Symmetric Solitary Waves Research Articles

Novel insights into the propagation of the generalized Boussinesq equation: Dynamics of bidirectional symmetric solitary waves

In this work, we explore the propagation of bidirectional solitary wave solutions in the generalized (2+1)-dimensional Boussinesq equation. The proposed model was recently introduced by Wazwaz and Kaur as an application to describe wave propagation in various multidimensional physical systems, including shallow water waves and similar phenomena. Via implementing a collection of effective approaches, we extract a diversity physical structures of the proposed model manifesting their propagation in a form of moving simultaneous symmetric bidirectional waves. The proposed methodologies will include the use of sine/cosine-function method, sinh/cosh-function method, tanh–coth expansion approach, and the rational form trial solutions including trigonometric and hyperbolic functions. By conducting graphical analyses, we illustrate the various physical structures and dynamic behaviors of this class of nonlinear partial differential equations. The results of this work are expected to facilitate the study of wave interactions, propagation dynamics, and soliton formation in multidimensional settings.

Features of propagation and properties of axially symmetric and azimuthal solitary waves on the curved surface of a half-space cold plasma

The analysis of the behavior features, wave characteristics, and spatial-wave structure of axial-symmetric and azimuthal solitary oscillations at the boundary of a cold semi-infinite plasma is carried out using a closed system of equations describing the interdependence of the curvature of the surface and the properties of the nonlinear surface charge of the plasma. The specificity of the nature of motion, spatial distribution, and interconnection of the parameters of wave formations of each type is described. Numerical calculations have been performed, and an approximate analytical solution has been found.

Spatial Dynamics and Solitary Hydroelastic Surface Waves

This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal fluid (of finite depth and in irrotational motion). The theory takes the form of a review of the Kirchgässner reduction to a finite-dimensional Hamiltonian system, highlighting the refinements in the theory over the years and presenting some novel aspects including the use of a higher-order Legendre transformation to formulate the problem as a spatial Hamiltonian system, and a Riesz basis for the phase space to complete the analogy with a dynamical system. The reduced system is to leading order given by the focussing cubic nonlinear Schrödinger equation, agreeing with the result of formal weakly nonlinear theory (which is included for completeness). We give a precise proof of the persistence of two of its homoclinic solutions as solutions to the unapproximated reduced system which correspond to symmetric hydroeleastic solitary waves.

Localized Symmetric and Asymmetric Solitary Wave Solutions of Fractional Coupled Nonlinear Schrödinger Equations

The existence of solutions with localized solitary wave structures is one of the significant characteristics of nonlinear integrable systems. Darboux transformation (DT) is a well-known method for constructing multi-soliton solutions, using a type of localized solitary wave, of integrable systems, but there are still no reports on extending DT techniques to construct such solitary wave solutions of fractional integrable models. This article takes the coupled nonlinear Schrödinger (CNLS) equations with conformable fractional derivatives as an example to illustrate the feasibility of extending the DT and generalized DT (GDT) methods to construct symmetric and asymmetric solitary wave solutions for fractional integrable systems. Specifically, the traditional n-fold DT and the first-, second-, and third-step GDTs are extended for the fractional CNLS equations. Based on the extended GDTs, explicit solutions with symmetric/asymmetric soliton and soliton–rogon (solitrogon) spatial structures of the fractional CNLS equations are obtained. This study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.

Traveling Wave Solutions for Time-Fractional mKdV-ZK Equation of Weakly Nonlinear Ion-Acoustic Waves in Magnetized Electron–Positron Plasma

In this paper, we discuss the time-fractional mKdV-ZK equation, which is a kind of physical model, developed for plasma of hot and cool electrons and some fluid ions. Based on the properties of certain employed truncated M-fractional derivatives, we reduce the time-fractional mKdV-ZK equation to an integer-order ordinary differential equation utilizing an adequate traveling wave transformation. Further, we derive a dynamical system to present bifurcation of the equation equilibria and show existence of solitary and kink singular wave solutions for the time-fractional mKdV-ZK equation. Furthermore, we establish symmetric solitary, kink, and singular wave solutions for the governing model by using the ansatz method. Moreover, we depict desired results at different physical parameter values to provide physical interpolations for the aforementioned equation. Finally, we introduce applications of the governing model in detail.

