Higher-order Breather Research Articles

New interaction of high-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions for the Hirota–Maccari system

The higher-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions of the Hirota–Maccari (HM) system by virtue of the Kadomtsev–Petviashvili (KP) hierarchy reduction method are investigated in this work. Through analyzing the structural characteristics of periodic-wave solutions, we attain the quasi-periodic W(M)-shaped waves and two kinds of breathers. The mixed solutions that consist of the quasi-periodic W(M)-shaped waves and breathers are constructed. Further, by taking the long wave limit on the periodic-wave solutions, the semi-rational solutions are derived, which illustrate the interaction of the rational soliton, lump, quasi-periodic wave and breather. Characteristics of these mixed solutions are discussed graphically and the corresponding generating conditions are given. Especially, a new bound-state interaction composed of lump and breather is generated under the velocity resonance mechanism. This newfangled pattern is a beautiful phenomenon for the HM system.

Multi-Solitons, Multi-Breathers and Multi-Rational Solutions of Integrable Extensions of the Kadomtsev–Petviashvili Equation in Three Dimensions

The celebrated Korteweg–de Vries and Kadomtsev–Petviashvili (KP) equations are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. The question of constructing integrable evolution equations in three-spatial dimensions has been one of the most important open problems in the history of integrability. Here, we study an integrable extension of the KP equation in three-spatial dimensions, which can be derived using a specific reduction of the integrable generalization of the KP equation in four-spatial and two-temporal dimensions derived in (Phys. Rev. Lett. 96, (2006) 190201). For this new integrable extension of the KP equation, we construct smooth multi-solitons, high-order breathers, and high-order rational solutions, by using Hirota’s bilinear method.

Degenerate soliton and breather solutions of the modified Korteweg–de Vries–Sine Gordon equation

For the modified Korteweg–de Vries–Sine Gordon (MKdV-SG) system, we have investigated the degenerate N-soliton solution and the degenerate higher-order breather solution via the bilinear method. The dynamic properties of these degenerate solutions are not only visually displayed through exquisite images, but also accurately described by the specific mathematical expressions. In addition, this paper also intuitively shows how to derive higher-order degenerate solutions from the N-soliton solution for the MKdV-SG equation.

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.

Studies on certain bilinear form, N-soliton, higher-order breather, periodic-wave and hybrid solutions to a (3+1)-dimensional shallow water wave equation with time-dependent coefficients

Studies on certain bilinear form, N-soliton, higher-order breather, periodic-wave and hybrid solutions to a (3+1)-dimensional shallow water wave equation with time-dependent coefficients

Higher-order breathers as quasi-rogue waves on a periodic background

We investigate higher-order breathers of the cubic nonlinear Schrödinger equation on a periodic elliptic background. We find that, beyond first order, any arbitrarily constructed breather on a disordered background generates a single-peaked solitary wave. However, on the periodic backgrounds, the so-called quasi-rogue waves are found more common. These are the quasiperiodic breathers that feature distorted side peaks. We construct such higher-order breathers out of constituent first-order breathers with commensurate periods (i.e., as higher-order harmonic waves). In addition to quasiperiodic, we also find fully periodic breathers, when their wavenumbers are harmonic multiples of the background and each other. But they are truly rare, requiring finely tuned parameters. Thus, on a periodic background, we arrive at the paradoxical conclusion that the higher-order quasi-rogue waves are rather common, while the truly periodic breathers are exceedingly rare.

