The study on peaked waves and nonlinear Jacobi elliptic function based solitons formulation with mixed dispersion to asymptotic scaling system

This work represents a novel integration of both frameworks in a single analytical context by using the modified extended (ME) mapping approach to get new, exact solutions of the (2+1)-dimensional Kadomtsev-Petviashvili and Shallow Water Wave equations. This model is essential for simulating multidimensional wave propagation processes, and it has several applications in shallow water hydrodynamics, plasma physics, nonlinear optics, and the investigation of long internal waves in stratified fluids. Bright soliton, dark soliton, singular soliton, and singular periodic, periodic, hyperbolic, exponential, Jacobi elliptic (JE), and Weierstrass elliptic doubly periodic solutions are among the many unique structures that are constructed in this study. The dynamical resilience of the generated solutions is evaluated by means of a linear stability study and bifurcation analysis. Crucially, the findings provide fresh information on nonlinear wave collision processes by demonstrating that the soliton interactions either behave elastically or preserve the soliton under particular conditions. The spatiotemporal dynamics and interactions of the resulting solutions are shown through a succession of 2D, 3D, and contour plots, which enhance the findings' physical understanding and application. A new theoretical understanding of nonlinear wave propagation in fluids and related physical systems is provided by this work.

This study presents a tripartite analytical framework for the (1+1)-dimensional nonlinear Klein–Fock–Gordon equation, a key model for spinless particles in relativistic quantum mechanics. The investigation begins with a Painlevé analysis showing that the equation is completely integrable via the Painlevé test by using Maple. Subsequently, classical Lie symmetry analysis is employed to derive the infinitesimal generators of the equation. A Lagrangian formulation is constructed for these generators, from which similarity variables are systematically obtained. This framework enables a complete similarity reduction, transforming the complex nonlinear partial differential equation into a more tractable ordinary differential equation. To solve this reduced ordinary differential equation and to obtain a spectrum of soliton solutions, we implement the new generalized exponential differential rational function method. This advanced technique utilizes a rational trial function based on the ith derivatives of exponentials, generating a diverse spectrum of closed-form soliton solutions through strategic choices of arbitrary constants. The novelty of this approach provides a unified framework for handling higher-order nonlinearities, yielding solutions such as multi-peakons and lump solitons, which are vividly characterized using Mathematica-generated 3D, 2D, and contour plots. These findings provide significant insights into nonlinear wave dynamics with potential applications in quantum field theory, nonlinear optics, plasma physics, etc.

A novel analytical method, termed the Variable Coefficient Second Degree Generalized Abel Equation Method, is presented in this paper. It has been specifically developed to address the complexities of the Gilson Pickering equation—an important nonlinear partial differential equation encountered in various physical applications. This equation is recognized as a general case of several significant nonlinear partial differential equations, including the Fornberg Whitham, Rosenau Hyman, and Fuchssteiner-Fokas-Camassa-Holm equations. Unlike conventional approaches, which are typically based on constant coefficient ordinary differential equations (ODEs) or auxiliary equations, a unique framework based on variable coefficient ODEs integrated within a subequation structure is introduced by this method.. By applying the method to the Gilson Pickering equation, new exact analytical solutions are derived. The correctness of the technique is validated, and its superior efficiency and robustness are highlighted through these results. The intricate dynamics of the equation are effectively captured, demonstrating the method’s suitability for modeling phenomena in fluid dynamics, nonlinear optics, and wave propagation. Furthermore, the potential application of the method to a broader class of nonlinear partial differential equations across mathematical physics is implied by its generalizability. The importance of advancing analytical tools to better understand and resolve complex physical systems is emphasized by the findings, thereby marking a significant contribution to the analytical study of nonlinear differential equations. Ultimately, an expansion of the repertoire of solution strategies available to researchers in applied mathematics and theoretical physics is achieved through this work.

In this paper, the dynamical behaviors of transformed nonlinear waves for the (2+1)-dimensional combined potential Kadomtsev-Petviashvili and B-type Kadomtsev-Petviashvili (pKP-BKP) equation are investigated, which can be used to reveal the nonlinear wave phenomena in nonlinear optics, plasma physics and hydrodynamics. The breath-wave and the lump solutions are constructed by means of the soliton solutions. The conversion mechanism for the breath-wave is systematically analyzed, which leads to several new kink-shaped nonlinear waves. The gradient relationships of these transformed waves are revealed by a Riemannian circle. Through the analysis of the nonlinear superposition between the periodic wave component and the kink solitary wave component, the dynamical characteristics including the formation mechanism, oscillation and locality for the nonlinear waves are investigated. The time-varying properties of transformed waves are shown by the study of time variables. By virtue of the two breath-wave solutions, several interactions including elastic and inelastic collisions between two nonlinear waves are studied. In particular, some transformed molecular waves encompassing the non-, semi- and full-transition modes are presented with the aid of velocity resonance. The results can help us further understand the complex nonlinear waves existing in the integrable systems.

