Evolution Of Solitary Waves Research Articles

Mixed neural operator learning on the solitary wave propagation over slope topography and inverse problem

This study proposes the mixed neural operator (MNO) learning framework, which further combines with the particle swarm optimization (PSO) to address challenges of solitary wave propagation over topography. The forward problem is defined as the evolution prediction of the solitary wave propagating over topography, while the inverse problem is defined as an optimization to identify the topography parameter based on the solitary wave elevation. Both the forward and inverse problems can be considered within a single framework and the dataset are provided by the classical Korteweg–de Vries (KdV) equation. The MNO framework is shown to simulate the evolution of solitary waves over topography, accurately capturing the wave elevation under different topographical conditions. By comparing with different neural operators, it is found that the U-shape neural operator is the most suitable for the KdV equation simulation. The coefficient of determination for the inverse problem based on the combination of MNO and PSO can reach 0.992, showing great potential of the approach in topography recognition. Finally, the proposed learning framework is preliminary applied to the prediction of the tsunami runup onto a complex beach, and a good agreement is also achieved between the direct simulation and the learning framework prediction.

A novel model for the fifth-order mKP equation describing (2+1)-dimensional shallow ocean solitary waves and their properties

In this paper, we develop a (2+1)-dimensional fifth-order modified Kadomtsev-Petviashvili (mKP) equation model using multiscale analysis and perturbation expansion, which is a novel model description of the shallow ocean solitary waves phenomenon, featuring stronger nonlinear terms and a higher-order compared to traditional models. We examine the conservation law within the model to further investigate the properties of solitary waves. Subsequently, we obtain the exact solutions of the model by employing the expansion method. We simulate the generation and propagation of solitary waves through appropriate parameter adjustments, deriving the solitary wave solutions, kink wave solutions, and periodic wave solutions. Additionally, we analyze the effects of different parameters on the propagation of solitary waves. Finally, we explore the evolution of solitary waves before and after generation under a given initial perturbation conditions. This study contributes to a deeper understanding of the intricate dynamics of shallow ocean solitary waves, shedding light on their behaviour and implications.

Effect of structural parameters of corrugated plate on the evolution of solitary waves and liquid film thickness distribution

The effect of wall distortion of the scrubbing-cooling tube on turbulent falling film is significant. The wall distortion was simplified by using corrugated plates; the effect of structural parameters (amplitude and wavelength) of corrugated plates on the action mechanism of solitary waves and the axial distribution of liquid film thickness were investigated. The results show that the Coanda effect decelerates and accelerates the flowing liquid film attached to the wall, which leads to the generation and extinction of solitary waves on the liquid film. The intensification of the local perturbation easily promotes the solitary waves to form slender water columns. Under the effect of Plateau-Rayleigh instability, the tails of the slender water columns break up to form tiny droplets. The mean liquid film thickness prediction model can accurately predict the mean liquid film thickness on corrugated plates (amplitude/wavelength = 0.02 ∼ 0.08) and the relative errors within ± 15%.

Integrable nonlinear perturbed hierarchies of NLS-mKdV equation and soliton solutions

We propose three spectral problems for NLS-mKdV equation by combining three integrable coupling ways. Then we obtain three nonlinear perturbation terms to derive three integrable nonlinear perturbed hierarchies of the NLS-mKdV equation. We proved the Lax integrability of the integrable nonlinear perturbed hierarchies. On the basis of a special orthogonal group, we prove the Liouville integrability of a third-order integrable nonlinear perturbed hierarchy of NLS-mKdV equation by deriving its bi-Hamiltonian structures. We build three Darboux matrices for constructing the Darboux transformations of the first two equations. As applications of the Darboux transformation, we present explicit solutions of these equations, three-dimensional plots, and density profiles the evolution of solitary waves.

