Cnoidal Research Articles

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.

Optical Solitons for a Concatenation Model by Trial Equation Approach

This paper addresses the newly proposed concatenation model by the usage of trial equation approach. The concatenation is a chain model that is a combination of the nonlinear Schrodinger’s equation, Lakshmanan–Porsezian–Daniel model as well as the Sasa–Satsuma equation. The recovered solutions are displayed in terms of dark solitons, singular solitons, cnoidal waves and singular periodic waves. The trial equation approach enables to recover a wide spectrum of solutions to the governing model. The numerical schemes give a visual perspective to the solutions derived analytically.

An ISPH with modified k–ε closure for simulating breaking periodic waves

The Incompressible Smoothed Particle Hydrodynamics (ISPH) method, solving the 2D RANS (Reynolds Averaged Navier-Stokes) equations with the modified k–ε turbulence closure developed by Wang and Liu (2020) [ Coastal Engineering , 157, 1–28], is further extended to periodic waves. The capability of this numerical model is demonstrated by applying it to three laboratory experiments: (1) Dynamics of surf-zone turbulence generated by a spilling breaker and a plunging breaker over a plane slope in small-scale wave flume experiments, (2) breaking wave generated turbulence over a stationary barred beach in large-scale wave flume experiments, and (3) spatial and temporal distributions of turbulence generated by bi-chromatic breaking waves over a submerged stationary barred slope in medium-scale wave flume experiments. In this new ISPH model, the gradient correction_ dynamically stabilized scheme is adopted to reduce numerical dissipation. The experimental data for the spilling breaker on a plane beach is used to calibrate the model. Once the model is calibrated, all the empirical coefficients in the turbulence closure model, including the stress limiter coefficient λ 3 , which is required in modifying the eddy viscosity, are fixed for the validation processes with other experimental data sets. All model-data comparisons are made in terms of free surface profile, mean velocity field and turbulent kinetic energy. The differences between experimental observations and numerical simulations are quantified by the percentage error. Overall, good agreement is observed for all experiments. This paper presents the first comprehensively validated 2D ISPH model with the modified k–ε turbulence closure, which can be applied to periodic wave breaking problems. • The Incompressible Smoothed Particle Hydrodynamics (ISPH) method is coupled with the modified k–ε turbulence closure equations to solve the 2D RANS (Reynolds Averaged Navier-Stokes) equations. The gradient correction_ dynamically stabilized scheme is adopted to reduce numerical dissipation. • The model has been checked with three laboratory experiments for periodic breaking waves. (1) Spilling and plunging cnoidal waves over a plane slope in small-scale wave flume experiments, (2) plunging periodic breaking wave over a stationary barred beach in large-scale wave flume experiments, and (3) bi-chromatic breaking waves over a submerged stationary barred slope in medium-scale wave flume experiments. • All model-data comparisons are made in terms of free surface profile, mean velocity field and turbulent kinetic energy. • The differences between experimental observations and numerical simulations are quantified by the percentage error. • Overall, good agreement is observed for all experiments. Good agreements are observed in terms of free surface profile, mean velocity field, vorticity field, turbulent kinetic energy and turbulent shear stress.

Long-wave response to laterally periodic reef-lagoon bathymetry

Longwave response in a reef-lagoon bathymetry adjacent to a shore is studied by performing a series of numerical experiments. We consider the problem such that the length scale of the lagoon is comparable to the reef. A series of offshore reefs is formed periodically in the alongshore direction, parallel to the shoreline; adjacent reefs are separated with aperture channels. We impose cnoidal waves for the incident longwave train with the length scale comparable to the reef and lagoon dimensions. Our results reveal that the incident waves amplify in the region behind the reef, and attenuate in the region directly exposed to the aperture; the resulting variation of wave amplification along the shore is found to be substantial. This somewhat counterintuitive outcome is caused by a combination of diffraction and refraction of the long waves interacting with the reef-lagoon bathymetry of comparable length scale. We also found that the pattern of the offshore wavefield may not identify the reef bathymetry characteristics; it is because the pattern of offshore wave formation is influenced by a complex combination of wave reflection from the reef-lagoon system. The induced residual currents are also examined. The results show that notable offshore-directed residual currents are formed in the aperture channel only when the reef height is high with respect to the ambient water depth and when the aperture width is narrow compared with the reef's alongshore breadth. Otherwise, the residual currents in the onshore direction are formed, although the magnitude is very small. It is demonstrated that the residual currents are induced by a combination of the gradients of the mean water surface and the time average of horizontal momentum flux.

