispersion coefficient. The second-order rogue waves and multi-rogue waves are also graphically discussed. It is observed that only parts of the second-order rogue wave approach the constant background, and the other parts move to the far distance with the undiminished amplitudes. The multi-rogue waves describe the interaction of several first-order rogue waves. We plot the interactions of two periodic, cubic- and s-shaped line rogue waves. The lump wave, which propagates stably in all directions, is also depicted.

General high-order rogue waves to nonlinear Schrödinger–Boussinesq equation with the dynamical analysis

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.

Generation mechanism of rogue waves for the discrete nonlinear Schrödinger equation

In this paper, we analyze the generation mechanism of rogue waves for the discrete nonlinear Schrödinger (DNLS) equation from the viewpoint of structural discontinuities. First of all, we derive the analytical breather solutions of the DNLS equation on a new nonvanishing background through the Darboux transformation (DT). Via the explicit expressions of group and phase velocities, we give the parameter conditions for existence of the velocity jumps, which are consistent with the derivation of rogue waves via the generalized DT. Furthermore, to verify such statement, we apply the Taylor expansion to the breather solutions and find that the first-order rogue wave can be obtained at the velocity-jumping point. Our analysis can help to enrich the understanding on the rogue waves for the discrete nonlinear systems.

Rogue waves for the coupled variable-coefficient fourth-order nonlinear Schrödinger equations in an inhomogeneous optical fiber

In this paper, investigation is made on the coupled variable-coefficient fourth-order nonlinear Schrödinger 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.

Rogue waves for a generalized nonlinear Schrödinger equation with distributed coefficients in a monomode optical fiber

Investigated in this paper is the generalized nonlinear Schrödinger equation with distributed coefficients, which describes the amplification or absorption of pulses propagating in a monomode optical fiber with distributed group-velocity dispersion and self-focusing Kerr nonlinearity. By virtue of the Kadomtsev-Petviashvili hierarchy reduction, we obtain the rogue waves based on rogue-wave solutions in terms of the Gramian under certain constraint. We study the effects of group-velocity dispersion, nonlinearity and amplification/absorption coefficients on the rogue waves with the help of figures. Amplitudes of the rogue waves are independent with the group-velocity dispersion and nonlinearity coefficients. The first-order rogue wave with an eye-shaped distribution density and the second-order rogue waves with the highest-peak amplitude and with the triple-peak structure are presented. Both the intermingled or separated composite rogue waves are derived. Periodic rogue waves are obtained and period of the periodic rogue wave increases with the period of the group-velocity dispersion. Furthermore, nonlinear tunneling of the rogue waves is observed: rogue waves get amplified when they reach to the dispersion barriers and recover their original shapes after passing through the barriers, while amplitudes of the rogue waves decrease inside the dispersion wells. Amplification/absorption coefficient influence the background and amplitude of the rogue wave, and three types of the backgrounds are discussed due to different amplification/absorption coefficients.

Rogue Waves and Lump Solitons of the (3+1)-Dimensional Generalized B-type Kadomtsev–Petviashvili Equation for Water Waves**Supported by the National Natural Science Foundation of China under Grant Nos. 11772017, 11272023, and 11471050, by the Open Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of Posts and Telecommunications), China (IPOC: 2017ZZ05), and by the Fundamental Research Funds for

In this paper, the (3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order, higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.

Localized waves of the coupled cubic–quintic nonlinear Schrödinger equations in nonlinear optics**Project supported by the Global Change Research Program of China (Grant No. 2015CB953904), the National Natural Science Foundation of China (Grant Nos. 11675054 and 11435005), the Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213), and the Natural Science Foundation of Hebei Province, China

We investigate some novel localized waves on the plane wave background in the coupled cubic–quintic nonlinear Schrödinger (CCQNLS) equations through the generalized Darboux transformation (DT). A special vector solution of the Lax pair of the CCQNLS system is elaborately constructed, based on the vector solution, various types of higherorder localized wave solutions of the CCQNLS system are constructed via the generalized DT. These abundant and novel localized waves constructed in the CCQNLS system include higher-order rogue waves, higher-order rogues interacting with multi-soliton or multi-breather separately. The first- and second-order semi-rational localized waves including several free parameters are mainly discussed: (i) the semi-rational solutions degenerate to the first- and second-order vector rogue wave solutions; (ii) hybrid solutions between a first-order rogue wave and a dark or bright soliton, a second-order rogue wave and two dark or bright solitons; (iii) hybrid solutions between a first-order rogue wave and a breather, a second-order rogue wave and two breathers. Some interesting and appealing dynamic properties of these types of localized waves are demonstrated, for example, these nonlinear waves merge with each other markedly by increasing the absolute value of α. These results further uncover some striking dynamic structures in the CCQNLS system.

