Nonlinear Dispersive Waves Research Articles

Assorted soliton solutions to the nonlinear dispersive wave models in inhomogeneous media

The Sharma-Tasso-Olver and Klein–Gordon equations are significant models to interpret plasma physics, relativistic physics, quantum mechanics, nonlinear optics, etc. In the present article, an advanced generalized approach, namely the generalized (G′/G)-expansion approach is scheduled to unfold certain wave solutions to the nonlinear evolution equations earlier described. The soliton solutions exposed are estimated in the form of hyperbolic, rational and trigonometric functions. The physical implications of the ascertained solutions are summed up by setting specific values of the integrated constraints and delineating the profiles to decode the physical phenomena. The numerical simulation of the solutions extracted is standard kink, rogue wave, bright-dark soliton, periodic soliton, breather type soliton, compaction, singular kink soliton, etc. This study affirms that the introduced scheme is robust, efficient in searching for nonlinear evolution equations, computer algebra compatible, and capable of finding further inclusive wave solutions.

A Comparative Analysis of Fractional-Order Kaup–Kupershmidt Equation within Different Operators

In this paper, we find the solution of the fractional-order Kaup–Kupershmidt (KK) equation by implementing the natural decomposition method with the aid of two different fractional derivatives, namely the Atangana–Baleanu derivative in Caputo manner (ABC) and Caputo–Fabrizio (CF). When investigating capillary gravity waves and nonlinear dispersive waves, the KK equation is extremely important. To demonstrate the accuracy and efficiency of the proposed technique, we study the nonlinear fractional KK equation in three distinct cases. The results are given in the form of a series, which converges quickly. The numerical simulations are presented through tables to illustrate the validity of the suggested technique. Numerical simulations in terms of absolute error are performed to ensure that the proposed methodologies are trustworthy and accurate. The resulting solutions are graphically shown to ensure the applicability and validity of the algorithms under consideration. The results that we obtain confirm that the proposed method is the best tool for handling any nonlinear problems arising in science and technology.

Shape-morphing reduced-order models for nonlinear Schrödinger equations

We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrodinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) The recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions.

A Class of Adaptive Multiresolution Ultra-Weak Discontinuous Galerkin Methods for Some Nonlinear Dispersive Wave Equations

In this paper, we propose a class of adaptive multiresolution (also called the adaptive sparse grid) ultra-weak discontinuous Galerkin (UWDG) methods for solving some nonlinear dispersive wave equations including the Korteweg--de Vries (KdV) equation and its two-dimensional generalization, the Zakharov--Kuznetsov (ZK) equation. The UWDG formulation, which relies on repeated integration by parts, was proposed for the KdV equation in [7]. For the ZK equation, which contains mixed derivative terms, we develop a new UWDG formulation. The $L^2$ stability is established for this new scheme on regular meshes, and the optimal error estimate with a novel local projection is obtained for a simplified ZK equation. Adaptivity is achieved based on multiresolution and is particularly effective for capturing solitary wave structures. Various numerical examples are presented to demonstrate the accuracy and capability of our methods.

The Classification of the Exact Single Travelling Wave Solutions to the Constant Coefficient KP-mKP Equation Employing Complete Discrimination System for Polynomial Method

The purpose of this article is to explore different types of solutions for the Kadomtsev-Petviashvili-modified Kadomtsev-Petviashvili (KP-mKP) equation which is termed as KP-Gardner equation, extensively used to model strong nonlinear internal waves in ( 1 + 2 )-dimensions on the stratified ocean shelf. This evolution equation is also used to describe weakly nonlinear shallow-water wave and dispersive interracial waves traveling in a mildly rotating channel with slowly varying topography. Introducing Liu’s approach regarding the complete discrimination system for polynomial and the trial equation technique, a set of new solutions to the KP-mKP equation containing Jacobi elliptic function have been derived. It is found that these analytical solutions numerically exhibit different nonlinear structures such as solitary waves, shock waves, and periodic wave profiles. The reliability and effectiveness are confirmed from the numerical graphs of the solutions. Finally, the existence and validity of the various topological structures of the solutions are confirmed from the phase portrait of the dynamical system. Based on this investigation, it is confirmed that the method is not only suited for obtaining the classification of the solutions but also for qualitative analysis, which means that it can also be extended to other fields of application.

