Nonlinear Dispersive Equations Research Articles

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.

Nonlinear Waves and Dispersive Equations

Nonlinear dispersive equations are models for nonlinear waves in a wide range of physical contexts. Mathematically they display an interplay between linear dispersion and nonlinear interactions, which can result in a wide range of outcomes from finite time blow-up to solitons and scattering. They are linked to many areas of mathematics and physics, ranging from integrable systems and harmonic analysis to fluid dynamics, geometry and general relativity and probability.

Deterministic Dynamics and Randomness in PDE

Over the last few years there has been spectacular progress in the study of parabolic SPDE, of nonlinear dispersive and wave equations and of probabilistic methods in PDE. An important direction connecting these three fields is the general question of how randomness affects the behavior of solutions to PDE. Research in recent years has been driven by the study of randomness in nonlinear evolution equations with a focus on the question of how to quantify the transport of such randomness under the nonlinear flow.

On local well-posedness of nonlinear dispersive equations with partially regular data

We revisit the local well-posedness theory of nonlinear Schrödinger and wave equations in Sobolev spaces Hs and H˙s, 0<s≤1. The theory has been well established over the past few decades under Sobolev initial data regular with respect to all spatial variables. But here, we reveal that the initial data do not need to have complete regularity like Sobolev spaces, but only partially regularity with respect to some variables is sufficient. To develop such a new theory, we suggest a refined Strichartz estimate which has a different norm for each spatial variable. This makes it possible to extract a different integrability/regularity of the data from each variable.

Applications of the invariant subspace method on searching explicit solutions to certain special-type non-linear evolution equations

We extend the invariant subspace method (ISM) to a class of Hamilton–Jacobi equations (HJEs) and a family of third-order time-fractional dispersive PDEs with the Caputo fractional derivative in this letter. More precisely, the complete classification is presented for such HJEs that admit invariant subspaces governed by solutions of the second-order and third-order linear ordinary differential equations (ODEs). Meanwhile, some concrete equations are derived for the construction of new exact solutions u(x,t)=∑i=1nCi(t)fi(x). Then a set of invariant subspaces of the considered third-order time-fractional non-linear dispersive equations are obtained. Based on the Laplace transform method (LTM) and applying several properties of the well known Mitta-Leffer (ML) function, the different types of explicit solutions of a family of third-order time-fractional dispersive PDEs are finally derived.

Analytical method for modal analysis of submerged electric motor stator

Modal analysis of the stator is important for reducing motor noise and diagnosing motor faults. An accurate analytical model is beneficial for the establishment of the relationship between the motor stator parameters and the natural frequencies, which will further aid in optimizing machine designs. Up to now, the existing modal analysis models are mainly aimed at onshore motors. When the motor is submerged in dense fluid, such as those used in the rim-driven thrusters, the vibration behaviors are quite different due to the high characteristic impedance of the fluid. In a bid to analyze the wet natural frequencies of motor stator, this paper proposes analytical models with one, three and five degrees of freedom for different application scenarios based on Donnell, Love and Soedel’s theories. The coupling between the structure and fluid is realized by the acoustic pressure load derived from the acoustic wave function. The resulting non-linear dispersion equations are then solved by Newton’s iterative method. The proposed models are applied to analyze the natural frequencies of motor stators of various sizes and with a wide range of thickness-to-radius and length-to-radius ratios. The results show that the analytical models are insensitive to the stator radius but sensitive to the thickness-to-radius and length-to-radius ratios. For thick and short motor stators, it is necessary to consider the shear deformation and rotatory inertia in the analytical model to compute the natural frequency accurately, especially when the fluid loading is taken into account.

Homotopy analysis of wave transformation over permeable seabeds and porous structures

Based on the Homotopy Analysis Method (HAM), a semi-analytical solution method is proposed to solve the modified mild slope equation (MMSE) for waves propagation over permeable seabeds and porous structures. The present method overcomes the mathematical difficulties in treating the nonlinear dispersion equations introduced by the permeability effect, which limits the previous semi-analytical solutions to impermeable seabed and structures only. The proposed method is applied to both 1-D and 2-D axisymmetric wave transformation problems. The method is validated by systematically comparing the semi-analytical solutions of wave amplitudes and reflection coefficients against available experimental data and numerical results, demonstrating the method's capability for predicting wave transformation over both impermeable and permeable seabeds. It is ready to show that, for impermeable conditions, the present semi-analytical solutions degenerate into the earlier solutions without considering the porous effect. The semi-analytical solution method is further extended to examine the porous effects on wave transformation and reflection in various topographical conditions.

The scattering map determines the nonlinearity

Using the two-dimensional nonlinear Schrödinger equation as a model example, we present a general method for recovering the nonlinearity of a nonlinear dispersive equation from its small-data scattering behavior. We prove that under very mild assumptions on the nonlinearity, the wave operator uniquely determines the nonlinearity, as does the scattering map. Evaluating the scattering map on well-chosen initial data, we reduce the problem to an inverse convolution problem, which we solve by means of an application of the Beurling–Lax Theorem.