Semirational rogue waves for the three-coupled fourth-order nonlinear Schrödinger equations in an alpha helical protein

Investigated in this paper are the three-coupled fourth-order nonlinear Schrödinger equations, which describe the dynamics of alpha helical protein with the interspine coupling at the higher order. We show that the representation of the Lax pair with Expressions (42)−(45) in Ref. [25] is not correct, because the three-coupled fourth-order nonlinear Schrödinger equations can not be reproduced by the Lax pair with Expressions (42)−(45) in Ref. [25] through the compatibility condition. Therefore, we recalculate the Lax pair. Based on the recalculated Lax pair, we construct the generalized Darboux transformation, and derive the first- and second-order semirational solutions. Through such solutions, dark-bright-bright soliton, breather-breather-bright soliton, breather soliton and rogue waves are analyzed. It is found that the rogue waves in the three components are mutually proportional. Moreover, three types of the semirational rogue waves consisting of the rogue waves and solitons are presented: (1) consisting of the first-order rogue wave and one soliton; (2) consisting of the first-order rogue wave and two solitons; (3) consisting of the second-order rogue wave and two solitons.

Darboux transformation of the coupled nonisospectral Gross–Pitaevskii system and its multi-component generalization

In this paper, we extend the one-component Gross–Pitaevskii (GP) equation to the two-component coupled GP system including damping term, linear and parabolic density profiles. The Lax pair with nonisospectral parameter and infinitely-many conservation laws of this coupled GP system are presented. Actually, the Darboux transformation (DT) for this kind of nonautonomous system is essentially different from the autonomous case. Consequently, we construct the DT of the coupled GP equations, besides, nonautonomous multi-solitons, one-breather and the first-order rogue wave are also obtained. Various kinds of one-soliton solution are constructed, which include stationary one-soliton and nonautonomous one-soliton propagating along the negative (positive) direction of x-axis. The interaction of two solitons and two-soliton bound state are demonstrated respectively. We get the nonautonomous one-breather on a curved background and this background is completely controlled by the parameter β. Using a limiting process, the nonautonomous first-order rogue wave can be obtained. Furthermore, some dynamic structures of these analytical solutions are discussed in detail. In addition, the multi-component generalization of GP equations are given, then the corresponding Lax pair and DT are also constructed.

Rogue waves with multiple compression points in nonlinear optical cavities

In this paper, we use the similarity transformation method to investigate the generation and the control of rogue wave solutions in nonlinear optical cavities described by a (2 + 1) -dimensional Lugiato-Lefever equation with varying coefficients. For a periodic distribution of the diffraction parameter, first-order and second-order rogue wave solutions with multiple compression points are generated. Our results reveal that multiple compression points depend on the values of the modulation amplitude of the diffraction parameter and are independent of the spatial distribution in the cavities.

Breather-to-soliton transformation rules in the hierarchy of nonlinear Schrödinger equations.

We study the exact first-order soliton and breather solutions of the integrable nonlinear Schrödinger equations hierarchy up to fifth order. We reveal the underlying physical mechanism which transforms a breather into a soliton. Furthermore, we show how the dynamics of the Akhmediev breathers which exist on a constant background as a result of modulation instability, is connected with solitons on a zero background. We also demonstrate that, while a first-order rogue wave can be directly transformed into a soliton, higher-order rogue wave solutions become rational two-soliton solutions with complex collisional structure on a background. Our results will have practical implications in supercontinuum generation, turbulence, and similar other complex nonlinear scenarios.

Rogue waves, breather-to-soliton transitions and modulational instability for the nonlinear Schrödinger equation with octic operator in an optical fiber

The nonlinear Schrödinger equation with octic operator in an optical fiber is investigated in this paper. Via the Darboux transformation with complex spectral parameter, we obtain the breathers and rogue waves for the equation. When the propagation directions of the two colliding breathers coincide, breather-to-soliton transitions could be observed, and breathers can be transited to two kinds of solitons, i.e., the M-shape and W-shape solitons. It is found that the coefficient of the eighth-order term η is necessary in the transition condition. Interactions between the solitons and breathers are shown to be elastic. By virtue of the separating function, rogue waves are seen to be separated into the so-called “triangular cascades” or “pentagram” patterns. Number of the separated first-order rogue waves seems to be related to the order of the rogue waves. Modulational instability for the generalized plane wave solutions is conducted, and we find that η does not affect the stability of the solutions.

