Nonlinear Dispersive Equations Research Articles

Local Existence and Uniqueness of Spatially Quasi-Periodic Solutions to the Generalized KdV Equation

Local Existence and Uniqueness of Spatially Quasi-Periodic Solutions to the Generalized KdV Equation

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.

On the proximity between the wave dynamics of the integrable focusing nonlinear Schrödinger equation and its non-integrable generalizations

The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the field of nonlinear dispersive equations. In this work, we investigate this topic in the context of focusing nonlinear Schrödinger (NLS) equations. In particular, we consider non-integrable counterparts of the (integrable) focusing cubic NLS equation, which are distinct generalizations of cubic NLS and involve a broad class of nonlinearities, with the cases of power and saturable nonlinearities serving as illustrative examples. This is a notably different direction from the one explored in other works, where the non-integrable models considered are only small perturbations of the integrable one. We study the Cauchy problem on the real line for both vanishing and non-vanishing boundary conditions at infinity and quantify the proximity of solutions between the integrable and non-integrable models via estimates in appropriate metrics as well as pointwise. These results establish that the distance of solutions grows at most linearly with respect to time, while the growth rate of each solution is chiefly controlled by the size of the initial data and the nonlinearity parameters. A major implication of these closeness estimates is that integrable dynamics emerging from small initial conditions may persist in the non-integrable setting for significantly long times. In the case of zero boundary conditions at infinity, this persistence includes soliton and soliton collision dynamics, while in the case of nonzero boundary conditions at infinity, it establishes the nonlinear behavior of the non-integrable models at the early stages of the ubiquitous phenomenon of modulational instability. For this latter and more challenging type of boundary conditions, the closeness estimates are proved with the aid of new results concerning the local existence of solutions to the non-integrable models. In addition to the infinite line, we also consider the cubic NLS equation and its non-integrable generalizations in the context of initial-boundary value problems on a finite interval. Apart from their own independent interest and features such as global existence of solutions (which does not occur in the infinite domain setting), such problems are naturally used to numerically simulate the Cauchy problem on the real line, thereby justifying the excellent agreement between the numerical findings and the theoretical results of this work.

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.

Testing by wave packets and modified scattering in nonlinear dispersive pde’s

Modified scattering phenomena are encountered in the study of global properties for nonlinear dispersive partial differential equations in situations where the decay of solutions at infinity is borderline and scattering fails just barely. An interesting example is that of problems with cubic nonlinearities in one space dimension. The method of testing by wave packets was introduced by the authors as a tool to efficiently capture the asymptotic equations associated to such flows, and thus establish the modified scattering mechanism in a simpler, more efficient fashion, and at lower regularity. In these expository notes we describe how this method can be applied to problems with general dispersion relations.

Unconditional well-posedness for the periodic Boussinesq and Kawahara equations

<abstract><p>In this article, we obtain new results on the unconditional well-posedness for a pair of periodic nonlinear dispersive equations using an abstract framework introduced by Kishimoto. This framework is based on a normal form reductions argument coupled with a number of crucial multilinear estimates.</p></abstract>

Lower bounds on the radius of spatial analyticity for the higher order nonlinear dispersive equation on the real line

Lower bounds on the radius of spatial analyticity for the higher order nonlinear dispersive equation on the real line

A Brief Historical Overview and Mathematical Equations of Soliton

bstract: A soliton is a strongly stable, nonlinear, self-reinforcing, localised wave packet that maintains its shape even after colliding with other similar localised wave packets and propagates freely at a constant speed. The medium's balanced cancellation of nonlinear and dispersive effects is responsible for its extraordinary stability. Dispersive effects are a characteristics of several systems where a wave frequency determines how fast it travels. Subsequently, it was shown that soliton solutions offer stable solutions for a broad range of weakly nonlinear dispersive partial differential equations that characterise physical systems. Solitons have a self-restoring quality that makes them desirable for long-distance, high-speed transmissions. The strain on networks is quickly reaching a critical threshold as transmission speeds over longer distances are now approaching 40 Gbit/sec. In the future, solitons might offer intriguing alternatives.