Blow-up criteria for a two-component nonlinear dispersive wave system

We investigate the blow-up phenomena for the two-component generalizations of nonlinear dispersive wave equation on the real line that includes the two-component Camassa-Holm system as well as the system for the hyperelastic-rod wave equation as special cases. We establish a blow-up criterion for system of coupled equations under certain natural initial profiles. The obtained criterion involves only the properties of the data u0 in a neighbourhood of a single point and consequently, it is local-in-space with respect to first component of the system under consideration.

Uniqueness of solutions to the spectral hierarchy in kinetic wave turbulence theory

In Eyink and Shi (2012) and Chibbaro et al. (2018), Eyink and Shi and Chibbaro et al., respectively, formally derived an infinite, coupled hierarchy of equations for the spectral correlation functions of a system of weakly interacting nonlinear dispersive waves with random phases in the standard kinetic limit. Analogously to the relationship between the Boltzmann hierarchy and Boltzmann equation, this spectral hierarchy admits a special class of factorized solutions, where each factor is a solution to the wave kinetic equation (WKE). A question left open by these works and highly relevant for the mathematical derivation of the WKE is whether solutions of the spectral hierarchy are unique, in particular whether factorized initial data necessarily lead to factorized solutions. In this article, we affirmatively answer this question in the case of 4-wave interactions by showing, for the first time, that this spectral hierarchy is well-posed in an appropriate function space. Our proof draws on work of Chen and Pavlović (2010) for the Gross–Pitaevskii hierarchy in quantum many-body theory and of Germain et al. (2020) for the well-posedness of the WKE.

Time difference between the 1854 CE Ansei\u2013Tokai and Ansei\u2013Nankai earthquakes estimated from distant tsunami waveforms on the west coast of North America

We estimated the time difference between the 1854 CE Ansei–Tokai and Ansei–Nankai earthquakes from tidal records of two tide gauge stations (San Francisco and San Diego) on the west coast of North America. The first signals of the Ansei–Tokai tsunami were apparent, whereas those of the Ansei–Nankai tsunami were obscured by the later waves of the Ansei–Tokai tsunami. Waveforms of the Ansei–Nankai tsunami simulated with nonlinear dispersive wave theory by assuming an origin time of 07:00 GMT on 24 December arrived earlier than in the observations. The normalized root mean square and the misfit between the simulated and observed waveforms of the Ansei–Nankai tsunami showed a time difference between them of approximately 0.4 h. This finding suggests that the actual origin time of the Ansei–Nankai tsunami was approximately 07:24 GMT on 24 December. A previous study estimated the origin time of the Ansei–Tokai tsunami to be about 00:30 GMT on 23 December. Thus, we concluded that the time difference between the 1854 CE Ansei–Tokai and Ansei–Nankai tsunamis was 30.9 h. Despite the significant difference in the time resolution between the seasonal timekeeping system used in Japan in 1854 and waveform digitization, our result is roughly in agreement with historical descriptions of the tsunamis, suggesting that such information can be effectively used to determine the origin times of historical earthquakes.

Nonlinear dispersion in wave-current interactions

<p style='text-indent:20px;'>Via a sequence of approximations of the Lagrangian in Hamilton's principle for dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonian models for describing wave-current interaction (WCI) in nonlinear dispersive wave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits <i>emergent singular solutions</i> for certain initial conditions. These singular solutions are identified with a singular momentum map for left action of the diffeomorphisms on a semidirect-product Lie algebra. This semidirect-product Lie algebra comprises vector fields representing horizontal current velocity acting on scalar functions representing wave elevation. We use computational simulations to demonstrate the dynamical interactions of the emergent wavefront trains which are admitted by this special subclass of Hamiltonians for a variety of initial conditions.</p><p style='text-indent:20px;'>In this paper, we investigate:</p><p style='text-indent:20px;'>(1) A variety of localised initial current configurations in still water whose subsequent propagation generates surface-elevation dynamics on an initially flat surface; and</p><p style='text-indent:20px;'>(2) The release of initially confined configurations of surface elevation in still water that generate dynamically interacting fronts of localised currents and wave trains. The results of these simulations show intricate wave-current interaction patterns whose structures are similar to those seen, for example, in Synthetic Aperture Radar (SAR) images taken from the space shuttle.</p>

