Dimensional Wave Equation Research Articles

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

The exact solutions to the generalized (2+1)-dimensional nonlinear wave equation

Due to the importance of the nonlinear partial differential equations in applied physics and engineering, many mathematicians and physicists are interesting to the nonlinear partial differential equations. One of the main tasks of studying the nonlinear partial differential equations is to obtain the exact solutions for the nonlinear partial differential equations. In this paper, we study the exact solutions of a generalized (2+1)-dimensional nonlinear partial differential equation. According to the modified hyperbolic function method and the traveling transformation method, the generalized (2+1)-dimensional nonlinear partial differential equation is changed into an ordinary differential equation. By taking the homogeneous balance between the highest order nonlinear terms and the highest order derivative terms, we change the differential equation into an algebraic system. By solving the algebraic system, we obtain the solutions for the nonlinear partial differential equation. At last, some solutions and their figures are given by specific parameters. All the solutions conclude the trigonometric function solutions, the positive hyperbolic function solutions, and the hyperbolic trigonometric function solutions. These solutions exhibit the dynamics of nonlinear waves and the solutions are useful for the study on interaction behavior of nonlinear waves in shallow water, plasma, nonlinear optics and Bose–Einstein condensates.

Dynamics characteristics of soliton structures of the new (3 + 1) dimensional integrable wave equations with stability analysis

This study investigates optical solitons solutions within the framework of the two new (3 + 1)-dimensional integrable wave equations by employing the modified Sardar sub-equation method. The importance of these suggested models span diverse fields including geophysics, seismology, medical imaging, photonics and sensor development. The derived solutions, meticulously verified using Mathematica, encompass a rich array of mathematical functions including trigonometric, hyperbolic, and exponential functions. Visualization techniques, such as 3D plots, 2D plots, density graphs, contour graphs and polar graphs, effectively illustrate the diverse behaviors exhibited by these soliton solutions. These behaviors include lump, periodic, kink, dark, bright, peakons, cuspons, compactons soliton waves and other complex phenomena. Moreover the modulation instability of the consider model is examined, and corresponding conditions are established. The soliton solutions highlight the efficiency and effectiveness of these method in discovering traveling wave solutions, presenting a valuable tool for addressing a range of nonlinear evolution equations arising in diverse scientific fields such as hydrodynamics, optical fiber communications, plasma physics, ocean engineering, nonlinear dynamics, and condensed matter physics.

Resonance solitons, soliton molecules and hybrid solutions for a ($$\mathbf{2+ 1}$$)-dimensional nonlinear wave equation arising in the shallow water wave

Resonance solitons, soliton molecules and hybrid solutions for a ($$\mathbf{2+ 1}$$)-dimensional nonlinear wave equation arising in the shallow water wave

Travelling-wave, Mixed-lump-kink and Mixed-rogue-wave-kink Solutions for an Extended (3+1)-dimensional Shallow Water Wave Equation in Oceanography and Atmospheric Science

Travelling-wave, Mixed-lump-kink and Mixed-rogue-wave-kink Solutions for an Extended (3+1)-dimensional Shallow Water Wave Equation in Oceanography and Atmospheric Science

Some Novel Fusion and Fission Phenomena for an Extended (2+1)-Dimensional Shallow Water Wave Equation

An extended (2+1)-dimensional shallow water wave (SWW) model which can describe the evolution of nonlinear shallow water wave propagation in two spatial and temporal coordinates, is systematically studied. The multi-linear variable separation approach is addressed to the extended (2+1)-dimensional SWW equation. The variable separation solution consisting of two arbitrary functions is obtained, by assumption, from a specific ansatz. By selecting these two arbitrary functions as the exponential and trigonometric forms, resonant dromion, lump, and solitoff solutions are derived. Meanwhile, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions, and periodic waves are studied with graphical and analytical methods. The results can be used to enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

Lump waves, bright-dark solitons and some novel interaction solutions in (3+1)-dimensional shallow water wave equation

Abstract The (3+1)-dimensional Geng equation is an extended version of the KdV model that describes the wave dynamics behavior of shallow water waves in complex applications. In this study, we discuss the (3+1)-dimensional Geng equation using the bilinear neural network method. By incorporating specific activation functions into the neural network model, new test functions are constructed. Using symbolic computational techniques and selecting appropriate parameters, we systematically obtain new meaningful exact solutions of some (3+1)-dimensional Geng equations, including dark lump solutions, three kinds of interaction solutions, and bright and dark soliton solutions. Furthermore, the results are visualized through diagrams of different categories, which intuitively demonstrate the evolution process and physical characteristics of the waves.