Fast magnetosonic solitonic structures in a quasi-neutral collision-free finite-beta plasma

The equations of the model of two fluids of a two component collision-free, isotropic and fully ionized plasma in thermal equilibrium consisting of singly-ionized ions and electrons in a uniform magnetic field with heated ions and electrons are treated. We focus our attention on determining the global hydromagnetic fast magnetosonic traveling solitonic waves for plane wave equations. Ordinary solitary waves, bifurcating from the zero wave number, exist for the angles of inclination θ of the magnetic field vector to the direction x of the wave propagation, located in some left neighborhood of π/2. A global family of solitary waves containing solitary waves of all possible amplitudes is numerically determined. The dynamic stability of these solitary waves is investigated by numerical methods: the problem of the evolution of solitary waves via the time dependent field equations of the quasi-neutral collision-free finite beta-plasma is analyzed. We also concern the domains in parameter space where traveling generalized and envelope solitary waves exist and compute the possible forms of these waves and of so-called multi-soliton waves with several humps. The evolution of a localized perturbation is also treated numerically.

Localized ion-acoustic solitons in collision with dusty plasma

In this research, we investigate the characters of localized waves arriving from an electron-ion plasma into an electron-ion-dusty plasma region. For this purpose, we consider the presence of dust particles as a weak perturbation in the medium. It is shown that the ion-acoustic wave propagates with greater speed and higher energy when it interacts with dust particles. In addition, an oscillatory shock wave creates at the border of two different plasma environments that propagates in the opposite direction of the initial wave velocity. Time evolution of solitary wave and characters of created shock profile is also studied.

Soliton solutions for the nonlocal reverse space Kundu–Eckhaus equation via symbolic calculation

The Kundu–Eckhaus (KE) equation describes the propagation of ultrashort femtosecond pulses in optical fibers. In this paper, the Hirota bilinear method is used to deal with nonlocal KE equation which is also called nonlocal integrable nonlinear Schrödinger equation with cubic and quintic nonlinearities. The N-soliton solution for the nonlocal KE equation is derived by virtue of symbolic calculation. Based on that, the exact solution expressions of two-soliton and three-soliton can be obtained. Several propagation situations are demonstrated and discussed for the combinatorial solutions of nonlocal KE equation under different parameters, including one periodic solitary wave evolution, two parallel, perpendicular and periodic solitary waves collision.

Atypical evolution of solitary wave propagating in nonlinear elastic medium

The atypical evolution of a solitary cylindrical wave that propagates in the nonlinear elastic medium and has the initial profile in the form of the Macdonald function is described and commented. In the analysis, the approximate method of restriction on the gradient of a deformation is used, and three first approximations are taken into account. Two examples of typical wave evolution — harmoniс and bell-shaped waves — are shown and commented, where the first three approximations are also taken into account. The numerical modeling showed that the atypical initial profile (profile without a hump) evolves atypically — the profile becomes essentially steeper, saving the concavity, and the wave bottom decreases almost two times.

Numerical modelling of solitary wave and structure interactions using level-set and immersed boundary methods by adopting adequate inlet boundary conditions

ABSTRACT Solitary waves are often used to replicate tsunami waves and, therefore, the study of their propagation and interactions with structures could help to understand the impacts of tsunami waves on infrastructures. This paper presents a numerical model that uses the level-set framework for simulating the propagation of solitary waves and their interactions with structures using the immersed boundary method. The model imposes a set of appropriate boundary conditions to prescribe the incoming solitary waves within the sharp-interface immersed boundary method. Turbulence is modelled using the large-eddy simulation. Experimental tests are conducted to measure the free-surface elevation and velocity field evolution of solitary waves, as they propagate downwave interacting with two different wall-mounted structures. Analysis of the computed and experimental results revealed interesting dynamics of primary vortex-tubes forming near the sharp edges of structures. It is shown that the primary vortices eventually split to form pairs of asymmetrical doublets.

Dispersive and hyperbolic models for non-hydrostatic shallow water flows and their application to steep forced waves modelling

We propose a hyperbolic system of first-order equations that approximates the 1D Nwogu model of the shallow water theory for non-hydrostatic unsteady flows. Solitary waves in the framework of these models are constructed and studied. The evolution of solitary waves on a mildly sloping beach is considered. We show that the solution of the hyperbolic system practically coincides with the corresponding solution of the Nwogu dispersive equations. Steep forced water waves generated by a harmonically oscillating rectangular tank are studied both experimentally and numerically. A comparison of the solutions of the modified Green–Naghdi and Nwogu equations with the obtained experimental data is made.