High-order rogue wave solutions of the classical massive Thirring model equations

The nth-order solutions of the classical massive Thirring model (MTM) equations are derived by using the n-fold Darboux transformation. These solutions are expressed by the ratios of the two determinants consisted of 2n eigenfunctions under the reduction conditions. Using this method, rogue waves are constructed explicitly up to the third-order. Three patterns, i.e., fundamental, triangular and circular patterns, of the rogue waves are discussed. The parameter μ in the MTM model plays the role of the mass in the relativistic field theory while in optics it is related to the medium periodic constant, which also results in a significant rotation and a remarkable lengthening of the first-order rogue wave. These results provide new opportunities to observe rouge waves by using a combination of electromagnetically induced transparency and the Bragg scattering four-wave mixing because of large amplitudes.

Conservation laws and rogue waves for a higher-order nonlinear Schrödinger equation with variable coefficients in the inhomogeneous fiber

Subpicosecond or femtosecond optical pulse propagation in the inhomogeneous fiber can be described by a higher-order nonlinear Schrödinger equation with variable coefficients, which is investigated in the paper. Via the Ablowitz-Kaup-Newell-Segur system and symbolic computation, the Lax pair and infinitely-many conservation laws are deduced. Based on the Lax pair and a modified Darboux transformation technique, the first- and second-order rogue wave solutions are constructed. Effects of the groupvelocity dispersion and third-order dispersion on the properties of the first- and second-order rouge waves are graphically presented and analyzed: The groupvelocity dispersion and third-order dispersion both affect the ranges and shapes of the first- and second-order rogue waves: The third-order dispersion can produce a skew angle of the first-order rogue wave and the skew angle rotates counterclockwise with the increase of the groupvelocity dispersion, when the groupvelocity dispersion and third-order dispersion are chosen as the constants; When the groupvelocity dispersion and third-order dispersion are taken as the functions of the propagation distance, the linear, X-shaped and parabolic trajectories of the rogue waves are obtained.

Discrete rational and breather solution in the spatial discrete complex modified Korteweg-de Vries equation and continuous counterparts.

In this paper, a spatial discrete complex modified Korteweg-de Vries equation is investigated. The Lax pair, conservation laws, Darboux transformations, and breather and rational wave solutions to the semi-discrete system are presented. The distinguished feature of the model is that the discrete rational solution can possess new W-shape rational periodic-solitary waves that were not reported before. In addition, the first-order rogue waves reach peak amplitudes which are at least three times of the background amplitude, whereas their continuous counterparts are exactly three times the constant background. Finally, the integrability of the discrete system, including Lax pair, conservation laws, Darboux transformations, and explicit solutions, yields the counterparts of the continuous system in the continuum limit.

Breathers and rogue waves for an eighth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber

An eighth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber is investigated in this paper. Through the Darboux transformation (DT) and the generalized DT, we obtain the breather and rogue wave solutions. Choosing different values of , , , , , , and , which are the coefficients of the group velocity dispersion, the third-, fourth-, fifth-, sixth-, seventh-, and eighth-order dispersions, respectively, we investigate their effects on the solutions, where x is the scaled propagation variable. Interaction between two kinds of the breathers is studied, i.e., the Akhmediev and Kuznetsov-Ma breathers, and we find that the interaction regions are similar to those of the second-order rogue waves. When , , , , , , and are all chosen as x, , , respectively, peaks of the rogue waves are found to have the parabolic, cubic, and quasi-parabolic shapes. The Akhmediev breathers are seen to pass through the Kuznetsov-Ma breathers who change their phases after the interaction. Rogue waves can be split into some first-order rogue waves or complex structures when the periodic and odd functions are chosen for , , and .

Darboux transformation and rogue wave solutions for the variable-coefficients coupled Hirota equations

We construct the Lax pair and Darboux transformation for the variable-coefficients coupled Hirota equations. Based on modulation instability and by taking the limit approach, we derive two types of Nth-order rogue wave solutions with different dynamic structures in compact determinant representations. The explicit first-order rogue wave solution is presented, prolific vector rogue-wave patterns such as the dark-bright, composite, three-sister, quadruple and sextuple rogue waves with multiple compression points are demonstrated. In particular, in contrast to the standard Peregrine combs, unusual vector rogue wave combs such as the dark-bright and composite rogue wave combs are revealed by choosing sufficiently large periodic modulation amplitudes. Further, some wave characteristics such as the difference between light intensity and continuous wave background, and pulse energy evolution of the dark rogue wave solution that features multiple compression points are discussed in detail.

