Breather Wave Research Articles

Study of Fifth-Order Nonlinear Caudrey–Dodd–Gibbon Model for Multiwave, M-Shaped Solitons, Lumps, Generalized Breathers, Manifold Periodic, Rogue Wave and Kink Wave Interactions

Numerous novel wave structures, such as lump, lump with one-kink, lump with two-kink soliton, generalized breather, Ma breather, Kuznetsov-Ma breather, Akhmediev breather, manifold periodic, and rogue wave solutions to [Formula: see text]-dimensional fifth-order nonlinear Caudrey–Dodd–Gibbon (FNCDG) model via symbolic computation are studied based on the ansatz function scheme. Studying the lump-type solution involves selecting the function as the generic quadratic polynomial function. The lump with one- and two-kink solutions is created by mixing a quadratic function with one or two exponential functions. By adding hyperbolic function with a quadratic function, the rogue wave solutions are manipulated. In addition, the multiwave, [Formula: see text]-shaped rational solitons and their interactions with kink waves are analytically evaluated. Furthermore, different 3D, 2D, and contour profiles are created in different dimensions by changing the parameter values.

Novel nonlinear wave transitions and interactions for (2+1)-dimensional generalized fifth-order KdV equation

The (2 + 1)-dimensional generalized fifth-order KdV (2GKdV) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKdV equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped (XS/YS), asymmetric M-Shaped (MS), W-Shaped (WS), Space-Curved (SC) and Oscillation M-Shaped (OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.

On the study of analytical soliton solutions and interaction aspects to the Estevez-Mansfield-Clarkson equation arising in diversity of fields

Abstract The beta fractional form of the Estevez-Mansfield-Clarkson equation is under consideration and this study is done with the assistance of methods such as modified F-expansion method and the logarithmic transformation. A variety of analytical solutions like bright, dark, mixed, singular, bright-dark, and combined solitons are extracted. Moreover, multi waves structures, interaction with double exponential form, breather waves, mixed type solutions as well as periodic cross kink solutions have been analyzed. The governing equation is converted into an ordinary differential equation by employing an appropriate wave transformation with the β-derivative in order to achieve the desired solutions. The applied approaches have substantial computational capability, enabling them to efficiently address exact solutions with high accuracy in these systems. The results indicate that the equation under investigation theoretically contains a substantial number of soliton solution structures. Additionally, in order to examine the behaviors of the solutions at various parameter values, we plot a variety of graphs that incorporate pertinent parameters. The results of this study have the potential to improve understanding of the nonlinear dynamic characteristics displayed by the specified system and to confirm the effectiveness of the techniques that have been implemented.

Nonlinear Structures of Dispersive Electrostatic Solitary Waves in a Multi‐Ion Partially Ionized Plasma

ABSTRACTThe nonlinear structure and dynamics of dispersive solitons and breather waves described by Korteweg‐de Vries and nonlinear Schrödinger equations are studied. The theoretical and numerical study of the generalized hydrodynamic equations, accounting for wave dissipation and particle production‐loss mechanism, are considered. The reductive expansion method has been used in the context of the instability problem of multi‐fluid dynamics, applied to the study of electrostatic solitons and ion‐acoustic waves. A nonlocal model of interacting solitary‐breather waves has been presented. Applications of the theory, concerning the ion streaming instability in the framework of plasma physics, are presented.

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.

Hirota [formula omitted]-operator forms, multiple soliton waves, and other nonlinear patterns of a 2D generalized Kadomtsev–Petviashvili equation

In the present paper, extensive research is conducted on a 2D generalized Kadomtsev–Petviashvili (2D-gKP) equation which models water waves with long wavelengths. The study starts by establishing Hirota D-operator forms of the governing equation using the Bell polynomial method (BPM). Through checking the three-soliton condition for the 2D-gKP equation, its integrability is then examined, and as a consequence, single-, double-, and (special) triple-soliton waves are derived from the modified Hirota method. In the end, after deriving lump and breather waves of the 2D-gKP equation through ansatz methods, a theoretical discussion along with its proof is presented. Attempts have been made to elucidate the physical features of nonlinear waves by displaying such waves in 2D and 3D postures. The paper’s findings contribute certainly to research on nonlinear waves of KP-type equations.

Generation of Solitary Waves with Analytical Solution for The (3+1)-dimensional pKP-BKP Equation and Reductions

In this study, new solitary wave solutions are obtained for the combination of the B-type Kadomtsev-Petviashvili (BKP) equation and the potential Kadomtsev-Petviashvili (pKP) equation, called the integrable (3+1)-dimensional coupled pKP-BKP equation, and its two reduced forms. For this purpose, the Bernoulli auxiliary equation method, which is an ansatz-based method, is used. As a result, kink, lump, bright soliton and breather wave solutions are observed. It is concluded that this method and the results obtained for the considered pKP -BKP equations are an important step for further studies in this field.

