Dispersive Wave Equations Research Articles

Exact solitary wave solutions for a coupled gKdV–Schrödinger system by a new ODE reduction method

AbstractA new method is developed for finding exact solitary wave solutions of a generalized Korteweg–de Vries equation with ‐power nonlinearity coupled to a linear Schrödinger equation arising in many different physical applications. This method yields 22 solution families, with . No solutions for were known previously in the literature. For , four of the solution families contain bright/dark Davydov solitons of the 1st and 2nd kind, obtained in recent literature by basic ansatz applied to the ordinary differential equation (ODE) system for traveling waves. All of the new solution families have interesting features, including bright/dark peaks with (up to) symmetric pairs of side peaks in the amplitude and a kink profile for the nonlinear part in the phase. The present method is fully systematic and involves several novel steps that reduce the traveling wave ODE system to a single nonlinear base ODE for which all polynomial solutions are found by symbolic computation. It is applicable more generally to other coupled nonlinear dispersive wave equations as well as to nonlinear ODE systems of generalized Hénon–Heiles form.

Analytic soliton solutions to the shallow water dispersive long gravity wave equations: the first integral approach in nonlinear physics

In this article, we investigate the (2+1)-dimensional dispersive long water wave equation and the (1+1)-dimensional Phi-four equation, which describe the behavior of long gravity waves with small amplitudes, long wave propagation in oceans and seas, coastal structures and harbor design, effects of wave motion on sediment transport, quantum field theory, phase transitions of matter, ferromagnetic systems, liquid-gas transitions, and the structure of optical solitons. We use the first integral technique and obtain new and generic solutions for the models under consideration. By setting definite values for the associated parameters, various types of richly structured solitons are generated. The solitons include kink, flat kink, bell-shaped, anti-bell-shaped, and singular kink formations. These solutions allow for a profound understanding of the behavior and properties of the phenomena, offering new insights and potential applications in the associated field. The first integral technique is simpler, directly integrates the models, and the solutions offer clear insights into the underlying phenomena without requiring intermediate steps, making it widely applicable to various other models, including nonlinear equations and those that are challenging to solve using other standard techniques.

Noether Symmetries of the Triple Degenerate DNLS Equations

In this paper, Lie symmetries and Noether symmetries along with the corresponding conservation laws are derived for weakly nonlinear dispersive magnetohydrodynamic wave equations, also known as the triple degenerate derivative nonlinear Schrödinger equations. The main goal of this study is to obtain Noether symmetries of the second-order Lagrangian density for these equations using the Noether symmetry approach with a gauge term. For this Lagrangian density, we compute the conserved densities and fluxes corresponding to the Noether symmetries with a gauge term, which differ from the conserved densities obtained using Lie symmetries in Webb et al. (J. Plasma Phys. 1995, 54, 201–244; J. Phys. A Math. Gen. 1996, 29, 5209–5240). Furthermore, we find some new Lie symmetries of the dispersive triple degenerate derivative nonlinear Schrödinger equations for non-vanishing integration functions Ki(t) (i=1,2,3).

Authentic fault models and dispersive tsunami simulations for outer-rise normal earthquakes in the southern Kuril Trench

The southern Kuril Trench is one of the most seismically active regions in the world. In this study, marine surveys and observations were performed to construct fault models for possible outer-rise earthquakes. Seismic and seafloor bathymetric surveys indicated that the dip angle of the outer-rise fault was approximately 50°–80°, with a strike that was slightly oblique to the axis of the Kuril Trench. The maximum fault length was estimated to be ~ 260 km. Based on these findings, we proposed 17 fault models, with moment magnitudes ranging from 7.2 to 8.4. To numerically simulate tsunami, we solved two-dimensional dispersive wave and three-dimensional Euler equations using the outer-rise fault models. The results of both simulations yielded identical predictions for tsunami with short-wavelength components, resulting in significant dispersive deformations in the open ocean. We also found that tsunami generated by outer-rise earthquakes were affected by refraction and diffraction because of the source location beyond the trench axis. These findings can improve future predictions of tsunami hazards.Graphical

Critical blow-up exponent for a doubly dispersive quasilinear wave equation

Critical blow-up exponent for a doubly dispersive quasilinear wave equation

Analysis on construction and evolution dynamics for multi-lump solutions of the dispersive long wave equations

Analysis on construction and evolution dynamics for multi-lump solutions of the dispersive long wave equations

Advancing computational models for wave dynamics applications with quartic trigonometric tension b-spline techniques