Studies on an extended (3+1)-dimensional variable-coefficient shallow water wave equation: Bilinear form, Pfaffian solutions and Solitonic interactions

Studies on an extended (3+1)-dimensional variable-coefficient shallow water wave equation: Bilinear form, Pfaffian solutions and Solitonic interactions

A Non-Local wave equation with General fractional derivatives and time delay

We analyze, spatially one dimensional, non-local elastic body with deformation measure given in the form of General fractional derivative of Riesz type with truncated power-law kernel. It is assumed that stress at the point x and time t depends on the strain at point ξ at time t−|x−ξ|/c, where c is a constant having dimension of time. We formulate spatially one dimensional wave equation for such a body and show the influence of the parameter c on the solution.

Soliton interactions, soliton bifurcations and molecules, breather molecules, breather-to-soliton transitions, and conservation laws for a nonlinear (3+1)-dimensional shallow water wave equation

Soliton interactions, soliton bifurcations and molecules, breather molecules, breather-to-soliton transitions, and conservation laws for a nonlinear (3+1)-dimensional shallow water wave equation

Variational approach for (3+1)-dimensional shallow water wave equation

This paper focuses on the variational approach for solving the (3+1)-dimensional shallow water wave equation. Some families of soliton-type solutions and periodic wave solutions are given, and some remarks about the variational formulation for the special (2+1)-dimensional shallow water wave equation in [Gu, et al. Results in Physics, 45 (2023) 106213] are provided. Numerical results are presented to show the propagation behaviour.

Localized excitation and fractal structures of a (2 + 1)-dimensional Longwater wave equation

Localized excitation and fractal structures of a (2 + 1)-dimensional Longwater wave equation

A novel analysis of Cole–Hopf transformations in different dimensions, solitons, and rogue waves for a (2 + 1)-dimensional shallow water wave equation of ion-acoustic waves in plasmas

This work investigates a (2 + 1)-dimensional shallow water wave equation of ion-acoustic waves in plasma physics. It comprehensively analyzes Cole–Hopf transformations concerning dimensions x, y, and t and obtains the dispersion for a phase variable of this equation. We show that the soliton solutions are independent of the different logarithmic transformations for the investigated equation. We also explore the linear equations in the auxiliary function f present in Cole–Hopf transformations. We study this equation's first- and second-order rogue waves using a generalized N-rogue wave expression from the N-soliton Hirota technique. We generate the rogue waves by applying a symbolic technique with β and γ as center parameters. We create rogue wave solutions for first- and second-order using direct computation for appropriate choices of several constants in the equation and center parameters. We obtain a trilinear equation by transforming variables ξ and y via logarithmic transformation for u in the function F. We harness the computational power of the symbolic tool Mathematica to demonstrate the graphics of the soliton and center-controlled rogue wave solutions with suitable choices of parameters. The outcomes of this study transcend the confines of plasma physics, shedding light on the interaction dynamics of ion-acoustic solitons in three-dimensional space. The equation's implications resonate across diverse scientific domains, encompassing classical shallow water theory, fluid dynamics, optical fibers, nonlinear dynamics, and many other nonlinear fields.

Data-driven wave solutions of (2+1)-dimensional nonlinear dispersive long wave equation by deep learning

In this paper, we introduce a deep learning framework to solve the (2+1)-dimensional nonlinear dispersive long wave equation. Using it, we study the data-driven solitary wave solutions, periodic wave solutions, kink wave solutions and various multiple soliton solutions for the equations with initial–boundary value conditions. On the one hand, we obtain the data-driven traveling wave solutions of the equations, which have high accuracy compared with corresponding exact traveling wave solutions. On the other hand, we set more complicated initial–boundary value problems of the equations, whose exact traveling wave solution cannot be easy to obtain, but our network still works well and gives its numerical solution. The results show that our deep learning algorithm is able to accurately capture the dynamical behavior of traveling waves of the (2+1)-dimensional nonlinear dispersive long wave equation.