A high-order linearly implicit energy-preserving Partitioned Runge-Kutta scheme for a class of nonlinear dispersive equations

&lt;abstract&gt;&lt;p&gt;In this paper, we design a novel class of arbitrarily high-order, linearly implicit and energy-preserving numerical schemes for solving the nonlinear dispersive equations. Based on the idea of the energy quadratization technique, the original system is firstly rewritten as an equivalent system with a quadratization energy. The prediction-correction strategy, together with the Partitioned Runge-Kutta method, is then employed to discretize the reformulated system in time. The resulting semi-discrete system is high-order, linearly implicit and can preserve the quadratic energy of the reformulated system exactly. Finally, we take the Camassa-Holm equation as a benchmark to show the efficiency and accuracy of the proposed schemes.&lt;/p&gt;&lt;/abstract&gt;

The local well-posedness for a stochastic system of nonlinear dispersive equations

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Global conservative solutions of the nonlocal NLS equation beyond blow-up

We consider the Cauchy problem for the integrable nonlocal nonlinear Schrödinger (NNLS) equation $ \mathrm{i}\partial_t q(x, t)+\partial_{x}^2q(x, t)+2\sigma q^{2}(x, t)\overline{q(-x, t)} = 0 $ with initial data $ q(x, 0)\in H^{1, 1}(\mathbb{R}) $. It is known that the NNLS equation is integrable and it has soliton solutions, which can have isolated finite time blow-up points. The main aim of this work is to propose a suitable concept for continuation of weak $ H^{1, 1} $ local solutions of the general Cauchy problem (particularly, those admitting long-time soliton resolution) beyond possible singularities. Our main tool is the inverse scattering transform method in the form of the Riemann-Hilbert problem combined with the PDE existence theory for nonlinear dispersive equations.

A homotopy inversion method for Rayleigh wave dispersion data

Inverting Rayleigh-wave dispersion data may provide reliable velocity structure. A lot of linearized inversion methods are developed to handle this problem, however, which may be trapped by local extrema (minima or maxima) or no convergence of the solution without propriate initial models. Although several nonlinear approaches are also proposed, most of which face amount of computational consumption. In order to keep the balance, we introduce the homotopy method (HM), which can transform the nonlinear Rayleigh-wave dispersion equation into a linear equation system, and has global optimized property and less computational burden. We apply the HM to synthetic dispersion datasets, including noise-free and contaminated data. Through the comparison with the Competitive Particle Swarm Optimization (CPSO), a nonlinear approach, the HM is also less dependent on the initial model and may have higher computational efficiency. Real datasets from ambient noise in Weiyuan, Sichuan province, China, are further employed to study the effectiveness and applicability of the HM. Similar structure of more sufficient abnormal magnitude and boundary consistence is derived, which shows the uplift of the Weiyuan anticline with most less Root Mean Square Error (RMSE). It is concluded that, the HM is a potential candidate for most geophysical inversion problems with global optimization and efficiency balance.

Low frequency electrostatic mode generated by electromagnetic waves in the Earth’s inner magnetosphere with two distinct electrons

The generation of low frequency electrostatic mode by parametric decay of electromagnetic waves (EMWs) in the Earth’s inner magnetosphere with exponentially truncated kappa distributed hot electrons and cold electrons is studied. Nonlinear dispersion equation for the parametric process is derived from kinetic theory. The parametric instability of EMWs decay into low frequency electrostatic normal mode (ion acoustic like wave modes and electron acoustic wave modes) and electrostatic quasi–mode in the Earth’s inner magnetosphere are numerically analyzed. It is shown that parametric instability occurs only when the EMW is sufficiently strong if the collisions between ions and electrons are taken into account. The growth rate and the threshold conditions of the decay instability depend on the concentration and distribution of hot electrons. Because they change the dispersion and the damping rate of normal mode, the collisional damping of sideband EMW. In addition, the excitation of electrostatic normal mode by parametric decay of EMWs is more difficult than the excitation of electrostatic quasi–mode. The growth rate of EMWs decaying into electrostatic quasi–mode is much larger than the ones of decaying into electrostatic normal mode. But the frequency of electrostatic quasi–mode corresponding to the maximum growth rate can be as low as a few tens Hz. The mechanism may excite the electrostatic mode with frequency comparable to those of the ultra–low frequency electric fields observed in the Earth’s inner magnetosphere.

Multiple lump solutions of the (2+1)-dimensional sawada-kotera-like equation

In this paper, 1-lump solution and 2-lump solution of a (2 + 1)-dimensional Sawada-Kotera-like equation are obtained by means of the Hirota’s bilinear method and long wave limit method. The propagation orbits, velocities and the collisions among waves are analyzed. By setting the parameter values, the dynamic characteristics of the obtained solutions are shown in 3D and density plots. These conclusions enrich the dynamical theory of higher-dimensional nonlinear dispersive wave equations.

The solutions of dissipation-dispersive models arising in material science

PurposeThe aim of this work is to obtain periodic waves of Eq. (1) via ansatz-based methods. So, the open questions are replied and the gap will be filled in the literature. Additionally, the comparison of the considered models (Eq. (1) and Eq. (2)) due to their performance. Although it is extremely difficult to find the exact wave solutions in Eq. (1) and Eq. (2) without any assumptions, the targeted solutions have been obtained with the chosen method.Design/methodology/approachMaterial science is the today's popular research area. So, the well-known model is the dissipation double dispersive nonlinear equation and, in the literature, open queries have been seen. The aim of this work is to reply open queries by obtaining wave solutions of the dissipation double dispersive model, double dispersive model and double dispersive model for Murnaghan's material via ansatz-based methods.FindingsThe results have been appeared for the first time in this communication work and they may be valuable for developing uses in material science.Originality/valueThe exact wave solutions of Eq. (1) and Eq. (2) without any assumptions have been obtained with via ansatz-based method. So, the open questions are replied and the gap will be filled in the literature.