Solitons, breathers and rogue waves for a sixth-order variable-coefficient nonlinear Schrödinger equation in an ocean or optical fiber

Under investigation in this paper is a sixth-order variable-coefficient nonlinear Schrodinger equation in an ocean or optical fiber. Through the Darboux transformation (DT) and generalized DT, we obtain the multi-soliton solutions, breathers and rogue waves. Choosing different values of \(\alpha\)(x), \(\beta\)(x), \(\gamma\)(x) and \(\delta\)(x), which are the coefficients of the third-, fourth-, fifth- and sixth-order dispersions, respectively, we investigate their effects on those solutions, where x is the scaled propagation variable. When \(\alpha\)(x), \(\beta\)(x), \(\gamma\)(x) and \(\delta\)(x) are chosen as the linear, parabolic and periodic functions, we obtain the parabolic, cubic and quasi-periodic solitons, respectively. Head-on and overtaking interactions between the two solitons are presented, and the interactions are elastic. Besides, with certain values of the spectral parameter \(\lambda\), a shock region between the two solitons appears, and the interaction is inelastic. Interactions between two kinds of the breathers are also studied, and we find that the interaction regions are similar to those of the second-order rogue waves. Rogue waves are split into some first-order rogue waves when \(\alpha\)(x), \(\beta\)(x), \(\gamma\)(x) and \(\delta\)(x) are the periodic or odd-numbered functions.

Solitons and rogue waves for a nonlinear system in the geophysical fluid

In this paper, we investigate a nonlinear system, which describes the marginally unstable baroclinic wave packets in the geophysical fluid. Based on the symbolic computation and Hirota method, bright one- and two-soliton solutions for such a system are derived. Propagation and collisions of the solitons are graphically shown and discussed with [Formula: see text], which reflects the collision between the wave packet and mean flow, [Formula: see text], which measures the state of the basic flow, and group velocity [Formula: see text]. [Formula: see text] is observed to affect the amplitudes of the solitons, and [Formula: see text] can influence the solitons’ traveling directions. By virtue of the generalized Darboux transformation, the first- and second-order rogue-wave solutions are derived. Properties of the first- and second-order rogue waves are graphically presented and analyzed: The first-order rogue waves are shown in the figures. [Formula: see text] has no effects on A, which is the amplitude of the wave packet, but with the increase of [Formula: see text], amplitude of B, which is a quantity measuring the correction of the basic flow, decreases. When [Formula: see text] is chosen differently, A and B do not keep their shapes invariant. With the value of [Formula: see text] increasing, amplitudes of A and B become larger. The second-order rogue wave is presented, from which we observe that with [Formula: see text] increasing, amplitude of B decreases, but [Formula: see text] has no effects on A. Collision features of A and B alter with the value of [Formula: see text] changing. When we make the value of [Formula: see text] larger, amplitudes of A and B increase.

Higher-order rogue waves with new spatial distributions for the (2 + 1) -dimensional two-component long-wave-short-wave resonance interaction system

In this paper, a two-component (2 + 1) -dimensional long-wave-short-wave (LWSW) system with nonlinearity coefficients, which describes the nonlinear resonance interaction between two short waves and a long wave, is studied. Via the Hirota's bilinear method and Pfaffian, N -order rogue waves for the LWSW system are constructed. Furthermore, correction of the N -order rogue waves is proved directly via the Pfaffian, which is cumbersome or inaccessible in other methods. Results of the first- and second-order rogue waves are presented: 1) For the first-order rogue waves, the two short-wave components are bright, while the long-wave component is dark. The position of maximum amplitude of the rogue wave is analyzed. Evolution process for the first-order rogue wave is also presented and discussed. 2) Choosing different forms of the elements defined in the Pfaffian, we obtain some kinds of the second-order rogue waves with new spatial distributions: when the elements defined in Pfaffian are the same as the first-order rogue waves, we find that the second-order rogue waves for the two short-wave components are split into two first-order rogue waves and the two bumps coexist and interact with each other; when we change the combination of the elements in Pfaffian, we find that the second-order rogue waves for the two short-wave components are split into three and four first-order rogue waves. 3) N -order rogue waves for a general M -component LWSW system are constructed.