Interaction of high-order breather, periodic wave, lump, rational soliton solutions and mixed solutions for reductions of the (4+1)-dimensional Fokas equation

The interaction of high-order breather, periodic-wave, lump, rational soliton solutions and mixed solutions for reductions of the (4+1)-dimensional Fokas equation are investigated by means of the Kadomtsev-Petviashvili (KP) hierarchy reduction method. Through analyzing the structural characteristics of periodic wave solutions, we find that evolution of the breather is decided by two characteristic lines. Interestingly, growing-decaying amplitude periodic wave and amplitude-invariant periodic wave are given through some conditions posed on the parameters. Some fascinating nonlinear wave patterns composed of high-order breathers and high-order periodic waves are shown. Furthermore, taking the long wave limit on the periodic-wave solutions, the semi-rational solutions composed of lumps, moving solitons, breathers, and periodic waves are obtained. Some novel dynamical processes are graphically analyzed. Additionally, we provide a new method to derive periodic-wave and semi-rational solutions for the (3+1)-dimensional KP equation by reducing the solutions of the (4+1)-dimensional Fokas equation. The presented results might help to understand the dynamic behaviors of nonlinear waves in the fluid fields and may provide some new perspectives for studying nonlinear wave solutions of high dimensional integrable systems.

Multiple breathers and high-order rational solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili equation

This paper is devoted to obtaining the multi-soliton solutions, high-order breather solutions and high-order rational solutions of the (3+1)-dimensional B-type Kadomtsev–Petviashvili (BKP) equation by applying the Hirota bilinear method and the long-wave limit approach. Moreover, the interaction solutions are constructed by choosing appropriate value of parameters, which consist of four waves for lumps, breathers, rouges and solitons. Some dynamical characteristics for the obtained exact solutions are illustrated using figures.

Further study of the localized solutions of the (2+1)-dimensional B-Kadomtsev–Petviashvili equation

In this paper, the localized solutions of the (2+1)-dimensional B-Kadomtsev–Petviashvili (BKP) equation, which is a useful physical model, are further studied. Firstly, by using the theory of Hirota bilinear operator, the corresponding N-soliton solutions are obtained. Then the localized solutions, which are the M-lump solutions, higher-order breathers and hybrid solutions, are also constructed by taking a long-wave limit and introducing some conjugation conditions. In the meanwhile, the dynamic behaviors of these obtained solutions are analyzed and shown graphically by the corresponding numerical simulations with specific parameters, which can greatly affect the solutions, such as the propagation properties.

Conservation laws and breather-to-soliton transition for a variable-coefficient modified Hirota equation in an inhomogeneous optical fiber

Optical fibers are used in the communications, biological sensors and chemical sensors. We investigate a variable-coefficient modified Hirota equation for the amplification or absorption of pulses propagating in an inhomogeneous optical fiber. With respect to the complex envelope of the optical field, we construct the infinitely-many conservation laws based on the existing Lax pair. According to the existing Darboux transformation, we derive the three-soliton solutions, the higher-order breather solutions and breather-to-soliton transition condition. Amplitudes of the two solitons change after the interaction, while velocities of them are unchanged via asymptotic analysis. When P ( z ) = 0 , interactions among the three parabolic or wavy solitons, interaction between the two parabolic or wavy or crooked breathers, and interactions among the three parabolic and wavy breathers are presented, where P ( z ) is related to the nonlinear focus length. Velocities of three solitons or two crooked breathers with P ( z ) ≠ 0 are different from those with P ( z ) = 0 . Based on the breather-to-soliton transition condition, when P ( z ) = 0 , parabolic or wavy multi-peak and M-shaped solitons are presented; when P ( z ) ≠ 0 , the crooked periodic wave and anti-dark soliton are shown.

General M-lump, high-order breather, and localized interaction solutions to (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation

General M-lump, high-order breather, and localized interaction solutions to (2 + 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation

New interaction of high-order breather solutions, lump solutions and mixed solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like equation

New interaction of high-order breather solutions, lump solutions and mixed solutions for (3+1)-dimensional Hirota–Satsuma–Ito-like equation

High-order breathers and semi-rational solutions of the (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation

In this paper, based on the Hirota bilinear method, the high-order breathers and interaction solutions between solitons and breathers of the (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation are investigated. The lump and semi-rational solutions are obtained by applying the long wave limit of the [Formula: see text]-soliton solution. Two types of semi-rational solutions are derived by choosing specific parameters, which are the mixture of the lump solution and solitons, and the mixture of the lump solution and breathers. Furthermore, the time evolution diagram illustrate the dynamic behavior of these 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.

