Dispersive Wave System Research Articles

An Efficient Local Artificial Boundary Condition for Infinite Long Finite Element Euler–Bernoulli Beam

To solve the wave propagation problems of the Euler–Bernoulli beam in an unbounded domain effectively and efficiently, a new local artificial boundary condition technology is proposed. It replaces the residual right-hand side of the truncated discrete equation with an equivalent linear algebraic system. First, the equivalent Schrodinger equation is discussed. Its artificial boundary condition is obtained by first rationalizing the Dirichlet-to-Neumann condition in the frequency domain with a Pade approximation and then inverse transforming each Pade term back into the time domain by introducing auxiliary degrees of freedom. Frequency shifting is employed such that it performs better near a prescribed frequency. Then, the artificial boundary condition of the finite element Euler–Bernoulli beam is obtained by simple algebraic manipulations on that of the corresponding Schrodinger equation. This method only makes local changes to the original truncated discrete dynamic system and thus is very efficient and easy to use. The accuracy of the proposed method can be improved by using more Pade terms and a proper shift frequency. The numerical example shows, with only a few additional degrees of freedom, the proposed artificial boundary condition effectively eliminates the spurious reflection. The idea of the proposed method can also be used in other dispersive wave systems.

Constructing artificial boundary condition of dispersive wave systems by deep learning neural network

To solve one dimensional dispersive wave systems in an unbounded domain, a uniform way to establish localized artificial boundary conditions is proposed. The idea is replacing the half-infinite interval outside the region of interest with a super element which exhibits the same dynamics response. Instead of designing the detailed mechanical structures of the super element, we directly reconstruct its stiffness, mass, and damping matrices by matching its frequency-domain reaction force with the expected one. An artificial neural network architecture is thus specifically tailored for this purpose. It comprises a deep learning part to predict the response of generalized degrees of freedom under different excitation frequencies, along with a simple linear part for computing the external force vectors. The trainable weight matrices of the linear layers correspond to the stiffness, mass, and damping matrices we need for the artificial boundary condition. The training data consists of input frequencies and the corresponding expected frequency domain external force vectors, which can be readily obtained through theoretical means. In order to achieve a good result, the neural network is initialized based on an optimized spring-damper-mass system. The adaptive moment estimation algorithm is then employed to train the parameters of the network. Different kinds of equations are solved as numerical examples. The results show that deep learning neural networks can find some unexpected optimal stiffness/damper/mass matrices of the super element. By just introducing a few additional degrees of freedom to the original truncated system, the localized artificial boundary condition works surprisingly well.

A robust way to justify the derivative NLS approximation

The derivative nonlinear Schrödinger (DNLS) equation can be derived as an amplitude equation via multiple scaling perturbation analysis for the description of the slowly varying envelope of an underlying oscillating and traveling wave packet in dispersive wave systems. It appears in the degenerated situation when the cubic coefficient of the similarly derived NLS equation vanishes. It is the purpose of this paper to prove that the DNLS approximation makes correct predictions about the dynamics of the original system under rather weak assumptions on the original dispersive wave system if we assume that the initial conditions of the DNLS equation are analytic in a strip of the complex plane. The method is presented for a Klein–Gordon model with a cubic nonlinearity.

Lie group analysis with the optimal system, generalized invariant solutions, and an enormous variety of different wave profiles for the higher-dimensional modified dispersive water wave system of equations

Lie group analysis with the optimal system, generalized invariant solutions, and an enormous variety of different wave profiles for the higher-dimensional modified dispersive water wave system of equations

A generalized dispersive water waves system: Conservation laws; symmetry reduction; travelling wave solutions; symbolic computation

Symmetry analysis is applied to the generalized dispersive water waves system, which is seen in many areas of physics including fluid dynamics and particle physics. The construction of new exact solutions and similarity reductions. A multiplier technique is then used to derive conservation laws. The generalized dispersive water waves system’s solutions were also attained using the most basic techniques. An integrated generalized dispersive water wave system is constructed using the Lie group analysis technique. Solitary wave and travelling wave solutions are among the outcomes.

The nonlinear vibration and dispersive wave systems with extended homoclinic breather wave solutions

Abstract This article investigates the extended homoclinic (heteroclinic) breather wave solutions and interaction periodic and dark soliton solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, breather, and soliton solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole–Hopf algorithm to obtain the exact solutions of the ( 2 + 1 2+1 )-dimensional modified dispersive water-wave system. The analytical treatment of extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are studied and plotted in four forms of density plots. A nonlinear vibration system will be studied. Employing appropriate mathematical assumptions, the novel kinds of the extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are derived and constructed in view of the combination of kink, periodic, and soliton for an extended homoclinic breather and also a combination of two kinks, periodic and dark soliton in terms of exponential, trigonometric, hyperbolic functions for interaction periodic and dark soliton of the governing equation. To achieve this, the illustrative example of the ( 2 + 1 2+1 )-D modified dispersive water-wave system is furnished to demonstrate the feasibility and reliability of the procedure applied in this research. The trajectory solutions of the traveling waves are offered explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested scheme with the other existing methods, the results state that the execution of this technique is succinct, extensive, and straightforward.