This paper presents a computational model using Quartic Trigonometric Tension B-spline (QT-TB) function with collocation approach for nonlinear dispersive wave equation. Shallow water wave modeling plays a crucial role in various fields, including coastal engineering, oceanography, and natural hazard assessment. The QT-TB function is employed for spatial derivatives, integrating the nonlinear term through the linearization technique. While maintaining most characteristics of traditional polynomial B-splines, this method improves numerical solutions, enhancing accuracy in modeling. The performance of the method is rigorously assessed across three benchmark problems, with results compared to those of prior studies employing identical parameters. Detailed numerical illustrations are presented. Graphical representations are utilized to illustrate the single solitary wave motion, dynamics of coupled solitary waves and dynamics of triplet solitary waves in this study. Numerical experiments validate the method's accuracy and efficiency in capturing RLW-driven wave dynamics, establishing it as a reliable tool for various applications. The presented work includes an analysis of the stability of the proposed scheme, employing the Fourier method and shows that its unconditional stable.

A Set of Accurate Dispersive Nonlinear Wave Equations

In this study, a set of accurate dispersive nonlinear wave equations is established, using the wave velocity and free surface elevation as variables. These equations improve upon previous equations in which the velocity potential is used as a variable by considering the rotational wave motion and by adding a second-order bottom slope term that applies to general situations, allowing the equations to consider the influence of rapidly changing, horizontal, two-dimensional bottom topographies. The problem of the inaccuracy of the integral calculations used in previous equations in nearshore areas is solved by approximating the integral calculations into differential calculations, and a set of coupled wave equations is established by keeping the free surface elevation and the horizontal velocity constant, thus allowing the calculation of nearshore wave-generated currents. The benefits of the current model are confirmed through comparisons with corresponding laboratory experimental findings and are illustrated through a comparison with the numerical outcomes of other pertinent models.

A class of fourth-order dispersive wave equations with exponential source

A class of fourth-order dispersive wave equations with exponential source

Bifurcation of the travelling wave solutions in a perturbed (1 + 1)-dimensional dispersive long wave equation via a geometric approach

Choosing ${\kappa }$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$ -dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa }\in \left (0,1-\frac {\sqrt 3}{3}\right )\cup \left (1+\frac {\sqrt 3}{3},2\right )$ , while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt {15}/3$ for $\kappa =0$ or $\kappa =2$ . The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.

Superconvergence analysis of a two-grid BDF2-FEM for nonlinear dispersive wave equation

The aim of this paper is to study the superconvergent behavior of a two-grid finite element method (FEM) with 2-step backward differential formula (BDF2) for nonlinear dispersive wave equation. By introducing an auxiliary variable q=ut for the original variable u, the problem is transformed into a parabolic system. With the help of the combination technique of the interpolation and Ritz projection, the superclose and superconvergent estimates for the above two variables in H1-norm are obtained under the lower regularity of the exact solution compared with the standard Galerkin FEM. Finally, some numerical results are provided to show the correctness of the theoretical predictions and good performance of the proposed method.

Lyapunov-Schmidt reduction in the study of bifurcation of periodic travelling wave solutions of a perturbed (1 + 1)−dimensional dispersive long wave equation

In this paper, the Lyapunov-Schmidt reduction is used to investigate the bifurcation of periodic travelling wave solutions of a perturbed (1+1)−dimensional dispersive long wave equation. We demonstrate that the bifurcation equation corresponding to the original problem is supplied by a nonlinear system of two cubic algebraic equations. As the bifurcation parameters change, this system has only one, three, or five regular real solutions. The linear approximation of the solutions to the main problem has been discovered.

Exploring solitary wave solutions to the simplified modified camassa-holm equation through a couple sophisticated analytical approaches

The simplified modified Camassa-Holm equation, a developed Korteweg-de Vries equation, is an integrable nonlinear dispersive water wave equation with potential applications in diverse fields such as shallow water waves, liquid drop patterning, water surface waves in channels, tsunamis in coastal regions, ocean engineering, etc. In this study, we derived solitary wave solutions to the described equation, employing two reliable approaches: the enhanced modified simple equation and the extended Kudryashov methods. The enhanced modified simple equation technique yields solutions with variable coefficients through a generalized wave transformation. On the other hand, the extended Kudryashov method exploits the solutions of the Riccati and Bernoulli differential equations. The determined results include diverse soliton solutions expressed in the form of trigonometric, hyperbolic, and rational functions. Notably, for specific parameter values, these solutions exhibit wave of distinct shapes, including kink, smooth kink, flat kink, periodic, singular periodic, one-sided kink, etc. To apparently explain the physical behavior and the impact of the fractional parameter in the solutions, we plotted three- and two-dimensional graphs. The obtained solutions have potential applications in oceanography, spanning offshore rigs, ocean gravity waves, and wave energy.