On the Cauchy problem for a nonlocal nonlinear Schrödinger equation

<p style='text-indent:20px;'>This paper considers the one-dimensional Schrödinger equation with nonlocal nonlinearity that describes the interactions of nonlinear dispersive waves. We obtain some the local well-posedness and ill-posedness result associated with this equation in the Sobolev spaces. Moreover, we prove the existence of standing waves of this equation. As corollary, we derive the conditions under which the solutions are uniformly bounded in the energy space.</p>

New wave behaviors of the (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation

In this study, the (3+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation that indicated the propagation of nonlinear dispersive waves in inhomogeneous media is given for consideration. The generalized exponential rational function method is used to seek some new exact solutions for the considered equation. The three-dimensional surfaces and two-dimensional graphs of the obtained solutions are plotted by choosing the appropriate values of the involving free parameters.

A regularized shallow-water waves system with slip-wall boundary conditions in a basin: theory and numerical analysis

The simulation of long, nonlinear dispersive waves in bounded domains usually requires the use of slip-wall boundary conditions. Boussinesq systems appearing in the literature are generally not well-posed when such boundary conditions are imposed, or if they are well-posed it is very cumbersome to implement the boundary conditions in numerical approximations. In the present paper a new Boussinesq system is proposed for the study of long waves of small amplitude in a basin when slip-wall boundary conditions are required. The new system is derived using asymptotic techniques under the assumption of small bathymetric variations, and a mathematical proof of well-posedness for the new system is developed. The new system is also solved numerically using a Galerkin finite-element method, where the boundary conditions are imposed with the help of Nitsche’s method. Convergence of the numerical method is analysed, and precise error estimates are provided. The method is then implemented, and the convergence is verified using numerical experiments. Numerical simulations for solitary waves shoaling on a plane slope are also presented. The results are compared to experimental data, and excellent agreement is found.

Bäcklund transformations, Lax pair and solutions of a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves

Burgers-type equations are considered as the models of certain phenomena in plasma astrophysics, ocean dynamics, atmospheric science and so on. In this paper, a Sharma-Tasso-Olver-Burgers equation for the nonlinear dispersive waves is studied. Based on the Painlevé-Bäcklund equations, one auto-Bäcklund transformation and two hetero-Bäcklund transformations are derived. Motivated by the Burgers hierarchy, a Lax pair is given. Via two hetero-Bäcklund transformations with different constant seed solutions, we find some multiple-kink solutions, complex periodic solutions, hybrid solutions composed of the lump, periodic and multiple kink waves. Then we discuss the influence of the coefficients of the above equation on such solutions. Via the auto-Bäcklund transformation with the nontrivial seed solutions, we obtain certain lump-type solutions, kink-type solutions and recurrence relation of the above equation.

Construction of abundant novel analytical solutions of the space–time fractional nonlinear generalized equal width model via Riemann–Liouville derivative with application of mathematical methods

Abstract The space–time fractional generalized equal width (GEW) equation is an imperative model which is utilized to represent the nonlinear dispersive waves, namely, waves flowing in the shallow water strait, one-dimensional wave origination escalating in the nonlinear dispersive medium approximation, gelid plasma, hydro magnetic waves, electro magnetic interaction, etc. In this manuscript, we probe advanced and broad-spectrum wave solutions of the formerly betokened model with the Riemann–Liouville fractional derivative via the prosperously implementation of two mathematical methods: modified elongated auxiliary equation mapping and amended simple equation methods. The nonlinear fractional differential equation (NLFDE) is renovated into ordinary differential equation by the composite function derivative and the chain rule putting together along with the wave transformations. We acquire several types of exact soliton solutions by setting specific values of the personified parameters. The proposed schemes are expedient, influential, and computationally viable to scrutinize notches of NLFDEs.

