Localized Wave Solutions Research Articles

Localized wave solutions to coupled variable-coefficient fourth-order nonlinear Schrödinger equations

This study investigates higher-order localized waves for coupled variable-coefficient fourth-order nonlinear Schrödinger equations, which are used to describe the simultaneous propagation of optical pulses in an inhomogeneous optical fiber. Based on the seed solutions and Lax pair, the Nth-order localized wave solutions are constructed. The interactions of rogue waves with dark–bright solitons are graphically analyzed via numerical simulation. The results are helpful for studying localized wave phenomena in nonlinear optics.

Superposition Formula of arbitrary functions to a (3+1)-demensional Boiti–Leon–Manna–Pempinelli equation

The (3+1)-dimensional Boiti–Leonon–Manna–Pempinelli equation was studied by using Hirota bilinear method. In this paper, we successfully derived the multi-solutions of this equation consisting of the superpositions of M arbitrary functions. A variety of new solutions can be obtained by choosing different values of the parameter M and test functions f in this paper, i.e., the localized solutions generated by M-positive quadratic functions, the mixed solutions consisting of the localized wave solutions and the kink wave solutions, and the mixed solutions consisting of a trigonometric function and exponential functions. Meanwhile, their corresponding physical plots were displayed to analyze their physical significance and dynamic properties. Some solutions from previous studies were found when M took a specific value and the test functions took particular functions. Thus, the received solutions are more general compared to the previous studies.

Dynamics of the breather and solitary waves in plasma using the generalized variable coefficient Gardner equation

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma.

Mixed localized matter wave solitons in Bose–Einstein condensates with time-varying interatomic interaction and a time-varying complex harmonic trapping potential

We study the generation of mixed localized matter wave solitons in the quasi-one-dimensional Gross–Pitaevskii equation with a complex potential (model equation) that describes the dynamics of Bose–Einstein condensates trapped in an harmonic potential when the loss/gain of condensate atoms is taken into account. Performing a modified lens-type transformation, the integrability condition is derived and the model equation is converted to a Kundu-like nonlinear Schrödinger equation. Through the linear stability analysis, the phenomenon of the modulational instability is analyzed and the criterion of the modulational instability is established. Based on the generalized perturbation (n,N−n)−fold Darboux transformation, new mixed localized wave solutions are presented and used for analyzing the generation of mixed matter-wave solitons in Bose–Einstein condensates with two-body interatomic interactions. We show that mixed localized waves generated with these exact solutions can change from a strong interaction to a weak interaction by choosing the parameters such as the similarity parameters, the spectrum parameter, or the seed solution parameter. Our results show that the spectrum parameter is useful for generating interesting structures which are beneficial to understanding the complex physical phenomena in Bose–Einstein condensates with atomic gain/loss.

On the dynamics of soliton solutions for the nonlinear fractional dynamical system: Application in ultrasound imaging

This article examines the third-order non-linear fractional Westervelt model by using a newly developed integration tool termed the new extended direct algebraic approach. The different wave structures to the considered model are secured in various forms, including bright, dark, and combo solitons. The application of wave structures is advantageous in the examination of sound wave propagation and high amplitude phenomena in the fields of medical imaging and therapy. These solutions are effective in facilitating ultrasound propagation in tissue, underwater acoustics, acoustic cavitation, acoustic levitation, and other related applications. The internal tissue of human beings can be visualized and studied through the use of ultrasound imaging technologies in the field of medicine. This technology possesses numerous uses in both industrial and medicinal sectors. There has been a growing interest in fractional nonlinear partial differential equations due to their ability to describe various complex occurrences and exhibit more dynamic architectures of localized wave solutions. On employing accurate parameter values, multiple graphical representations are generated to provide the visual depiction of the acquired results. The results of this study indicate that the chosen methodology effectively improves nonlinear dynamical processes. The findings indicate that the selected methodology demonstrates efficacy, feasibility, and versatility when applied to intricate systems across several domains, with a special emphasis on ultrasonic imaging. The findings indicate that the system exhibits a potentially significant abundance of soliton structures.