Global Well-Posedness for H^{-1}(mathbb {R}) Perturbations of KdV with Exotic Spatial Asymptotics

Given a suitable solution V(t, x) to the Korteweg–de Vries equation on the real line, we prove global well-posedness for initial data u(0,x) in V(0,x) + H^{-1}(mathbb {R}). Our conditions on V do include regularity but do not impose any assumptions on spatial asymptotics. We show that periodic profiles V(0,x)in H^5(mathbb {R}/mathbb {Z}) satisfy our hypotheses. In particular, we can treat localized perturbations of the much-studied periodic traveling wave solutions (cnoidal waves) of KdV. In the companion paper Laurens (Nonlinearity. 35(1):343–387, 2022. https://doi.org/10.1088/1361-6544/ac37f5) we show that smooth step-like initial data also satisfy our hypotheses. We employ the method of commuting flows introduced in Killip and Vişan (Ann. Math. (2) 190(1):249–305, 2019. https://doi.org/10.4007/annals.2019.190.1.4) where Vequiv 0. In that setting, it is known that H^{-1}(mathbb {R}) is sharp in the class of H^s(mathbb {R}) spaces.

A new reductive perturbation formalism for ion acoustic cnoidal waves

A new formalism for the derivation of the cnoidal wave solution is presented by introducing a new set of initial conditions and a subacoustic moving frame. The resulting set of solutions is also constructed in a way that ensures the conservation of number density. The solutions are illustrated in a variety of graphical representations, and the effect of the amplitude's magnitude on the cnoidal wave form is presented. Interestingly, it is shown that the wavelength of the solutions decreases with amplitude. In addition, it is shown that the small-amplitude cnoidal wave solutions converge to linear waves in the small-amplitude limit.

Numerical investigation of solute transport into deformable marine sediments driven by ocean waves

Most existing studies do not consider the effect that the deformability of submarine sediments may have on solute transport under the action of ocean waves. In this study a two-dimensional numerical model based on Biot’s poro-elastic consolidation theory and an advection–dispersion equation is established to investigate the transport of non-reactive solute into a deformable porous seabed under the influence of ocean waves. The numerical model is then validated against experimental and analytical results in order to demonstrate its accuracy and robustness. The simulation results indicate that ignoring the deformability of seabed soil and associated seepage force due to ocean waves may lead to a significant underestimation of the rate of solute transport in shallow layers of marine sediment, particularly fine sand or silt. The effects of parameters such as the shear modulus, non-dimensional mechanical dispersion coefficient (related to particle size and permeability), dimensionless sediment thickness, wave steepness and the relative water depth have on solute transport into deformable sediments were investigated. The results indicate that the ratio of the longitudinal coefficient of mechanical dispersion between deformable and non-deformable sediments that can represent the enhancement of solute migration due to soil deformation may reach up to 85. This becomes more pronounced as the shear modulus and permeability of sediments decrease and the depth of water increases when the sediment thickness is less than 0.8 times the wavelength. The rate of solute transport in deformable sediments reaches its maximum when the sediment thickness is about 0.2 times the wavelength. In fact, as the relative water depth increases from 0.1 to 0.3, the enhancement of solute migration due to wave-induced soil deformation increases by about 60%-80%. In comparison to linear waves, shallow water waves, e.g., the first-order cnoidal waves, can increase the transport of solute more due to its strong non-linear behaviour, particularly in deformable seabed.

On the analytical approximations to the nonplanar damped Kawahara equation: Cnoidal and solitary waves and their energy

In this work, the non-integrable nonplanar (cylindrical and spherical) damped Kawahara equation (ndKE) is solved and analyzed analytically. The ansatz method is implemented for analyzing the ndKE in order to derive some high-accurate and more stable analytical approximations. Based on this method, two-different and general formulas for the analytical approximations are derived. The obtained solutions are applied for studying the distinctive features for both cylindrical and spherical dissipative dressed solitons and cnoidal waves in a complex plasma having superthermal ions. Moreover, the accuracy of the obtained approximations is numerically examined by estimating the global maximum residual error. Also, a general formula for the nonplanar dissipative dressed solitons energy is derived in detail. This formula can recover the energy of the nonplanar dissipative dressed solitons, the planar dressed solitons, the planar damped dressed solitons, and the nonplanar dressed solitons. Both the suggested method and obtained approximations can help a large sector of authors interested in studying the nonlinear and complicated phenomena in various fields of science such as the propagating of nonlinear phenomena in physics of plasmas, nonlinear optics, communications, oceans, and seas.