Compact ETDRK scheme for nonlinear dispersive wave equations

Compact ETDRK scheme for nonlinear dispersive wave equations

Rapid phase-resolved prediction of nonlinear dispersive waves using machine learning

The phase-resolved prediction of ocean waves suffers from two major challenges that have so far hindered solutions in real time or faster: (i) The reconstruction of the nonlinear and dispersive waves: how to calculate the state of the wavefield from measurable wave amplitude data, and (ii) Rapid solution: how to integrate equations in a timely manner. Here, we overcome these challenges at once through an advanced machine learning technique based on spatiotemporal patches of the time history of surface wave elevation data in the domain. The proposed time-series analysis uses a new concept of convolution that instead of constructing kernels from filters, implements random patches of spatiotemporal data as the input to predict the next patch. We demonstrate that the proposed methodology can accurately make predictions for ocean surface gravity waves more than two orders of magnitude faster than numerically solving governing equations.

Undular bores generated by fracture.

Undular bores, or dispersive shock waves, are nonstationary waves propagating as oscillatory transitions between two basic states, in which the oscillatory structure gradually expands and grows in amplitude with distance traveled. In this work we report an important mechanism of generation of nonlinear dispersive shock waves in solids. We demonstrate, using high-speed pointwise photoelasticity, the generation of undular bores in solid (polymethylmethacrylate) prestrained bars by natural and induced tensile fracture. For the distances relevant to our experiments, the viscoelastic extended Korteweg-de Vries equation is shown to provide very good agreement with the key observed experimental features for suitable choice of material parameters, while some local features at the front of the bore are also captured reasonably well by the linearization near the nonzero prestrain level. The experimental and theoretical approaches presented open avenues and analytical tools for the study and applications of dispersive shock waves in solids.

On the rate of error growth in time for numerical solutions of nonlinear dispersive wave equations

We study the numerical error in solitary wave solutions of nonlinear dispersive wave equations. A number of existing results for discretizations of solitary wave solutions of particular equations indicate that the error grows quadratically in time for numerical methods that do not conserve energy, but grows only linearly for conservative methods. We provide numerical experiments suggesting that this result extends to a very broad class of equations and numerical methods.

Bright-dark wave envelopes of nonlinear regularized-long-wave and Riemann wave models in plasma physics

The nonlinear regularized-long-wave (RLW) and the Riemann wave (RW) models are physically significant in plasma physics and in further study of nonlinear dispersive waves, namely shallow water, ion-acoustic and magneto-sound waves in plasmas, anharmonic lattice and pressure waves in liquid-gas bubble mixtures, tidal and tsunami waves in rivers and oceans, longitudinal electromagnetic wave propagation in plasma, etc. The advanced generalized (G'/G)-expansion scheme is facilitated in this article for retrieving the bright-dark exact solitary wave solutions generated in the form of hyperbolic, trigonometric, and rational structures to the above-stated models. To figure out the internal contrivance of the solutions, 3D and contour plots concerning bright parabolic wave shape, compacton wave shape, bell-shaped soliton, kink wave type soliton, dark anti-parabolic wave shape, anti-compacton wave shape, bright propagation of solitary wave shape solutions are depicted for definite free parametric values. The characteristic feature of the wave envelops fluctuates taking into consideration the changes of free parameters and they are essentially dominated by the linear and nonlinear impact.

Chirped periodic and solitary waves in nonlinear negative index materials

Propagation of ultrashort pulses at least a few tens of optical cycles in duration through a negative index material is investigated theoretically based on the generalized nonlinear Schrödinger equation with pseudo-quintic nonlinearity and self-steepening effect. Novel periodic waves of different forms are shown to exist in the system in the presence of higher-order effects. It is found that such periodic structures exhibit an interesting chirping property which depends on the light field intensity. The nonlinearity in pulse chirp is found to be caused by the presence of self-steepening effect in the negative index medium. The solutions also comprise dark, bright, and kink solitary waves. Stability of the solitary wave solutions are proved analytically using the theory of nonlinear dispersive waves. The stability of the solutions is numerically studied under finite initial perturbations.