In this study, we examine the third-order fractional nonlinear Schrödinger equation (FNLSE) in \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(1+1)$$\\end{document}-dimensional, by employing the analytical methodology of the new extended direct algebraic method (NEDAM) alongside optical soliton solutions. In order to better understand high-order nonlinear wave behaviors in such systems, the researched model captures the physical and mathematical properties of nonlinear dispersive waves, with applications in plasma physics and optics. With the aid of above mentioned approach, we rigorously assess the novel optical soliton solutions in the form of dark, bright–dark, dark–bright, periodic, singular, rational, mixed trigonometric and hyperbolic forms. Additionally, stability assessments using conserved quantities, such as Hamiltonian property, and consistency checks were used to validate the solutions. The dynamic structure of the governing model is further examined using chaos, bifurcation, and sensitivity analysis. With the appropriate parameter values, 2D, 3D, and contour plots can all be utilized to graphically show the data. This work advances our knowledge of nonlinear wave propagation in Bose–Einstein condensates, ultrafast fibre optics, and plasma physics, among other areas with higher-order chromatic effects.

This study investigates the Kaup–Newell equation, a nonlinear Schrödinger-type model with important applications in plasma physics and nonlinear optics, particularly in modeling sub-picosecond pulses. By employing the modified direct algebraic method and the advanced [Formula: see text]-expansion method, we derive a broad spectrum of soliton solutions, including hyperbolic, trigonometric, shock, singular, and mixed types. These solutions, validated using Maple software, enhance the understanding of nonlinear wave behavior. Additionally, the energy balance method is applied to examine traveling wave dynamics, yielding periodic solution in cosine form. Graphical representations in 2D, 3D, and contour plots are provided to visualize the solution profiles. The results not only demonstrate the efficacy of the proposed methods but also contribute novel insights into the Kaup–Newell model, with potential applicability to a wider class of nonlinear partial differential equations.

This manuscript deals with the Fourth-order Boussinesq water wave equation, which is integrable and possesses soliton solutions. Boussinesq water wave equation is a vital tool for investigating nonlinear phenomena in various waves and shallow water phenomena in fluid dynamics, such as diffraction, refraction, weak nonlinearity, and shoaling. Along with fluid dynamics, it is essential in many disciplines of physics, including the transmission of long waves in shallow waters, vibrations in a nonlinear string, acoustics, laser optics, and one-dimensional nonlinear lattice waves. The Generalized Arnous approach, the new Kudryashov method, and the Modified Sub-equation method are applied to this objective. The resultant diverse solutions consist of trigonometric and hyperbolic functions. These approaches generate accurate analytical curves for soliton waves, which comprise kink, bright, and dark waves. The graphical aspects of the produced solutions are investigated using 3D-surface graphs, 2D-line graphs, and contour and polar plots, in addition to theoretical derivations. This work is novel in its integrated use of three symbolic methods to derive a broad spectrum of exact soliton solutions for the fourth-order Integrated Boussinesq water wave equation, including compound and hybrid waveforms. The inclusion of the graphical visualization, stability analysis, and open source code resources further strengthens its contribution to nonlinear wave modeling.

Solitons, nonlinear wave transitions and characteristics of quasi-periodic waves for a (3+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff equation in fluid mechanics and plasma physics

In this paper, we succeed at discovering the new exact wave solutions to the truncated M-fractional complex three coupled Maccari’s system by utilizing the Sardar sub-equation scheme. The obtained solutions are in the form of trigonometric and hyperbolic forms. These solutions have many applications in nonlinear optics, fiber optics, deep water-waves, plasma physics, mathematical physics, fluid mechanics, hydrodynamics and engineering, where the propagation of nonlinear waves is important. Achieved solutions are verified with the use of Mathematica software. Some of the achieved solutions are also described graphically by 2-dimensional, 3-dimensional and contour plots with the help of Maple software. The gained solutions are helpful for the further development of a concerned model. Finally, this technique is simple, fruitful and reliable to handle nonlinear fractional partial differential equations (NLFPDEs).

