Higher-order Breather Research Articles

Superposition of soliton, breather and lump waves in a non-painlevé integrabale extension of the Boiti-Leon-Manna-Pempinelli equation

Abstract In this paper, we derive general Nth-order Pfaffian solutions for a (3 + 1)-dimensional non-Painlevé integrable extension of the Boiti-Leon-Manna-Pempinelli (BLMP) equation. Specifcally, we obtain N-soliton, higher-order breather, higher-order lump and hybrid solutions, and explore the superpositions of Y-shaped and X-shaped soliton-breather waves. Moreover, we construct bilinear Bäcklund transformations, Lax pairs, and conservation laws using Bell polynomials. Finally, we identify a similar equation in the literature and demonstrate that it represents another non-Painlevé integrable extension of the BLMP equation.

High-order solitary waves, fission, hybrid waves and interaction solutions in the nonlinear dissipative (2+1)-dimensional Zabolotskaya-Khokhlov model

The Zabolotskaya-Khokhlov model (ZKm) is a widely used nonlinear model in the fields of sound, ultrasound, and shock waves. The aims of this paper stems from its examination and rectification of earlier results concerning the N-soliton solutions of nonlinear dissipative (2+1)-dimensional ZKm. By recognizing and incorporating the non-zero values of the dispersion coefficient , this study addresses a significant omission in current research. The findings enhance the comprehension of higher-order soliton behaviors, encompassing bifurcation solitons, higher-order breathers, rogue waves, periodic lumps, and their interactions, which are crucial for both theoretical studies and practical applications in areas like nonlinear optics and fluid dynamics. Subsequent detailed numerical simulations are conducted to elucidate the complex behaviors of the obtained solutions. This thorough exploration provides crucial insights into the intricate patterns exhibited by the nonlinear dissipative (2+1)-dimensional ZKm under different conditions, enhancing our understanding of the underlying physical phenomena.

Conversion mechanisms and transformed waves for the (3 + 1)-dimensional nonlinear equation

In this paper, we focus on investigating the (3 + 1)-dimensional nonlinear equation which is used to describe the propagation of waves in the shallow water. The study begins with the application of the Hirota bilinear method to obtain N-soliton solution. Building on this foundation, the research delves into the construction of first-order breather wave by imposing complex conjugate constraints on the parameters of two solitons. Further analysis of the characteristic lines of breathers leads to the derivation of conversion conditions. Under this specific condition, a series of nonlinear transformed waves are presented, including quasi-kink solitons, W-shaped kink solitons, oscillation W-shaped kink solitons, multipeaks solitons, quasi-periodic waves, and line rogue waves. Each of these transformed waves exhibits unique structural and dynamic properties, enriching the understanding of wave behavior in higher-dimensional nonlinear systems. The study also explores the nonlinear superposition mechanism between solitary wave and periodic wave. This mechanism elucidates the formation process of nonlinear waves, explaining how their locality and oscillatory characteristics emerge from the superposition of different wave components. Moreover, the geometric properties of the two characteristic lines of the waves are analyzed to understand the time-varying nature of the transformed waves. This temporal analysis is crucial for predicting the evolution and interaction of these waves over time. Finally, the research extends to the higher-order breather wave and explores the interactions among various waves. These interactions reveal the complex dynamics that may arise in the (3 + 1)-dimensional nonlinear systems and provide deeper insights into the interactions among different wave structures.

Higher-order breather and interaction solutions to the [formula omitted]-dimensional Mel’nikov equation

In this paper, we construct the high-order breather and interaction solutions of the (3+1)-dimensional Mel’nikov equation using the KP hierarchy reduction approach and express them in a concise determinant form. Our solutions show that the two breathers, two periodic waves, and the hybrid mode of the breather and periodic wave are all mutually parallel. Furthermore, by examining the long wave limit of the periodic wave solutions, a variety of rational solutions (lumps) and mixed solutions are obtained. Notably, the interaction between the lump and breather is found to be elastic. These novel results provide deeper insights into the interactions among different solution types.