Propagation of regularity and decay of solutions to nonlinear dispersive equations

Propagation of regularity and decay of solutions to nonlinear dispersive equations

On the Properties of Difference Schemes for Solving Nonlinear Dispersive Equations of Increased Accuracy. I. The Case of One Spatial Variable

On the Properties of Difference Schemes for Solving Nonlinear Dispersive Equations of Increased Accuracy. I. The Case of One Spatial Variable

Dynamical behavior of fractional nonlinear dispersive equation in Murnaghan’s rod materials

The primary objective of this study aims to carry out a more thorough investigation into a fractional nonlinear double dispersive equation that is used to represent wave propagation in an elastic, inhomogeneous Murnaghan’s rod. By Murnaghan’s rod, we mean the materials, which include the constitutive constant, Poisson ratio, and Lame’́s coefficient, are considered to be compressible in nature forming up the elastic rod. To solve the fractional version of Murnaghan’s rod problem, we employed β-fractional and M-Truncated fractional derivative. Regarding the extraction of polynomial and rational function solutions of the Murnaghan’s rod problem, which degenerate into several wave solutions including solitary, soliton (dromions), as well as periodic wave solutions. We employ the well-known unified and new auxiliary equation methods of nonlinear sciences. A finite series of certain functions satisfying an ordinary differential equation of first order, second degree is used to represent the projected solution. Based on the given approach, numerous types of solutions for exponential, hyperbolic, and trigonometric functions are generated. In this research study, the behavior of a dynamical planer system has been examined by giving various values to parameters and by depicting every possible situation as a phase portrait. The sensitivity analysis, where the soliton wave velocity and wave number parameters influence the water wave singularity, is demonstrated using the wave profiles of the constructed dynamical structural system. With the use of graphs, we have simulated the solitons to determine their kinds. All of solutions found in this manuscript is been confirmed through back substituting them into the original model using computational software.

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.

Performance of affine-splitting pseudo-spectral methods for fractional complex Ginzburg-Landau equations

We evaluate the performance of novel numerical methods for solving one-dimensional nonlinear fractional dispersive and dissipative evolution equations. The methods are based on affine combinations of time-splitting integrators and pseudo-spectral discretizations using Hermite and Fourier expansions. We show the effectiveness of the proposed methods by numerically computing the dynamics of soliton solutions of the standard and fractional variants of the nonlinear Schrödinger equation (NLSE) and the complex Ginzburg-Landau equation (CGLE), and by comparing the results with those obtained by standard splitting integrators. An exhaustive numerical investigation shows that the new technique is competitive when compared to traditional composition-splitting schemes for the case of Hamiltonian problems both in terms accuracy and computational cost. Moreover, it is applicable straightforwardly to irreversible models, outperforming high-order symplectic integrators which could become unstable due to their need of negative time steps. Finally, we discuss potential improvements of the numerical methods aimed to increase their efficiency, and possible applications to the investigation of dissipative solitons that arise in nonlinear optical systems of contemporary interest. Overall, the method offers a promising alternative for solving a wide range of evolutionary partial differential 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.

Multilinear oscillatory integrals and estimates for coupled systems of dispersive PDEs

We establish sharp global regularity of a class of multilinear oscillatory integral operators that are associated to nonlinear dispersive equations with both Banach and quasi-Banach target spaces. As a consequence we also prove the (local in time) continuous dependence on the initial data for solutions of a large class of coupled systems of dispersive partial differential equations.