Localized wave solutions and localized-kink solutions to a (3+1)-dimensional nonlinear evolution equation

Localized wave solutions and localized-kink solutions to a (3+1)-dimensional nonlinear evolution equation

Localized wave solutions and interactions of the (2+1)-dimensional Hirota–Satsuma–Ito equation

This paper studies the (2+1)-dimensional Hirota–Satsuma–Ito equation. Based on an associated Hirota bilinear form, lump-type solution, two types of interaction solutions, and breather wave solution of the (2+1)-dimensional Hirota–Satsuma–Ito equation are obtained, which are all related to the seed solution of the equation. It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons, and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton. Furthermore, the breather wave solution is also obtained by reducing the two-soliton solutions. The trajectory and period of the one-order breather wave are analyzed. The corresponding dynamical characteristics are demonstrated by the graphs.

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.

DYNAMICS OF OPTICAL WAVE PROFILES TO THE FRACTIONAL THREE-COMPONENT COUPLED NONLINEAR SCHRÖDINGER EQUATION

This paper explores the truncated [Formula: see text]-fractional three-component coupled nonlinear Schrödinger (tc-CNLS) equation, that regulates the behavior of optical pulses in optical fibers. These equations are utilized in various scientific and engineering fields, including nonlinear fiber optics, electromagnetic field waves, and signal processing through optical fibers. The study of multi-component NLS equations has gained significant attention due to their ability to elucidate various complex physical phenomena and exhibit more dynamic structures of localized wave solutions. The freshly invented integration tools, known as the fractional modified Sardar subequation method (MSSEM) and fractional enhanced modified extended tanh-expansion method (eMETEM), are employed to ensure the solutions. The study focuses on extracting various types of optical solitons, including bright, dark, singular, bright-dark, complex, and combined solitons. Optical soliton propagation in optical fibers is currently a subject of great interest due to the multiple prospects for ultrafast signal routing systems and short light pulses in communications. In nonlinear dispersive media, optical solitons are stretched electromagnetic waves that maintain their intensity due to a balance between the effects of dispersion and nonlinearity. Furthermore, hyperbolic, periodic and exponential solutions are generated. The utilized methodology is effective in explaining fractional nonlinear partial differential equations (FNLPDEs) as it offers pre-existing solutions and additionally derives novel exact solutions by mixing outcomes from various procedures. Furthermore, we plot the visualizations of solutions by plotting 3D, 2D, and contour graphs with the corresponding parameter values. The findings of this paper can improve the understanding of the nonlinear dynamical behavior of a specific system and demonstrate the efficacy of the methodology used. We anticipate that our study will provide substantial benefits to a considerable group of engineering model experts. The findings demonstrate the efficacy, efficiency, and applicability of the computational method employed, particularly in dealing with intricate systems.

Various types of discrete exact localized wave solutions and dynamical analysis for the discrete complex modified Korteweg-de Vries equation

Under consideration is the second-order integrable discretization of complex modified Korteweg-de Vries (mKdV) equation which is regarded as the discrete counterpart of the mKdV equation having an essential role in describing the propagation behavior of water waves and acoustic waves in nonlinear media. First of all, based on the known linear spectral problem, the discrete generalized (n,N−n)-fold Darboux transformation is constructed to derive various types of discrete exact localized wave solutions, including soliton and semi-rational soliton solutions on vanishing background, breather, rogue wave and hybrid interaction solutions on plane wave background, and rational soliton solution on constant background, and the relevant evolution structures are studied graphically. Secondly, the asymptotic analysis is used to discuss the elastic interaction for two-soliton solutions and limit states for rational soliton solutions. Finally, the numerical simulation is utilized to investigate the dynamical behavior and propagation stability of some exact solutions. The findings presented in this paper may contribute to explaining physical phenomena described by the mKdV equation.

Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones

In this paper, we study data-driven localized wave solutions and parameter discovery in the massive Thirring (MT) model via the deep learning in the framework of physics-informed neural networks (PINNs) algorithm. Abundant data-driven solutions including soliton of bright/dark type, breather and rogue wave are simulated accurately and analyzed contrastively with relative and absolute errors. For higher-order localized wave solutions, we employ the extended PINNs (XPINNs) with domain decomposition to capture the complete pictures of dynamic behaviors such as soliton collisions, breather oscillations and rogue-wave superposition. In particular, we modify the interface line in domain decomposition of XPINNs into a small interface zone and introduce the pseudo initial, residual and gradient conditions as interface conditions linked adjacently with individual neural networks. Then this modified approach is applied successfully to various solutions ranging from bright-bright soliton, dark-dark soliton, dark-antidark soliton, general breather, Kuznetsov-Ma breather and second-order rogue wave. Experimental results show that this improved version of XPINNs reduce the complexity of computation with faster convergence rate and keep the quality of learned solutions with smoother stitching performance as well. For the inverse problems, the unknown coefficient parameters of linear and nonlinear terms in the MT model are identified accurately with and without noise by using the classical PINNs algorithm.

Bilinear auto-Bäcklund transformations and the hybrid localized wave solutions for the ($$3+1$$)-dimensional B-type Kadomtsev–Petviashvili equation

Bilinear auto-Bäcklund transformations and the hybrid localized wave solutions for the ($$3+1$$)-dimensional B-type Kadomtsev–Petviashvili equation

Interacting Solitons, Periodic Waves and Breather for Modified Korteweg–de Vries Equation

We theoretically demonstrate a rich and significant new families of exact spatially localized and periodic wave solutions for a modified Korteweg–de Vries equation. The model applies for the description of different nonlinear structures which include breathers, interacting solitons and interacting periodic wave solutions. A joint parameter which can take both positive and negative values of unity appeared in the functional forms of those closed form solutions, thus implying that every solution is determined for each value of this parameter. The results indicate that the existence of newly derived structures depend on whether the type of nonlinearity of the medium should be considered self-focusing or defocusing. The obtained nonlinear waveforms show interesting properties that may find practical applications.

Two-component generalized nonlinear Schrödinger equations and their soliton and breather solutions

In this paper, a cluster of two-component generalized nonlinear Schrödinger equations is generated in terms of zero-curvature equation and polynomial expansion of the spectral parameter, which can be used to illustrate effects on various nonlinear phenomena. After that, explicit representations of classical Darboux transformation and generalized perturbation (n,M)-fold Darboux transformation of the above equations are constructed. Localized wave solutions including soliton solutions, degenerate soliton solutions, soliton solutions interacting with degenerate soliton solutions, breather solutions and degenerate breather solutions are subsequently acquired via the classical Darboux matrix. Meanwhile, degenerate soliton solutions are acquired by the generalized perturbation (n,M)-fold Darboux transformation. Analyses of these solutions are shown through a series of figures ultimately.

Novel resonant multi-soliton solutions of time fractional coupled nonlinear Schrödinger equation in optical fiber via an analytical method

The aim of this article is to investigate the time fractional coupled nonlinear Schrödinger equation (TFCNLSE) which can be used to describe the interaction among the modes in nonlinear optics and Bose–Einstein condensation. The TFCNLS equation plays a crucial role in soliton wavelength division multiplexing and pulse propagation through a two-mode optical fiber. For the sake of getting different analytical solutions, we capitalize modified version of the extended tanh-expansion method. Fractional nonlinear partial differential equations (FNLPDEs) have become popular because they can explain complex physical phenomena and have dynamic structures of localized wave solutions. The study obtained various soliton solutions including bright, dark, periodic, and singular solitons. The results are presented in graphical form with appropriate parameter values to aid visualization. Such solutions are critical for explaining some mesmerizing and complex physical phenomena. In addition, some results from earlier studies are generalized which justify the novelty of the current study. This study proves that the computational method used is efficient, brief, and widely applicable in finding analytical solutions to diverse complex nonlinear problems arising in the recent era of nonlinear optics, applied sciences and engineering.