Nonlinear waves and transitions mechanisms for (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation

In this paper, state transition waves are investigated in a (2+1)-dimensional Korteweg–de Vries-Sawada-Kotera-Ramani equation by analyzing characteristic lines. Firstly, the N-soliton solutions are given by using the Hirota bilinear method. The breather and lump waves are constructed by applying complex conjugation limits and the long-wave limit method to the parameters. In addition, the transition condition of breather and lump wave are obtained by using characteristic line analysis. The state transition waves consist of quasi-anti-dark soliton, M-shaped soliton, oscillation M-shaped soliton, multi-peak soliton, W-shaped soliton, and quasi-periodic wave soliton. Through analysis, when solitary wave and periodic wave components undergo nonlinear superposition, it leads to the formation of breather waves and transformed wave structures. It can be used to explain the deformable collisions of transformation waves after collision. Furthermore, the time-varying property of transformed waves are studied using characteristic line analysis. Based on the high-order breather solutions, the interactions involving breathers, state transition waves, and solitons are exhibited. Finally, the dynamics of these hybrid solutions are analyzed through symbolic computations and graphical representations.

Dynamics of heterotypic soliton, high-order breather, M-lump wave, and multi-wave interaction solutions for a ($$3+1$$)-dimensional Kadomtsev–Petviashvili equation

Dynamics of heterotypic soliton, high-order breather, M-lump wave, and multi-wave interaction solutions for a ($$3+1$$)-dimensional Kadomtsev–Petviashvili equation

Evolution of multi-solitons and interaction behaviors of lump to a (2+1) dimensional generalized shallow water wave model

In this paper, the trajectory equations of 1-lump before and after collision with high-order solitons and the degradation of some novel breather waves are studied in the (2+1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation(gCBS). Firstly, we derive N-solitons for the gCBS equation by the Hirota bilinear form. With the help of N-solitons, we obtain M-lump as well as high-order breather based on the long-wave limit technique and the parametric conjugate method. Secondly, we construct many hybrid waves, such as the hybrid wave between breather and lump. Thirdly, the interaction phenomenon of lump-N-solitons(N → ∞) is investigated, and the theory of its existence is given and proved. Besides, the different degeneracies of double and single breather are discussed. Finally, we also present a large number of two-dimensional and three-dimensional images to better illustrate these nonlinear evolutionary behaviors.

Transmission mode transformation of rotating controllable beams induced by the cross phase.

In this paper, complex-variable sine-Gaussian cross-phase (CVSGCP) beams are proposed, and the transmission dynamics properties of the CVSGCP beams in strongly nonlocal nonlinear media are investigated. CVSGCP beams can produce a variety of mode transformation characteristics during transmission. The roles of parameters in the sine and cross-phase terms of the initial light field expression in the evolution of light intensity modes, phase, and beam width are analyzed in detail, and it is proved that the effect of cross phase is to cause the beams to rotate. The control of different modes can be achieved by selecting suitable parameters, which have certain advantages in the practical application of CVSGCP beams. CVSGCP beams can be regarded as generalized high-order breathers because light intensity modes and beam width show periodic oscillation distribution during transmission. The typical evolution characteristics of the CVSGCP beams are verified by numerical simulation. Strongly nonlocal nonlinear optical media can be mathematically equivalent to a variety of optical systems, such as gradient index potential wells and resonant potential wells, so the conclusions in this paper can also be extended to these equivalent optical systems.

Interactions of certain localized waves for an extended (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid mechanics

Currently, there is much interest in the study on the localized waves and their interactions in fluid mechanics. An extended (3+1)-dimensional Kadomtsev-Petviashvili equation is considered here. By means of the Hirota method, we obtain the N-soliton solutions, with N as a positive integer. The higher-order breather solutions are obtained from the N-soliton solutions through the complex conjugated transformations. Making use of the long-wave limit method, we determine the higher-order lump solutions via the N-soliton solutions. Besides, some hybrid solutions are presented. Three kinds of the localized waves, namely, the solitons, breathers and lumps, along with their interactions, are investigated via the above solutions. Amplitudes, shapes and velocities of those localized waves remain invariant after the interactions, which indicate that the interactions are elastic. Fluid-mechanically meaningful, all our results rely on the coefficients in that equation.