Wellposedness for a (1+1)-dimensional wave equation with quasilinear boundary condition

We consider the linear wave equation on with initial conditions and a nonlinear Neumann boundary condition at x = 0. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where f is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at x = 0 have enough regularity to solve the initial value problem with their own initial data.

On the unique ergodicity for a class of 2 dimensional stochastic wave equations

We study the global-in-time dynamics for a stochastic semilinear wave equation with cubic defocusing nonlinearity and additive noise, posed on the 2 2 -dimensional torus. The noise is taken to be slightly more regular than space-time white noise. In this setting, we show existence and uniqueness of an invariant measure for the Markov semigroup generated by the flow over an appropriately chosen Banach space. This extends a result of the second author [Comm. Math. Phys. 377 (2020), pp. 1311–1347] to a situation where the invariant measure is not explicitly known.

Deep neural networks learning forward and inverse problems of two-dimensional nonlinear wave equations with rational solitons

In this paper, we investigate the forward problems on the data-driven rational solitons for the (2+1)-dimensional Kadomtsev–Petviashvili-I (KP-I) equation and spin-nonlinear Schrödinger (spin-NLS) equation via the deep neural networks learning. Moreover, the inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations by optimizing the chosen loss functions related to the considered nonlinear wave equations.

Covariance models and Gaussian process regression for the wave equation. Application to related inverse problems

In this article, we consider the general task of performing Gaussian process regression (GPR) on pointwise observations of solutions of the 3 dimensional homogeneous free space wave equation. In a recent article, we obtained promising covariance expressions tailored to this equation: we now explore the potential applications of these formulas. We first study the particular cases of stationarity and radial symmetry, for which significant simplifications arise. We next show that the true-angle multilateration method for point source localization, as used in GPS systems, is naturally recovered by our GPR formulas in the limit of the small source radius. Additionally, we show that this GPR framework provides a new answer to the ill-posed inverse problem of reconstructing initial conditions for the wave equation from a limited number of sensors, and simultaneously enables the inference of physical parameters from these data. We finish by illustrating this “physics informed” GPR on a number of practical examples.

A Generalized (3+1)-Dimensional Nonlinear Wave Equation in Liquid with Gas Bubbles: Symmetry Reductions; Exact Solutions; Conservation Laws

The analysis of a generalised (3+1)-dimensional nonlinear wave equation that simulates a variety of nonlinear processes that occur in liquids including gas bubbles will be performed. After some cosmetic adjustments to the underlying equation, this generalised (3+1)-dimensional nonlinear wave equation naturally degenerates into the (3+1)-dimensional Kadomtsev-Petviashvili equation, the (3+1)-dimensional nonlinear wave equation, and the Korteweg-de Vries equation. To completely investigate this fundamental equation, a clear and rigorous technique is used. In order to obtain innovative symmetry reductions, group invariant solutions, conservation laws, and eventually kink wave solutions, the Lie symmetry, multiplier, and simplest equation methods are used. Complex waves and their dealing dynamics in fluids can be well imitated by the verdicts.

Analytical study on two new (3 + 1)-dimensional Painlevé integrable equations: Kink, lump, and multiple soliton solutions in fluid mediums

In this work, two new (3 + 1)-dimensional integrable wave equations are investigated. The complete Painlevé integrability of the two suggested equations will be investigated using Mathematica. We employ the method of Hirota to formally derive two sets of multiple soliton solutions for the two suggested models. Additionally, using symbolic computation with Maple, we provide a variety of lump solutions for the two suggested models. Other exact solutions of distinct structures, such as periodic, singular, and many other physical nonlinear structures, will be determined. We should mention here that the proposed two new models will assist many authors that are working in the field of fluids and plasma physics, in understanding the scenarios of the nonlinear waves that arise in different physical systems. Also, this study will contribute to understanding the nature of nonlinear waves that arise in the seas and oceans.