Regarding the shallow water in an ocean via a Whitham-Broer-Kaup-like system: hetero-Bäcklund transformations, bilinear forms and [formula omitted] solitons

Considering the water waves, people have investigated many systems. In this paper, what we study is a Whitham-Broer-Kaup-like system for the dispersive long waves in the shallow water in an ocean. With respect to the water-wave horizontal velocity and deviation height from the equilibrium of the water, we construct (A) two branches of the hetero-Bäcklund transformations, from that system to a known constant-coefficient nonlinear dispersive-wave system, (B) two branches of the bilinear forms and (C) two branches of the M-soliton solutions, with M as a positive integer. Results rely upon the oceanic shallow-water coefficients in that system.

The nonlinear vibration and dispersive wave systems with cross-kink and solitary wave solutions

This paper investigates the cross-kink and solitary wave solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, cross-kink and solitary wave solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole–Hopf algorithm to get the exact solutions of the [Formula: see text]-dimensional modified dispersive water-wave system. The analytical treatment of cross-kink, solitary wave solutions is studied and plotted in three forms of two, density and three-D plots. A nonlinear vibration system will be investigated. Employing appropriate mathematical assumptions, the novel kinds of the cross-kink and solitary wave solutions are derived and constructed in view of the combination of kink, periodic and soliton for cross-kink and also a combination of two kinks in terms of exponential functions for solitary of the governing equation. To achieve this, the illustrative example of the [Formula: see text]-D modified dispersive water-wave system is furnished to exhibit the feasibility and reliability of the procedure utilized in this research. The trajectory solutions of the traveling waves are presented explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested technique with the other existing schemes, the results present that the execution of this technique is concise, simple and straightforward.

Oceanic long-gravity-water-wave investigations on a variable-coefficient nonlinear dispersive-wave system

Oceanic water waves are actively investigated. One example is a variable-coefficient nonlinear dispersive-wave system modelling the long gravity water waves in a shallow oceanic environment. The system describes the surface velocity of the water wave and the wave elevation. Making use of symbolic computation, we perform the Painlevé analysis and work out two sets of the bilinear forms, two sets of the N-soliton solutions and one set of the similarity reductions for the aforementioned system, with N being a positive integer. We also graphically discuss those soliton solutions. What we accomplish should rely on the variable coefficients. This paper could be of some use for the future oceanic studies.

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.

Exact solutions of a class of nonlinear dispersive long wave systems via Feng's first integral method

<abstract> In this paper, eight groups of exact solutions for the (1+1)-dimensional and (2+1)-dimensional nonlinear dispersive long wave systems are found respectively via Feng's first integral method. It is shown that there are some similarities in the expressions of the solutions of (1+1)-dimensional and (2+1)-dimensional DLWEs, while there exist some differences in their dimensions and their physical significance. Finally, some graphs are presented to show these features, which also show the effectiveness of the proposed method. </abstract>

Oceanic studies via a variable-coefficient nonlinear dispersive-wave system in the Solar System

Abstract People pay attention to the oceans crossing the Solar System: the Earth, Enceladus and Titan. Hereby, on a variable-coefficient nonlinear dispersive-wave system for the shallow oceanic environment, (A) nonlinear-water-wave symbolic computation and Bell polynomials lead to two hetero-Backlund transformations and (B) nonlinear-water-wave symbolic computation gives rise to a similarity reduction, both depending on the variable coefficients representing the wave elevation and surface velocity of the water wave. This paper might be of some use for the future oceanic studies on the Solar System.

Interaction properties of solitons for a couple of nonlinear evolution equations

We study one- and two-soliton solutions for the Cahn–Allen (CA) equation and the Brethorton equation. The CA equation has broad spectrum of applications especially in anti-phase boundary motion and it is used in phase-field models. While the Brethorton equation is a model for dispersive wave systems, it is used to find the resonant nonlinear interaction among three linear modes. We use the Hirota bilinear method to obtain one- and two-soliton solutions to the CA equation and the Brethorton equation.

Dispersive focusing in fractional Korteweg–de Vries-type equations

In this paper we study dispersive enhancement of a wave train in systems described by the fractional Korteweg–de Vries-type equations of the form where the operator Dm{u} is written in the Fourier space, αn, βm are arbitrary constants and n, m being rational numbers (positive or negative). Using both approximate and exact solutions of these wave equations we describe constructively the process of dispersive focusing. It is based on a time-reversing approach with the expected rogue wave chosen as the initial condition for a solution of these equations. We demonstrate the qualitative difference in the shape of the focused wavetrains for various n and m. Our results can be used for prediction of the rogue wave appearance arising in many types of weakly nonlinear and weakly dispersive wave systems in physical context.