Novel Approximate Analytical Solutions to the Nonplanar Modified Kawahara Equation and Modeling Nonlinear Structures in Electronegative Plasmas

In this investigation, the nonplanar (spherical and cylindrical) modified fifth-order Korteweg–de Vries (nmKdV5) equation, otherwise known as the nonplanar modified Kawahara equation (nmKE), is solved using the ansatz approach. Two general formulas for the semi-analytical symmetric approximations are derived using the recommended methodology. Using the obtained approximations, the nonplanar modified Kawahara (mK) symmetric solitary waves (SWs) and cnoidal waves (CWs) are obtained. The fluid equations for the electronegative plasmas are reduced to the nmKE as a practical application for the obtained solutions. Using the obtained solutions, the characteristic features of both the cylindrical and spherical mK-SWs and -CWs are studied. All obtained solutions are compared with each other, and the maximum residual errors for these approximations are estimated. Numerous researchers that are interested in studying the complicated nonlinear phenomena in plasma physics can use the obtained approximations to interpret their experimental and observational findings.

Simulation Studies on the Dissipative Modified Kawahara Solitons in a Complex Plasma

In this work, a damped modified Kawahara equation (mKE) with cubic nonlinearity and two dispersion terms including the third- and fifth-order derivatives is analyzed. We employ an effective semi-analytical method to achieve the goal set for this study. For this purpose, the ansatz method is implemented to find some approximate solutions to the damped mKE. Based on the proposed method, two different formulas for the analytical symmetric approximations are formally obtained. The derived formulas could be utilized for studying all traveling waves described by the damped mKE, such as symmetric solitary waves (SWs), shock waves, cnoidal waves, etc. Moreover, the energy of the damped dressed solitons is derived. Furthermore, the obtained approximations are used for studying the dynamics of the dissipative dressed (modified Kawahara (mK)) dust-ion acoustic (DIA) solitons in an unmagnetized collisional superthermal plasma consisting of inertia-less superthermal electrons and inertial cold ions as well as immobile negative dust grains. Numerically, the impact of the collisional parameter that arises as a result of taking the ion-neutral collisions into account and the electron spectral index on the profile of the dissipative structures are examined. Finally, the analytical and numerical approximations using the finite difference method (FDM) are compared in order to confirm the high accuracy of the obtained approximations. The achieved results contribute to explaining the mystery of several nonlinear phenomena that arise in different plasma physics, nonlinear optics, shallow water waves, oceans, and seas, and so on.

Analysis of solitary waves in fluid-filled thin-walled electroactive tubes

The influence of an electric field on the propagation of axially symmetric solitary waves in a fluid-filled thin-walled tube made of a dielectric elastomer is analysed. The electric field is applied by means of soft flexible electrodes lining the internal and external surfaces of the tube and connected to a power source with a given constant difference of potential. For a specific hyperelastic constitutive law, numerical results are obtained and discussed.

Formation of Asymmetric Electron Acoustic Double Layers in the Earth's Inner Magnetosphere

AbstractThe Van Allen Probes have observed both symmetric and asymmetric bipolar electric field structures in the Earth's inner magnetosphere. In general, the symmetric bipolar structures are identified as electron‐phase space holes, whereas the asymmetric structures are interpreted as electron acoustic double layers (EADLs). The generation mechanism of these EADLs is not entirely understood yet. We have modeled the EADLs observed on 13 November 2012 by Van Allen Probe‐B. We performed a fluid simulation of the EADLs and tracked their formation and evolution in the simulation. We found that the localized depletion and enhancement in the electron populations act as a perturbation to excite the symmetric bipolar electron acoustic solitary waves, which later evolve into the EADLs. The Ponderomotive force is found to be the main driver behind transformation of the symmetric electron acoustic solitary waves to EADLs via formation of the electron acoustic shocks.