Bilinear form, solitons, breathers, lumps and hybrid solutions for a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation

In this work, we study a (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation for the nonlinear dispersive waves in an inhomogeneous medium. Bilinear form and N-soliton solutions are derived, where N is a positive integer. The higher-order breather and lump solutions are constructed based on the N-soliton solutions. Hybrid solutions comprising the solitons and breathers, breathers and lumps, as well as solitons and lumps are worked out. Amplitudes and velocities of the one solitons as well as periods of the first-order breathers are investigated. Amplitudes of the first-order lumps reach the maximum and minimum values at certain points given in the paper. Interactions between any two of those waves are discussed graphically.

High-order breather, M-kink lump and semi-rational solutions of potential Kadomtsev–Petviashvili equation

N-kink soliton and high-order synchronized breather solutions for potential Kadomtsev–Petviashvili equation are derived by means of the Hirota bilinear method, and the limit process of high-order synchronized breathers are shown. Furthermore, M-lump solutions are also presented by taking the long wave limit. Additionally, a family of semi-rational solutions with elastic collision are generated by taking a long-wave limit of only a part of exponential functions, their interaction behaviors are shown by three-dimensional plots and contour plots.

Novel localized wave solutions of the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation

Based on a special transformation that we introduce, the N-soliton solution of the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation is constructed. By applying the long wave limit and restricting certain conjugation conditions to the related solitons, some novel localized wave solutions are obtained, which contain higher-order breathers and lumps as well as their interactions. In particular, by choosing appropriate parameters involved in the N-solitons, two interaction solutions mixed by a bell-shaped soliton and one breather or by a bell-shaped soliton and one lump are constructed from the 3-soliton solution. Five solutions including two breathers, two lumps, and interaction solutions between one breather and two bell-shaped solitons, one breather and one lump, or one lump and two bell-shaped solitons are constructed from the 4-soliton solution. Five interaction solutions mixed by one breather/lump and three bell-shaped solitons, two breathers/lumps and a bell-shaped soliton, as well as mixing with one lump, one breather and a bell-shaped soliton are constructed from the 5-soliton solution. To study the behaviors that the obtained interaction solutions may have, we present some illustrative numerical simulations, which demonstrate that the choice of the parameters has a great impacts on the types of the solutions and their propagation properties. The method proposed can be effectively used to construct localized interaction solutions of many nonlinear evolution equations. The results obtained may help related experts to understand and study the interaction phenomena of nonlinear localized waves during propagations.

Breather stripes and radial breathers of the two-dimensional sine-Gordon equation

We revisit the problem of transverse instability of a 2D breather stripe of the sine-Gordon (sG) equation. A numerically computed Floquet spectrum of the stripe is compared to analytical predictions developed by means of multiple-scale perturbation theory showing good agreement in the long-wavelength limit. By means of direct simulations, it is found that the instability leads to a breakup of the quasi-1D breather in a chain of interacting 2D radial breathers that appear to be fairly robust in the dynamics. The stability and dynamics of radial breathers in a finite domain are studied in detail by means of numerical methods. Different families of such solutions are identified. They develop small-amplitude spatially oscillating tails (“nanoptera”) through a resonance of higher-order breather’s harmonics with linear modes (“phonons”) belonging to the continuous spectrum. These results demonstrate the ability of the 2D sG model within our finite domain computations to localize energy in long-lived, self-trapped breathing excitations.

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 $$h_4$$ .

General high-order breather, lump, and semi-rational solutions to the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation

In this work, we investigate the (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation. Based on its bilinear form, the [Formula: see text]th-order breather solutions of the gBK equation are successful given by taking appropriate parameters. Furthermore, the [Formula: see text]th-order lump solutions of the gBK equation are obtained via the long-wave limit method. In addition, the semi-rational solutions are generated to reveal the interaction between lump solutions, soliton solutions, and breather solutions.