Computational simulations of propagation of a tsunami wave across the ocean

The 2D regularized long-wave equation is a mathematical model used to describe the behavior of water waves in two dimensions, taking into account nonlinear and dispersive effects. To obtain soliton wave solutions, computational methods have been employed to solve this equation. Solitons are localized wave solutions that maintain their shape and speed over long distances, making them useful in various applications, such as optical fiber communication and modeling water waves. This study utilized the Khater II, septic-B-spline, and Adomian decomposition techniques to solve the 2D regularized long-wave equation and obtain soliton wave solutions. The results showed a wide range of soliton solutions, including bright and dark solitons as well as soliton pairs and chains. Furthermore, we investigated the impact of the regularization parameter on the soliton solutions and observed that it affects the soliton speed and shape. Our findings demonstrate how computers can be utilized to obtain new soliton wave solutions for the 2D regularized long-wave equation, which have implications for various fields, such as oceanography, engineering, and physics. By understanding how solitons behave within the context of this equation, we can create more accurate models of water waves and other physical systems.

A variety of M-truncated optical solitons to a nonlinear extended classical dynamical model

This article secures optical pulses modeled by higher order generalized extended classical nonlinear Schrödinger equation (GECNLSE). The studied equation is discussed by the assistance of truncated M-fractional derivative and composed of self-steepening, and stimulated Raman scattering effects in nonlinear optical fibers. Because of their versatility in explaining a wide range of complicated physical events and their richer dynamical structures of localized wave solutions, the NLSE equations have received increased attention. The solutions are extracted with the assistance of modified Sardar sub-equation method (MSSEM), one of the newest integration methods. A variety of optical solitons are extracted like, bright, dark, combo, and singular soliton solutions. Furthermore, periodic, exponential, and hyperbolic type solutions are obtained. Assuming the right values for the parameters, various graph shapes are sketched to provide information about the visual format of the earned results. This paper’s findings support the efficacy of the approach taken in enhancing nonlinear dynamical behavior. We believe this research will be of interest to a wide variety of engineers that work with engineering models. Findings show the effectiveness simplicity, and generalizability of the chosen computational approach, even when applied to complicated systems.

Localized wave solutions to the vector nonlinear Schrödinger equation with nonzero backgrounds

The vector nonlinear Schrödinger (NLS) system is studied through the generalized Darboux transformation (DT). We begin with the nonzero seed solution in the exponential form of evolution variable and explore the eigenfunction of the ‐component NLS equation. With the formulae of the eigenfunction, localized wave solutions of the two‐component and three‐component NLS equations are studied. By choosing different free parameters, some special solutions and the dynamic evolutions are presented, including breather–rogue wave, breather fission, breather fusion, and breather–soliton.

Vector localized and periodic waves for the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation

Dynamical properties of vector localized and periodic waves hold significant importance in the study of physical systems. In this work, we investigate the matrix Hirota equation with sign-alternating nonlinearity via the binary Darboux transformation. For the two interacting components, we construct the binary Darboux transformation formulas, vector localized, and periodic wave solutions. Via those solutions, different kinds of nonlinear waves can be achieved, including rogue waves, solitons, positons, and periodic waves. When the imaginary part of the spectral parameter is not zero, eye-shaped rogue waves appear in one component, and the twisted rogue wave pairs in the other component. As the spectral parameter is real, we derive distinct forms of vector localized and periodic waves on the non-zero background, such as the vector solitons, positons, periodic waves, breathers on the periodic wave background, and rational solitons. These results may be valuable in this investigation of nonlinear waves in physical systems.

Localized wave solutions and mixed interaction structures in the AB system

For AB system, in the first place we calculate the modulational instability for the possible reason of the formation of localized wave. Secondly, based on the generalized (n,N−n)-fold Darboux transformation, we construct order-n degenerate soliton solutions and their interaction solutions sitting on zero background of this system. Specifically, we discuss the three cases of degenerate soliton with interaction solutions obtained above in detail: (a) For order-1 degenerate soliton solutions, we show that they can be decomposed into two order-1 solitons with variable “phase shift”. (b) For higher-order degenerate soliton solutions, we achieve the bright-dark transformation of these solutions by changing the positive and negative of eigenvalues λi. Then a number of the corresponding structural maps are clearly displayed. (c) For interaction solutions with soliton and degenerate soliton solutions, we study the collision under various structures, and these collision characteristics are significantly different from those before. At last, we also sum up various mathematical features of degenerate soliton and mixed interaction solutions.