Highly dispersive optical solitons and solitary wave solutions for the (2 + 1)-dimensional Mel’nikov equation in modeling interaction of long waves with short wave packets in two dimensions

In this paper, the optical soliton and solitary wave solutions of the [Formula: see text]-dimensional Mel’nikov equation are investigated using the Kudryashov [Formula: see text] function technique. The Kudryashov [Formula: see text] function approach has various features that significantly facilitate symbolic computing, particularly for highly dispersive nonlinear equations. In computations, this approach has the benefit of not requiring the use of a certain function form. This approach gives an algorithm that is straightforward, efficient, and simple for finding solitary wave solutions. In addition, this approach is very influential and reliable when it comes to discovering hyperbolic function solutions of nonlinear equations. Many new hyperbolic function solutions have been obtained from the governing equation by using this technique. In addition, numerous types of soliton solutions describing various structures of optical solitons are retrieved. Using this method, breather, W-shaped, bell shaped, and bright soliton solutions have been generated from the governing equation. From the obtained results, it can be asserted that the applied approach may be a useful tool for addressing more highly nonlinear problems in various fields. By choosing particular values for the relevant parameters, the dynamic features of some breather, W-shaped, bell shaped and bright soliton solutions to the [Formula: see text]-dimensional Mel’nikov equation have been displayed in 3D, 2D and contour graphs.

Local well-posedness and spatial analyticity for the generalized Zakharov–Kuznetsov equation

The initial value problem for the generalized Zakharov–Kuznetsov equation on R2 is shown to be local well-posed in spaces of functions which are analytic on a strip without shrinking the width of the strip in time. The proof mainly relies upon the local smoothing effect, a multi-dimensional maximal function estimate and Bourgain-type spaces. These techniques are significant for derivative nonlinear dispersive equation in low regularity spaces. In addition, under the boundedness assumption of a suitable Sobolev norm the generalized Zakharov–Kuznetsov equation is proved to be Gevrey-class analysis. Especially, we obtain an explicit lower bound on the possible decreasing rate of the uniform radius of analyticity of a solution starting from analytic initial data.

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.

Discovering optical soliton solutions in the Biswas–Milovic equation through five innovative approaches

Purpose:In recent years, depending on the development of models containing highly dispersive nonlinear equations, especially in the field of nonlinear optics, it has been important to obtain the solutions of these equations with analytical methods. This study presents some effective, reliable, and easily applicable solution techniques based on the well-known extended auxiliary equation. Besides, these techniques were applied to the Biswas–Milovic equation. Methodology:First, in order to achieve the ordinary differential equation structure of the Biswas–Milovic equation, the complex wave transform was given and utilized. Then, the algorithms, constraints of each proposed approach, and key points for their implementation were presented. In addition, the relations and transitivities between the methods are also interpreted. Second, analytical optical soliton solutions were obtained by applying the proposed techniques. Lastly, graphical simulations of optical soliton types represented by some solution functions are made. Findings:It has been shown that all of the proposed methods yield effective results in solving the Biswas–Milovic equation and can be used not only for nonlinear evolution equations but also for highly dispersive nonlinear optical equations. It has been observed that the methods have the capacity to be employed easily and do not require tedious calculus and algebraic manipulations. Originality:The work in the article has not been presented before.

Two approaches for the stability of a nonlocal time-delayed fourth-order dispersive model

The primary concern of the current article is to investigate the well-posedness and stability problem of a nonlinear dispersive equation of order four and with a nonlocal time-delayed term. In order to deal with such a problem, two different approaches are used. The first approach consists in using the Lyapunov method to prove the well-posedness of the system and the exponential stability of the zero solution. The latter is established, provided that the time-delay has small values and an interior damping control acts on the equation. In turn, the second approach permits to get the well-posedness outcome as well as another stability finding by invoking Schauder Theorem. The advantage of the second method is the lack of need for any damping control. To conclude, our findings show that the dispersive equation under consideration possesses a unique stable solution despite the occurrence of a nonlocal time-delay term in the nonlinearity of the equation, with or without the presence of a damping control. These results extent and complement those of [1,2,10,21,28], where no local term takes place.