Study on the head-on collisions of two-dimensional trains of solitons

The head on collisions of trains of solitons induced by a two-dimensional submerged elliptical cylinder at critical speed in shallow water are studied based on velocity potential theory. The boundary value problems are solved through boundary element method (BEM). The nonlinear free surface boundary conditions are satisfied. The mixed Euler–Lagrangian method is adopted to track the free surface through a time stepping scheme. The effects of thickness and velocity of the elliptical cylinder on the evolution of solitary waves have been investigated. Two sets of solitons are truncated from these trains of solitary waves. The head-on collisions of these solitons have been simulated. The wave profiles and velocity fields during collision have been analysed. The propagation of solitary waves is the transmissions of kinetic energy and the collision processes are the results of the dynamic balance of potential energy and kinematic energy.

The Role of Modified Boltzmann Relation With Electron Inertia on the Evolution of Solitary Waves Derived by Energy Integral Method Along With the Comparative Studies on Soliton Dynamics in Generalized Multi-Component Plasma

The nonlinear solitary waves in multi-component plasmas coexisting with negative ions, relativistic, and nonrelativistic electrons have been investigated under the influence of a modified Boltzmann relation with electron inertia $m_{\text {e}}$ (mass of electrons $\ne 0$ ) by Sagdeev potential and perturbation methods. The supersonic and subsonic compressive solitons are shown to be dependable on higher negative ion concentrations and the direction of wave propagation ( $k_{x} $ ). The condition of characteristic representation of the new Boltzmann relation appears to generate substantially high amplitude rarefactive solitons based on the direction of propagation, negative to positive ion mass ratio $r$ (>1), and high Mach number $M$ . The relativistic effects in non-Boltzmann electrons causing an increase in mass are responsible for the change in the growth of solitons in plasma.

Numerical simulation of internal wave propagation over a variable topography

The Boussinesq-type of equation is considered here as a model for describing interfacial wave dynamics in a two-layer fluid system. The equation, which is derived under rigid-lid assumption, has first order nonlinear term and dispersion term, and it holds for interfacial wave with long wavelength and small amplitude, relative to depth. The second-order Mac-Cormack scheme is implemented to solve this Boussinesq model. The numerical scheme is validated by simulating solitary wave as well as monotonic bore. Finally, the evolution of solitary wave propagating over a variable bathymetry with a shelf is examined.

Landau degeneracy effect on ion beam driven degenerate magneto plasma: Evolution of hypersonic soliton

The effect of quantized magnetic field and trapped electron on the characteristics of ion acoustic solitary wave in degenerate dense plasmas driven by ion beams is carried out. The evolution of solitary wave in such plasmas is described by deriving Zakharov–Kuznetsov (Z–K) equation considering classical magnetized positive ions and ion beams including the magnetically quantized trapped electron. Under suitable combination of physically admissible parameters, the existence conditions as well as nature of soliton propagation in such plasma system are examined. Three different modes of wave propagation namely fast, slow and ion acoustic modes are found to propagate in such plasma system. The degeneracy parameters such as magnetic quantization and temperature are found to have a decisive role on the solitonic characteristics of the plasma environment. Depending on the choice of degenerate parameters, the hypersonic soliton/solitary wave propagation is also observed and its nature with the relevant physical parameters is discussed.

Dynamics of Waves in the Cubically Nonlinear Model for Mutually Penetrating Continua

In this report we study the mathematical model for mutually penetrating continua. This model consists of the wave equation describing the carrying medium and the equation for oscillators forming the oscillating inclusion. Prescribing the constitutive equation of the carrying medium and kinetics of oscillator’s dynamics for model in question, the cubic nonlinearity is accounted. We are interested in the structure of wave solutions obeying the dynamical system of Hamiltonian type. This allows us to determine the peculiarities of the phase space of dynamical system, namely, the relation describing the homoclinic trajectories, the division of phase plane into the parts with equivalent orbits’ behavior, the conditions of bifurcations. To simulate the wave dynamics, we construct the three level finite-difference numerical scheme and study the evolution of solitary waves, their pair interactions and stability. Propagation of periodic waves is modeled as well.