Upper bound on the slope of steady water waves with small adverse vorticity

We consider the angle of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. There is an upper bound of 31.15° in the irrotational case [1] and an upper bound of 45° in the case of favorable vorticity [13]. On the other hand, if the vorticity is adverse, the profile can become vertical. We prove here that if the adverse vorticity is sufficiently small, then the angle still has an upper bound which is slightly larger than 45°.

Bound on the Slope of Steady Water Waves with Favorable Vorticity

We consider the angle $\theta$ of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. Although $\theta$ may surpass 30$^\circ$ for some irrotational waves close to the extreme wave, Amick [Ami87] proved that for any irrotational wave the angle must be less than 31.15$^\circ$. Is the situation similar for periodic or solitary waves that are not irrotational? The extreme Gerstner wave has infinite depth, adverse vorticity and vertical cusps ($\theta = 90^\circ$). Moreover, numerical calculations show that even waves of finite depth can overturn if the vorticity is adverse. In this paper, on the other hand, we prove an upper bound of 45$^\circ$ on $\theta$ for a large class of waves with favorable vorticity and finite depth. In particular, the vorticity can be any constant with the favorable sign. We also prove a series of general inequalities on the pressure within the fluid, including the fact that any overturning wave must have a pressure sink.

New hydroelastic solitary waves in deep water and their dynamics

A numerical study of fully nonlinear waves propagating through a two-dimensional deep fluid covered by a floating flexible plate is presented. The nonlinear model proposed by Toland (Arch. Rat. Mech. Anal., vol. 289, 2008, pp. 325–362) is used to formulate the pressure exerted by the thin elastic sheet. The symmetric solitary waves previously found by Guyenne & Părău (J. Fluid Mech., vol. 713, 2012, pp. 307–329) and Wang et al. (IMA J. Appl. Maths, vol. 78, 2013, pp. 750–761) are briefly reviewed. A new class of hydroelastic solitary waves which are non-symmetric in the direction of wave propagation is then computed. These asymmetric solitary waves have a multi-packet structure and appear via spontaneous symmetry-breaking bifurcations. We study in detail the stability properties of both symmetric and asymmetric solitary waves subject to longitudinal perturbations. Some moderate-amplitude symmetric solitary waves are found to be stable. A series of numerical experiments are performed to show the non-elastic behaviour of two interacting stable solitary waves. The large response generated by a localised steady pressure distribution moving at a speed slightly below the minimum of the phase speed (called the transcritical regime in the literature) is also examined. The direct numerical simulation of the fully nonlinear equations with a single load reveals that in this range the generated waves are of finite amplitude. This includes a perturbed depression solitary wave, which is qualitatively similar to the large response observed in experiments. The excitations of stable elevation solitary waves are achieved by applying multiple loads moving with a speed in the transcritical regime.

Solitary Water Waves of Large Amplitude Generated by Surface Pressure

We consider exact nonlinear solitary water waves on a shear flow with an arbitrary distribution of vorticity. Ignoring surface tension, we impose a non-constant pressure on the free surface. Starting from a uniform shear flow with a flat free surface and a supercritical wave speed, we vary the surface pressure and use a continuation argument to construct a global connected set of symmetric solitary waves. This set includes waves of depression whose profiles increase monotonically from a central trough where the surface pressure is at its lowest, as well as waves of elevation whose profiles decrease monotonically from a central crest where the surface pressure is at its highest. There may also be two waves in this connected set with identical surface pressure, only one of which is a wave of depression.

Electron beam-associated symmetric electrostatic solitary waves on the separatrix of magnetic reconnection: multi-spacecraft analysis

We present an in situ evidence of electron beam-associated symmetric bipolar electrostatic solitary waves (ESWs) on the current sheet-side of the separatrix of the magnetic reconnection in the near-Earth magnetotail by multi-spacecraft observation of Cluster. Within one spin period, 42 cases of symmetric ESWs are continuously observed during 2 s by SC2 while other spacecrafts do not “detect” them. And the Plasma Electron and Current Experiment (PEACE) spinPAD mode data exhibits unidirectional electron beam antiparallel to the ambient field, and no electron beam-like distribution is found by other spacecrafts without ESW observation. Though the electron beam is strongly associated with the ESWs in observation by multiple spacecraft differentiation, however, the relationship between the counter-directed electron beam and the simultaneously observed ESWs remains unclear and open to the next study.