The excitation of high-order localized waves in (3+1)-dimensional Kudryashov-Sinelshchikov equation

The aim of this work is to explore the excitation of high-order localized waves in the (3+1)-dimensional Kudryashov-Sinelshchikov equation, which is used to describe the dynamic of liquid with gas bubble. First of all, classical N-soliton solutions are constructed by means of Hirota bilinear form and symbolic calculation. What’s more, the high-order breather waves are derived through the degeneration process of the N-soliton solutions with conjugate parameter. Then, high-order lump waves are constructed by taking long wave limit technique on N-soliton solutions. Finally, the high-order mixed localized waves involving resonant Y-type solitons, high-order breather waves and high-order lump waves are obtained by utilizing some comprehensive methods. Abundant dynamical and evolutionary behaviors of these results are investigated specifically, some figures are presented to shed light on the nonlinear phenomena hidden in the high-order localized waves vividly.

A study of (2+1)-dimensional variable coefficients equation: Its oceanic solitons and localized wave solutions

In this work, shallow ocean-wave soliton, breather, and lump wave solutions, as well as the characteristics of interaction between the soliton and lump wave in a multi-dimensional nonlinear integrable equation with time-variable coefficients, are investigated. The Painlevé analysis is used to verify the integrability of this model. Based on the bilinear form of this model, we use the simplified Hirota's method obtained from the perturbation approach and various auxiliary functions to construct the aforementioned solutions. Besides, the interaction between the soliton and lump wave solutions is also examined. In addition, by imposing specific constraint conditions on the N-soliton solutions, we further derive higher-order breather solutions. To show the physical characteristics of this model, several graphical representations of the discovered solutions are established. These graphs show that the time-variable coefficients result in a variety of novel dynamic behaviors that differ significantly from those for integrable equations with constant coefficients. The acquired results are useful for the study of shallow water waves in fluid dynamics, marine engineering, nonlinear sciences, and ocean physics.

Nonlinear localized waves and their interactions for a (2+1)-dimensional extended Bogoyavlenskii-Kadomtsev–Petviashvili equation in a fluid

In fluid dynamics, a (2+1)-dimensional extended Bogoyavlenskii-Kadomtsev-Petvi-ashvili equation is hereby investigated. Bilinear form and N-soliton solutions are determined with the Hirota method, where N is a positive integer. N-soliton solutions are used to build the higher-order breather and lump solutions with the complex parameter relation and long-wave-limit method. Elastic and inelastic interactions between the two breathers and elastic interaction between the two lumps are constructed and graphically depicted. We find that the coefficients of the weak dispersion terms have an effect on the velocities and amplitudes of the breathers and lumps. Based on the hybrid solutions containing the breathers, lumps and solitons, several interactions of those nonlinear localized waves are graphically illustrated, including the fission of the breather and soliton, elastic or inelastic interaction between the one breather and one soliton, elastic interaction among one breather and two solitons, as well as elastic interaction between the one lump and one breather.

Multi solitons, bifurcations, high order breathers and hybrid breather solitons for the extended modified Vakhnenko–Parkes equation

This manuscript delves into a comprehensive investigation of significant exact solutions pertaining to the extended modified Vakhnenko–Parkes equation (emVPE), with a focus on novel solutions. By taking into account various kinds of auxiliary functions, we are able to generate a number of high order solitons, bifurcations, and breathers solutions for the given model. The study of multiple solitons, multiple bifurcation solitons, high order breathers, and hybrid breathers are all part of our current research. Our proposed results are validated through graphical visualizations.

Novel multiple solitons, their bifurcations and high order breathers for the novel extended Vakhnenko–Parkes equation

This paper presents a rigorous exploration of the extended Vakhnenko–Parkes-Equation (eVPE), unveiling a rich tapestry of multi solitons, bifurcations, and high-order breathers. Notably, it achieves an impressive feat by deriving solitons up to sixth order and breathers up to third order, pushing the boundaries of the equation’s solutions. These intricate solutions are not only derived but also vividly brought to life through numerical visualizations, underscoring the profound implications of the our work. Through insightful discussions, the manuscript effectively underscores the significance of these findings, opening new avenues for understanding nonlinear dynamics and their applications across various scientific domains.