New variable separation solutions and localized waves for (2+1)-dimensional nonlinear systems by a full variable separation approach

A full variable separation approach is firstly proposed for (2+1)-dimensional nonlinear systems by extending the well-established multilinear variable separation approach through the assumption that the expansion function is composed of full variable separated functions, namely, functions with respect to only one spacial or temporal argument. Taking the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation, the Nizhnik-Novikov-Veselov equation and the dispersive long wave equation as examples, new full variable separation solutions are obtained with several arbitrary one dimensional functions. Especially, a common formula for some suitable physical quantities is discovered. By taking the arbitrary functions in different explicit expressions, the solutions can be used to describe plentiful novel nonlinear localized waves, which might be non-travelling waves as the spacial and temporal variables are fully separated into different functions. In particular, some new hybrid solitary waves, which can pulsate periodically, appear and/or decay with an adjustable lifetime, are discovered through the on-site interactions between a doubly periodic wave and a ring soliton, a four-humped dromion and a four-humped lump, and a doubly periodic wave and a cross type solitary wave. Nonlinear wave structures and their dynamical behaviours are discussed and graphically displayed in detail.

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.

Exponential Runge-Kutta Parareal for non-diffusive equations

Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration. It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods. However, like many parallel-in-time methods it can fail to converge when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use of exponential integrators within the Parareal iteration. Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators. This work begins with an introduction to exponential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems. The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. These experiments demonstrate that exponential Parareal methods offer improved time-to-solution compared to serial exponential integrators when solving certain non-diffusive equations.

Resolving entropy growth from iterative methods

We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers’ equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.

An efficient fourth-order three-point scheme for solving some nonlinear dispersive wave equations

In this work, based on the value computed by the fourth-order implicit Padé scheme for the first derivative, a fourth-order three-point compact scheme for approximating the third derivative is suggested and applied to numerically solve some nonlinear dispersive wave equations including the Korteweg de Vires (KdV) equation and the two-dimensional Zakharov–Kuznetsov (ZK) equation. Spectral properties of the proposed scheme are analyzed and compared with other high-order schemes, and it is found that the proposed scheme has satisfactory numerical properties similar to high-order compact schemes. An optimized boundary closure scheme for the approximation of the first derivative is proposed to improve numerical stability. Numerical results of KdV equation and ZK equation obtained by the present scheme match very well with analytic solutions and available numerical results and showcase excellent performance of the scheme to solve nonlinear problems. And owing to the explicit formula for the second and the third derivative, the present scheme is efficient and consumes noticeable less computation time for solving the KdV equation compared to existing three-point high-order compact schemes.

Time-domain Green’s function associated to the interface problem for the Klein–Gordon equation

In this work we present a representation formula for the Green’s function associated with the source or interface problem for the Klein–Gordon equation from a known result on solution for the non-dispersive wave equation. The results can extend to more general situations, in particular stratified media where the Cagniard-de Hoop method fails for reasons of non-homogeneity of the amplitude and the phase of the exponential in the Laplace transform. Examples and numerical applications are given in two dimensions for the interface problem associated to a dispersive wave equation where the reflected and incident fields are obtained. The calculation of the transmitted field is omitted for the sake of simplicity of the paper. Such a representation formula can be useful to study other qualitative properties of dispersive waves from those known on non-dispersive ones.

Breather interactions in a three-layer fluid

In a three-layer system with equal upper and lower layer thicknesses that are sufficiently thin and with the same density difference across each interface, breathers have been shown to exist using fully nonlinear governing equations. These breathers are well modelled by theoretical solutions of the mKdV equation, provided the interfaces between the layers do not cross a critical depth. The soliton-like characteristics of fully nonlinear breathers, in particular how two breathers interact, have yet to be explored. Using numerical simulations, this study addresses this shortcoming by studying fully nonlinear overtaking collisions of two breathers in a three-layer symmetric stratification. We apply the fully nonlinear and strongly dispersive FDI-3s internal wave equations, based on a variational principle, in a three-layer system. When the amplitude is small, the analytic breathers fit the wave shapes of the overtaking collision breathers. We find that the larger the upper and lower layer thicknesses are, provided they are below the critical thickness, the more the breathers behave like solitons. We show that an overtaking collision of two breathers is close to elastic.