Rossby solitary waves excited by the unstable topography in weak shear flow

A new forced KdV equation including topography is derived and the numerical solutions are given. The topographic variable should be related with the temporal and spatial function, which is called unstable topography. The physical features of the solitary waves about the mass and energy are discussed by theoretical analysis. In further studies, the pseudo-spectral numerical methods are used to discuss the evolution of solitary wave generated by the topography when meridional wave number $$m=1$$ ; in a similar way, we analyze the solitary wave when meridional wave number $$m=2$$ . At last, we make the comparison for the characteristics of waves between $$m=1$$ and $$m=2$$ , the wave of meridional number $$m=1$$ plays a leading role.

Evolution of a primary pulse in the granular dimers mounted on a linear elastic foundation: An analytical and numerical study.

In this exposition we consider the wave dynamics of a one-dimensional periodic granular dimer (diatomic) chain mounted on a damped and an undamped linear elastic foundation (otherwise called the on-site potential). It is very well known that periodic granular dimers support solitary wave propagation (similar to that in the homogeneous granular chains) for a specific discrete set of mass ratios. In this work we present the analytical investigation of the evolution of solitary waves and primary pulses in granular dimers when they are mounted on on-site potential with and without velocity proportional foundation damping. We invoke a methodology based on the multiple time-scale asymptotic analysis and partition the dynamics of the perturbed dimer chain into slow and fast components. The dynamics of the dimer chain in the limit of large mass mismatch (auxiliary chain) mounted on on-site potential and foundation damping is used as the basis for the analysis. A systematic analytical procedure is then developed for the slowly varying response of the beads and in estimating primary pulse amplitude evolution resulting in a nonlinear map relating the relative displacement amplitudes of two adjacent beads. The methodology is applicable for arbitrary mass ratios between the beads. We present several examples to demonstrate the efficacy of the proposed method. It is observed that the amplitude evolution predicted by the described methodology is in good agreement with the numerical simulation of the original system. This work forms a basis for further application of the considered methodology to weakly coupled granular dimers which finds practical relevance in designing shock mitigating granular layers.

Ion acoustic kinetic Alfvén rogue waves in two temperature electrons superthermal plasmas

The propagation properties of ion acoustic kinetic Alfven (IAKA) solitary and rogue waves have been investigated in two temperature electrons magnetized superthermal plasma in the presence of dust impurity. A nonlinear analysis is carried out to derive the Korteweg-de Vries (KdV) equation using the reductive perturbation method (RPM) describing the evolution of solitary waves. The effect of various plasma parameters on the characteristics of the IAKA solitary waves is studied. The dynamics of ion acoustic kinetic Alfven rogue waves (IAKARWs) are also studied by transforming the KdV equation into nonlinear Schrodinger (NLS) equation. The characteristics of rogue wave profile under the influence of various plasma parameters ( $\kappa_{c}$ , $\mu_{c}$ , $\sigma $ , $\theta$ ) are examined numerically by using the data of Saturn’s magnetosphere (Schippers et al. 2008; Sakai et al. 2013).

ANALYTICAL APPROXIMATE SOLUTION FOR THE FORCED KORTEWEG-DE VRIES (FKDV) ON CRITICAL FLOW OVER A HOLE USING HOMOTOPY ANALYSIS METHOD

Free surface flows in a two-dimensional channel past over a hole is studied using shallow water forced Korteweg-de Vries (fKdV) equation. The forcing term of fKdV equation represents the hole shaped bottom topography. Froude number (Fr), which represents the ratio of flow speed to the wave speed, will also be used in solving fKdV equation. The fKdV equation is solved using Homotopy Analysis Method (HAM). HAM is an approximate analytical technique used to obtain series of solutions for the nonlinear problems where HAM has an auxiliary parameter coto adjust and control the convergence region of the series solution. Solitary wave solutions are obtained from the series of solutions of HAM and wave flows are observed at particular time. The HAM solution shows the hole shaped bottom topography plays an important role in determining the evolution of solitary waves.

Multi-Symplectic Method for the Zakharov-Kuznetsov Equation

AbstractA new scheme for the Zakharov-Kuznetsov (ZK) equation with the accuracy order ofis proposed. The multi-symplectic conservation property of the new scheme is proved. The backward error analysis of the new multi-symplectic scheme is also implemented. The solitary wave evolution behaviors of the Zakharov-Kunetsov equation is investigated by the new multi-symplectic scheme. The accuracy of the scheme is analyzed.