Stability of Symmetric Solitary Wave Solutions of a Forced Korteweg-de Vries Equation and the Polynomial Chaos

Stability of Symmetric Solitary Wave Solutions of a Forced Korteweg-de Vries Equation and the Polynomial Chaos

Multilump Symmetric and Nonsymmetric Gravity-Capillary Solitary Waves in Deep Water

Multilump gravity-capillary solitary waves propagating in a fluid of infinite depth are computed numerically. The study is based on a weakly nonlinear and dispersive partial differential equation (PDE) with weak variations in the spanwise direction, a model derived by Akers and Milewski [Stud. Appl. Math., 122 (2009), pp. 249--274]. For a two-dimensional fluid, this model agrees qualitatively well with the full Euler equations for the bifurcation curves, wave profiles, and dynamics of solitary waves. Fully localized solitary waves are then computed for three-dimensional fluids. New symmetric lump solutions are computed by using a continuation method to follow the branch of elevation waves. It is then found that the branch of elevation waves has multiple turning points from which new solutions, consisting of multiple lumps separated by smaller oscillations, bifurcate. Nonsymmetric solitary waves, which also feature a multilump structure, are computed and found to appear via spontaneous symmetry-breaking bi...

Asymmetric gravity–capillary solitary waves on deep water

AbstractWe present new families of gravity–capillary solitary waves propagating on the surface of a two-dimensional deep fluid. These spatially localised travelling-wave solutions are non-symmetric in the wave propagation direction. Our computation reveals that these waves appear from a spontaneous symmetry-breaking bifurcation, and connect two branches of multi-packet symmetric solitary waves. The speed–energy bifurcation curve of asymmetric solitary waves features a zigzag behaviour with one or more turning points.

Existence of solitary waves in nonlocal nematic liquid crystals

In this paper, we study the solitary waves in nonlocal nematic liquid crystals. By applying the Mountain Pass Theorem and the Krasnoselskii genus theory, we prove some existence and multiplicity results of radially symmetric solitary waves.

Exact and approximate solutions for optical solitary waves in nematic liquid crystals

The equations governing optical solitary waves in nonlinear nematic liquid crystals are investigated in both (1+1) and (2+1) dimensions. An isolated exact solitary wave solution is found in (1+1) dimensions and an isolated, exact, radially symmetric solitary wave solution is found in (2+1) dimensions. These exact solutions are used to elucidate what is meant by a nematic liquid crystal to have a nonlocal response and the full role of this nonlocal response in the stability of (2+1) dimensional solitary waves. General, approximate solitary wave solutions in (1+1) and (2+1) dimensions are found using variational methods and they are found to be in excellent agreement with the full numerical solutions. These variational solutions predict that a minimum optical power is required for a solitary wave to exist in (2+1) dimensions, as confirmed by a careful examination of the numerical scheme and its solutions. Finally, nematic liquid crystals subjected to two different external electric fields can support the same solitary wave, exhibiting a new type of bistability.

On effective solutions of the nonlinear Schrödinger equation

Cubic nonlinear Schrödinger type equation with specific initial-boundary conditions in the infinite domain is considered. The equation is reduced to an equivalent system of partial differential equations and studied in the case of solitary waves. The system is modified by introducing new functions, one of which belongs to the class of functions of negligible fifth order and vanishing at infinity exponentially. For this class of functions the system is reduced to a nonlinear elliptic equation which can be solved analytically, thereby allowing us to present nontrivial approximated solutions of nonlinear Schrödinger equation. These solutions describe a new class of symmetric solitary waves. Graphics of modulus of the corresponding wave function are constructed by using Maple.