Retrieval of lump, breather, interactions, and rogue wave solutions to the fractional complex paraxial wave dynamical model with sensitivity analysis

In this research, the Hirota bilinear method and the modified Sardar sub-equation (MSSE) techniques are used to investigate the generation and detection of soliton structures in the fractional complex paraxial wave dynamical (FPWD) model together with Kerr media. By employing the aforementioned techniques, we derive lump and different exact solitary wave solutions for the selected model, which has not been documented in previous literature. We manifested some novel lump soliton solutions, including the homoclinic breather wave, periodic cross rational wave, the M-shaped interaction with rogue and kink waves, the M-shaped rational solution, the M-shaped rational solution with one and two kink waves, and multi-wave solutions. Furthermore, for intellectual curiosity, we also amalgamated the rich spectrum of soliton solutions such as W-shape, periodic, dark, bright, combo, rational, exponential, mixed trigonometric, and hyperbolic soliton wave solutions inherent in the FPWD equation. We also undertake sensitivity analysis to examine the resilience of the selected model in the face of variations in initial circumstances and parameters, which provides insights into the system’s sensitivity to perturbations. Furthermore, we investigate the ramifications of these findings for a variety of physical systems, including optics, fluid dynamics, and plasma physics. These findings are to gain a better knowledge of nonlinear wave phenomena and fresh insights into the dynamics of complex systems by combining the Hirota bilinear technique and the MSSE method.

Hybrid soliton, breather waves and solution molecules of the (2+1)-dimensional generalized fifth-order KdV equation

In this paper, we take the (2+1)-dimensional generalized fifth-order KdV (fKdV) equation as an example. We obtain many wave solutions, including hybrid soliton, breather waves and solution molecules. In order to solve hybrid soliton, we give two transformation forms. For Transform one, real and complex parameters lead to different wave solutions. Soliton solutions are obtained when the parameter values are real, but some breather waves appear when the parameter values are complex. When the parameter value changes, the number of solitons and the position of the waves will also change. For Transform two, we assign the parameter values to real values and obtain the corresponding two-, three- and four-soliton solutions. The most important thing is that, on the basis of Transform one, we add different constraint conditions according to different coordinate systems, and finally get a very interesting phenomenon: soliton molecules. In addition, we give not only the three-dimensional plots but also the density plots. In general, these results will enrich the existing literature on the (2+1)-dimensional generalized fKdV equation and help to understand this kind of KdV equation.

Collision dynamics between breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation

In this paper, the collision dynamics of breather and lump-type localized waves in the (3+1)-dimensional shallow water wave equation are investigated in detail. Firstly, the auto-Bäcklund transformation and the linear representation of the (3+1)-dimensional shallow water wave equation are derived in virtue of the truncated Painlevé expansion method, which provide convenience in solving the (3+1)-dimensional shallow water wave equation. Secondly, based on the linear representation and the principle of linear superposition, the rational solutions in the exponential and polynomial forms are constructed. Tuning the free parameters of the rational solutions, localized waves of various patterns are obtained such as breather, lump-type localized waves and their hybrid structure. The anomalous inelastic interaction phenomenons of breather and lump-type localized waves are exhibited. Thirdly, combining the large-time behaviors of solution with the velocity relationship of localized waves, the dynamical properties and the classification of localized wave solutions are discussed in detail. Finally, we discuss the bound state of breather and lump-type localized waves under the velocity resonance condition, three different types of lump-breather molecules are displayed. The obtained results further enrich the structures and dynamical behaviors of localized waves. It is expected that the interaction phenomena taking place in the (3+1)-dimensional shallow water wave equation will be helpful in predicting or controlling some related shallow water wave phenomena.

Soliton-based modeling of nano-ionic currents in transmission line

Many nonlinear evolution equations, such as the nano-ionic currents (NIC) equation, are used extensively in many scientific and technological domains particularly in nanoelectronics and bioelectronics. The mathematical modeling of NIC phenomena is vital for understanding their behavior and optimizing device performance. Our research leverages an array of mathematical methods, including multi-wave analysis, periodic wave solutions, lump soliton dynamics, breather wave phenomena, homoclinic breathers, M-shaped waveforms, and rogue wave analysis. Additionally, our investigation encompasses the exploration of single kink and double kink configurations, interactions between periodic and kink waves, interaction between M shaped with kink and rogue, interaction between M shaped with one kink, interaction between M shaped with kink and periodic, interaction between M shaped with two kinks as well as periodic wave interactions with lump waves. To further emphasize the structure of solutions derived from particular parameter choices, we include three-dimensional, two-dimensional, streamplot, and contour graphs.