Viewing the Solar System via a variable-coefficient nonlinear dispersive-wave system

Oceans crossing the Solar System attract people’s attention: the Earth, Enceladus, and Titan. On a variable-coefficient nonlinear dispersive-wave system for the shallow oceanic environment, our symbolic computation yields two non-auto-Backlund transformations and auto-Backlund transformations with some solitons, with regard to the wave elevation and surface velocity of the water wave, which depend on the variable coefficients. This paper could be of some use for the future oceanic studies in the Solar System.

Magnetohydrodynamic Richtmyer–Meshkov instability under an arbitrarily oriented magnetic field

The effect of an initially uniform magnetic field of arbitrary orientation on the Richtmyer–Meshkov instability in Hall-magnetohydrodynamics (MHD) and ideal MHD is considered. Attention is restricted to the case where the initial density interface has a single-mode sinusoidal perturbation in amplitude and is accelerated by a shock traveling perpendicular to the interface. An incompressible Hall-MHD model for this flow is developed by solving the relevant impulse-driven linearized initial value problem. The ideal MHD theory is naturally obtained by taking the limit of vanishing ion skin depth. It is shown that the out-of-plane magnetic field component normal to both the impulse and the interface perturbation does not affect the evolution of the flow. For all field orientations other than strictly out-of-plane, the growth of interface perturbations is suppressed. However, the suppression is most effective for near tangential fields but becomes less effective with increasing ion skin depth and Larmor radius. The modeled suppression mechanism is transport of vorticity along magnetic field lines via Alfvén fronts in ideal MHD, and via a dispersive wave system in Hall-MHD. Oscillation of the interface growth rate is caused by a continuous phase change of the induced velocities at the interface due to vorticity transport parallel to the perturbation direction in ideal MHD, while it can also result from interfacial vorticity production associated with the ion cyclotron effect in Hall-MHD with a finite Larmor radius. The limiting flow behavior of a large ion-skin-depth is explored. To assess the accuracy and appropriateness of the incompressible model, its ideal MHD predictions are compared to the results of the corresponding shock-driven nonlinear compressible simulations.

Anomalous Correlators in Nonlinear Dispersive Wave Systems

An extension to wave turbulence theory shows that in a system of nonlinear waves, correlations among the waves arise, causing ``ghost'' excitations that lead to coherent structures in physical space.

Failure of the N‐wave interaction approximation without imposing periodic boundary conditions

AbstractThe N‐wave interaction (NWI) system appears as an amplitude system in the description of N nonlinearly interacting and linearly transported wave packets in dispersive wave systems such as the water wave problem or problems in nonlinear optics. The purpose of this paper is twofold. First we give a new simplified proof for the failure of the NWI approximation in case of resonances which are located at integer multiples of a basic wave number k0 in the original ‐spatially periodic dispersive wave system. Secondly, we give a first rigorous proof that an amplitude system fails in the description of an original system, without imposing periodic boundary conditions on the original system.

The KdV approximation for a system with unstable resonances

The KdV equation can be derived via multiple scaling analysis for the approximate description of long waves in dispersive systems with a conservation law. In this paper, we justify this approximation for a system with unstable resonances by proving estimates between the KdV approximation and true solutions of the original system. By working in spaces of analytic functions, the approach will allow us to handle more complicated systems without a detailed discussion of the resonances and without finding a suitableenergy.

Impulse-driven Richtmyer-Meshkov instability in Hall-magnetohydrodynamics

We utilize the incompressible, Hall-magnetohydrodynamics (MHD) model for conducting fluids to investigate the effect of Hall current on the stability of an impulsively accelerated, perturbed density interface, or contact discontinuity (CD) separating two fluids in the presence of a background magnetic field. This is used as a simple model, in a conducting fluid, of a Richtmyer-Meshkov type flow that is characterized in a neutral fluid by a shock-wave-density-interface interaction. Two versions of the Hall-MHD equations are explored. In the first, the ions are treated as an incompressible fluid but the electron gas retains its compressibility, while for the second version the incompressible limit for both species is invoked. The linearized equations of motion are first formulated for a sinusoidal interface perturbation and then solved as an initial-value problem using a Laplace transform method with general numerical inversion but with analytical inversion for some limiting-parameter cases. While the field equations are identical for both Hall-MHD models, the CD-jump conditions differ leading to qualitatively similar but quantitatively different CD dynamics. For both models, the presence of the magnetic field is found to suppress the incipient interfacial growth associated with neutral-gas, Richtmyer-Meshkov instability (RMI). When the ion skin depth is finite, the vorticity dynamics that drive the suppression of the RMI differ markedly from the ideal MHD, RMI flow. On the interface, the Hall-MHD description allows the presence of a tangential slip velocity which leads to finite circulation deposition. Away from the interface, vorticity is produced by the perturbed magnetic fields and transported to infinity by a dispersive wave system. This leads to decay of the velocity slip at the interface with the effect that interface growth remains bounded but distorted by damped oscillations associated with the ion cyclotron effect. The flow behavior for several limits of the ion skin depth and the Larmor radius is explored. Specific comparisons with the results from both models against ideal MHD are presented.