In this paper, by using the Hirota bilinear method, we investigate elastic and resonant interactional solutions for the (2+1)-dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation, which is a fundamental model equation governing diverse nonlinear phenomena in fluid dynamics, plasma physics, nonlinear optics and so on. The introduced logarithmic transformation with a function of y facilitates the creation of more intricate and adaptable solution frameworks. We study three different cases of soliton interactions: the interactions between one breather and one kink soliton, the interactions between one breather and two kink solitons, and the interactions between two breathers. Through comprehensive asymptotic analysis and graphical representations, we elucidate the dynamic characteristics and fundamental properties of these nonlinear wave solutions. These solutions could uncover some novel nonlinear interactional patterns and shed light on the theoretical comprehension for the localized wave dynamics in higher-dimensional nonlinear systems.

A unified concatenation model for plasma physics: Integrability and soliton solutions

Odd-soliton solutions and inelastic interaction for the differential–difference nonlinear Schrödinger equation in nonlinear optics

This article investigates the stochastic Davey–Stewartson equations influenced by multiplicative noise within the framework of the Ithat{o} calculus. These equations are of significant importance because they extend the nonlinear Schrödinger equation into higher dimensions, serving as fundamental models for nonlinear phenomena in plasma physics, nonlinear optics, and hydrodynamics. This paper is motivated by the need to understand how random fluctuations affect soliton behavior in nonlinear systems. This is particularly relevant in applications such as turbulent plasma waves and optical fibers, where noise can significantly impact wave propagation. We employ the modified extended direct algebraic method for finding exact stochastic soliton solutions to the stochastic Davey–Stewartson equations. The study derives a class of exact stochastic soliton solutions, including dark, singular, rational, and periodic waves. MATLAB is used to provide visual representations of these stochastic soliton solutions through 3D surface plots, contour plots, and line plots. These solutions offer essential insights into how random disturbances influence nonlinear wave systems, particularly in turbulent plasma waves and optical fibers. To the best of our knowledge, the application of the modified extended direct algebraic method to the stochastic Davey–Stewartson equations with multiplicative noise, along with the subsequent analysis of the stabilizing effects on dark, singular, rational, and periodic stochastic soliton solutions is novel. The study demonstrates how multiplicative Brownian motion regulates these wave structures, providing new information on the impact of noise on higher-dimensional nonlinear systems.

In this paper, the Jacobi elliptic function expansion technique is used to obtain the exact solutions of the sixth order (3+1)-dimensional Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation. Modulation instability is also discussed for this equation. The main purpose is to find novel exact solutions to this equation by means of a finite series expansion of degree n in terms of Jacobi elliptic functions. Single and combined Jacobi elliptic function solutions are obtained. The JEFE method is found to be highly effective for exact analytical solutions of nonlinear partial differential equations and its flexibility permits the development of several variations for specific problem types. The studied equation is reduced to nonlinear ordinary differential equation of integer order by using the traveling wave transformation. We observe that the solutions obtained are precise, and include periodic wave solutions, quasi-periodic wave solutions and solitary waves. Oscillatory phenomena in systems such as plasma physics and optics can be described by periodic wave solutions. Quasi periodic solutions occur in complex systems with multiple interacting frequencies, which are important in turbulence and nonlinear resonance. Solitary waves (solitons) are stable, localized waves that are critical to fluid dynamics, nonlinear optics, and plasma physics, and that model stable wave propagation in many applications. In addition, graphical representations of some solutions are presented to show the direct viewing analysis of the solutions. The results confirm that the proposed technique is a powerful tool for solving a large variety of NPDEs in mathematical physics, and may have applications to other nonlinear evolution equations.

In this paper, the nonlinear coupled system of partial differential equations named the Wu-Zhang system investigated by applying the new auxiliary equation method. The Wu-Zhang system help us to investigate the various nonlinear wave propagation phenomena physically including the width and amplitude of solitons, physically form of shock, traveling and solitary wave structures in fiber optics, fluid dynamics, plasma physics, nonlinear optics, these nonlinear wave equations play significant role in these phenomenas. In this regard, the dispersive long wave is described by the Wu-Zhang system, from which a number of solitons and solitary wave structure are formally extracted as an accomplishment. On the basis of the computational program Mathematica, soliton and many other solitary wave results have been obtained with the ability to use an analytical approach. Consequently, various solutions in solitons and solitary waves are generated in rational, trigonometric, and hyperbolic functions and displayed within contour, two-dimension and three-dimension plotting by using the numerical simulation. The soliton solutions are obtained including bright and dark solitons, anti-kink wave solitons, peakon bright and dark solitons, kink wave soliton, periodic wave solitons, solitary wave structure, and other mixed solitons. In order to comprehend the significance of investigating various nonlinear wave phenomena in engineering and science, including soliton theory, nonlinear optics, fluid mechanics, material energy, water wave mechanics, mathematical physics, signal transmission, and optical fibers, all research outcomes are required. With precise analytical results, shed light that the applied approach to be more powerful, dependable, and accurate.