Multiple rogue wave, double-periodic soliton and breather wave solutions for a generalized breaking soliton system in (3 + 1)-dimensions

We focused on solitonic phenomena in wave propagation which was extracted from a generalized breaking soliton system in (3 + 1)-dimensions. The model describes the interaction phenomena between Riemann wave and long wave via two space variable in nonlinear media. Abundant double-periodic soliton, breather wave and the multiple rogue wave solutions to a generalized breaking soliton system by the Hirota bilinear form and a mixture of exponentials and trigonometric functions are presented. Periodic-soliton, breather wave and periodic are studied with the usage of symbolic computation. In addition, the symbolic computation and the applied methods for governing model are investigated. Through three-dimensional graph, density graph, and two-dimensional design using Maple, the physical features of double-periodic soliton and breather wave solutions are explained all right. The findings demonstrate the investigated model’s broad variety of explicit solutions. All outcomes in this work are necessary to understand the physical meaning and behavior of the explored results and shed light on the significance of the investigation of several nonlinear wave phenomena in sciences and engineering.

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.

Integrability and multiple kinks, lumps, and breathers solutions to an extended (3+1)-dimensional Calogero-Bogoyavlenskii-Schiff fluid model

This study focuses on analyzing a newly constructed extended (3+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) fluid model. The Painlevé test is employed to verify the integrability of this newly extended model. We demonstrate that the inclusion of additional terms does not kill the integrability of the standard model. Hirota’s bilinear approach is employed to formally determine multiple soliton \\kink solutions. In addition, we rigorously investigate the particular conditions of the parameters to provide lump solutions. In contrast to lump solutions, we obtain breather wave solutions without any requirement for constraints on the used parameters. Various techniques, including the family of tanh and tan procedures, are used to derive different traveling wave solutions with differing physical structures. The obtained solutions are examined and numerically discussed for several arbitrary functions.

Analytical solutions to (modified) Korteweg–de Vries–Zakharov–Kuznetsov equation and modeling ion-acoustic solitary, periodic, and breather waves in auroral magnetoplasmas

Analytical solutions to (modified) Korteweg–de Vries–Zakharov–Kuznetsov equation and modeling ion-acoustic solitary, periodic, and breather waves in auroral magnetoplasmas

Various dynamic behaviors for the concatenation model in birefringent fibers

This study explores various wave phenomena related to the concatenation model, which is characterized by the inclusion of the Kerr law of nonlinearity in birefringent fibers. Several distinct auxiliary functions and logarithmic transformation are utilized to formulate various analytical solutions, including multi-wave solutions, two solitary wave solutions, breather waves, periodic cross kink solutions, Peregrine-like rational solutions, and three-wave solutions. To demonstrate the influence of different parameters on the interaction of the obtained solutions, some figures are provided to vividly display these transmission and interaction characteristics.

Exploration of soliton structures in the Hirota–Maccari system with stability analysis

In this research, the modified extended tanh-function (METF) and the extended Jacobi elliptic function expansion (EJEFE) techniques are used to investigate the generation and detection of soliton structures in the Hirota–Maccari (HM) model. Consequently, we obtain soliton solutions with advanced structures, including singular bright soliton, dark soliton, periodic waves, breather waves, periodic breather waves, and multiple bright and dark breather waves. In addition, a lump-type breather wave is also included in the presented solutions. Stability analysis of the obtained solutions is addressed by employing the Hamiltonian technique. 3D surfaces and 2D visuals of the outcomes are represented with the help of a computer application. These findings contribute to understanding nonlinear wave phenomena with potential applications in optics, fluid dynamics, and plasma physics.

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.

The localized excitation on the Jacobi elliptic function periodic background for the Gross–Pitaevskii equation

In this paper, the nonlinear wave solutions for Gross–Pitaevskii equation on the periodic wave background are investigated by Darboux-Bäcklund transformation, from which the soliton and breather wave solutions on the Jacobi elliptic cn and dn functions backgrounds are derived. The corresponding evolutions and dynamical properties of nonlinear wave solutions under different parameters are discussed. These results reported in this paper may raise the possibility of related experiments and potential applications in nonlinear science fields, such as nonlinear optics, oceanography and so on.

WITHDRAWN: Lump, lump with damped oscillation, double cross kinky lump with oscillation, collision of kinky and breather wave solutions to a broadened Hietarinta-like equation

One of the most significant solutions to linear and nonlinear partial differential equations is a lump soliton as it has seismic wave nature. Realizing seismic natures of such lump and oscillating lump wave, one can take initiative to mitigate its harmful energies by opposite external periodic forces. The main purpose of this study is to demonstrate various lump solutions from kinky breather-wave. In this regard, the Hirota bilinear scheme and the enhanced homoclinic test procedure are utilized to get kinky breather-wave solutions with second-order linear dispersive components that can be produced by a fourth-order nonlinear term of the Hietarinta-like equation. By making particular values of certain parameters, we derived bright- dark lump solitons from kinky-breather propagation wave solutions in a specific scenario. The cross-kinky wave in breather form, lump with damped oscillation, kink with damped oscillation, as well as lump solutions are also obtained from the Hirota bi-linear scheme constructing breather waves. A set of 3D, 2D and density graphs are used to investigate such dynamical behaviors of model numerically. These patterns are also illustrated numerous clarifications for others prototypes in the purviews of nonlinear science and engineering.