Dynamics of dissipative structures in coherently-driven Kerr cavities with a parabolic potential

By means of a modified Lugiato–Lefever equation model, we investigate the nonlinear dynamics of dissipative wave structures in coherently-driven Kerr cavities with a parabolic potential. This potential stabilizes the system dynamics, leading to the generation of robust dissipative solitons in the positive detuning regime, and of higher-order solitons in the negative detuning regime. In order to understand the underlying mechanisms which are responsible for these high-order states, we decompose the field on the basis of linear eigenmodes of the system. This permits to investigate the resulting nonlinear mode coupling processes. By increasing the external pumping, one observes the emergence of high-order breathers and chaoticons. Our modal content analysis reveals that breathers are dominated by modes of corresponding orders, while chaoticons exhibit proper chaotic dynamics. We characterize the evolution of dissipative structures by using bifurcation diagrams, and confirm their stability by combining linear stability analysis results with numerical simulations. Finally, we draw phase diagrams that summarize the complex dynamics landscape, obtained by varying the pump, the detuning, and the strength of the potential.

Bilinear form, auto-Bäcklund transformations, Pfaffian, soliton, and breather solutions for a (3 + 1)-dimensional extended shallow water wave equation

Study of the water waves remains central to fluid physics, ocean dynamics, and engineering. In this paper, a (3 + 1)-dimensional extended shallow water wave equation is investigated via symbolic computation. Bilinear form and two kinds of the bilinear auto-Bäcklund transformations with some solutions are given via the Hirota method. The Nth-order Pfaffian solutions are worked out by means of the Pfaffian technique, where N is a positive integer. N-soliton solutions are exported through the Nth-order Pfaffian solutions. By virtue of the asymptotic analysis, elastic and inelastic interactions between the two solitons on some periodic backgrounds are discussed. Interaction among the three solitons is illustrated graphically. The higher-order breather solutions are investigated via the complex parameter relation. Elastic and inelastic interactions between the two breathers on the periodic backgrounds are depicted graphically. Hybrid solutions consisting of the solitons and breathers are obtained. Interaction between the one soliton and one breather on a periodic background is presented.

Higher-order breather, lump and hybrid solutions of (2 + 1)-dimensional coupled nonlinear evolution equations with time-dependent coefficients

This paper investigates a class of (2 + 1)-dimensional coupled nonlinear evolution equation with time-dependent coefficients in an inhomogeneous medium via the Hirota bilinear method. Combining the long wave limit method and complex conjugate transform, the higher-order breather and lump solutions are initially constructed. Furthermore, hybrid solutions among N-soliton, lump and breather solutions are derived by linear constraints on the parameters. Meanwhile, the dynamic evolution behaviour of some special concrete solutions under different time-dependent coefficients is presented visually in the form of images.

Higher-order breathers and breather interactions for the AB system in fluids

Higher-order breathers and breather interactions for the AB system in fluids

Pfaffian solutions and nonlinear waves of a (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt system in fluid mechanics

Fluid mechanics is concerned with the behavior of liquids and gases at rest or in motion, where the nonlinear waves and their interactions are important. Hereby, we study a (3 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt system in fluid mechanics. We determine a bilinear form of that system via the Hirota method. Nth-order Pfaffian solutions are obtained via the Pfaffian technique and our bilinear form, where N is a positive integer. Based on the Nth-order Pfaffian solutions, we derive the N-soliton, higher-order breather, and hybrid solutions. Using those solutions, we present the (1) elastic interaction between the two solitary waves with a short stem, (2) elastic interaction between the two solitary waves with a long stem, (3) fission between the two solitary waves, (4) fusion between the two solitary waves, (5) one breather wave, (6) elastic interaction between the two breather waves, (7) fission between the two breather waves, (8) fusion among the one breather wave and two solitary waves, and (9) elastic interaction between the one breather wave and one solitary wave.

Propagation characteristics of cosine-Gaussian cross-phase beams in strongly nonlocal nonlinear media

In this paper, the propagation expression of cosine-Gaussian cross-phase beams (CGCPBs) is derived based on the nonlocal nonlinear Schrödinger equation and the propagation characteristics of CGCPBs in strongly nonlocal nonlinear media (SNNM) are researched. It is worth noting that the mode of CGCPBs can be controlled by modifying the parameters of cosine term and cross phase term. CGCPBs can form generalized high-order breathers, which signifies that the multi-peak light intensity pattern and the beam width evolve periodically during the transmission process. Apart from that, the impact of the parameters on the light intensity mode, the phase and the beam width is discussed by changing the parameters in the initial incident transmission expression. The research content of this paper provides reference value for controlling beams.