Periodic solutions of coupled Boussinesq equations and Ostrovsky-type models free from zero-mass contradiction.

Coupled Boussinesq equations are used to describe long weakly nonlinear longitudinal strain waves in a bi-layer with soft bonding between the layers (e.g., a soft adhesive). From a mathematical viewpoint, a particularly difficult case appears when the linear long-wave speeds in the layers are significantly different (high-contrast case). The traditional derivation of the uni-directional models leads to four uncoupled Ostrovsky equations for the right- and left-propagating waves in each layer. However, the models impose a "zero-mass constraint"; i.e., the initial conditions should necessarily have zero mean, restricting the applicability of that description. Here, we bypass the contradiction in this high-contrast case by constructing the solution for the deviation from the evolving mean value, using asymptotic multiple-scale expansions involving two pairs of fast characteristic variables and two slow time variables. By construction, the Ostrovsky equations emerging within the scope of this derivation are solved for initial conditions with zero mean, while initial conditions for the original system may have non-zero mean values. Asymptotic validity of the solution is carefully examined numerically. We apply the models to the description of counter-propagating waves generated by solitary wave initial conditions, or co-propagating waves generated by cnoidal wave initial conditions, as well as the resulting wave interactions, and contrast with the behavior of the waves in bi-layers when the linear long-wave speeds in the layers are close (low-contrast case). One local (classical) and two non-local (generalized) conservation laws of the coupled Boussinesq equations for strains are derived and used to control the accuracy of the numerical simulations.

Higher-order gap resonance and heave response of two side-by-side barges under Stokes and cnoidal waves

Coupled piston-mode fluid response and the heave motion of two identical barges in side-by-side configuration is studied under finite-depth and shallow-water waves using a two-dimensional fully nonlinear numerical wave tank. To understand possible critical responses of the gap flow and the floating barge, regular-wave conditions which are able to excite up to 5th-order nonlinear gap resonance and also the resonant heave motion of the barge are considered. In shallow-water waves, high-frequency oscillations, featured by secondary peaks in the time histories, are observed for both wave elevation in the gap and the heave motion of the barge. The shallow-water wave-induced 4th- or 5th-order gap resonance can be equally crucial as the 1st- and 2nd-order resonances due to finite-depth waves. At higher-order gap resonance, the higher-harmonic heave motion of the barge is negligibly small, in contrast to the gap-flow response. Compared with fixed barges, the free-heave motion of an upstream barge tends to increase the wave elevation in the gap in most of the resonant conditions, except at 1st-order gap resonance where the gap response is greatly reduced. When the resonant heave motion of a floating barge, either located upstream or downstream, is excited, significant barge motion is observed. However, the relative motion between the gap flow and the floating barge is seen to be very small, ascribed by small phase difference between the two. The present study suggests that the effects of heave motion and water-depth should be carefully considered in the design of side-by-side marine operations, and hiding the small bunkering ships behind the large receiving ships is regarded as a preferred arrangement during the bunkering operations in offshore and coastal environments.

Diffraction and Refraction of Nonlinear Waves by the Green–Naghdi Equations

Abstract Diffraction and refraction of nonlinear shallow water waves due to uneven bathymetry are studied by use of the Green–Naghdi (GN) equations in three dimensions. A numerical wave tank consisting of deep, transitional, and shallow regions is created. Various forms of three-dimensional bathymetry, consisting of ramps with nonuniform profiles and large slopes, are used to connect the deep-water side of the tank to the shallow water shelf. A wavemaker is placed at the upwave side of the domain, capable of generating solitary and cnoidal waves of the GN equations. A numerical wave absorber is located downwave of the domain to minimize the wave reflection back into the domain. The system of equations is solved numerically in time domain by use of a second-order finite-difference approach for spatial discretization, and in a boundary-fitted coordinate system, and by use of the modified Euler method for time marching. Results include solitary and cnoidal wave surface elevation and particle velocities and are compared with the existing solutions where possible. Overall, very good agreement is observed. Discussion is provided on the nonlinearity and dispersion effects on the wave diffraction and refraction by the various forms of the ramps, as well as on the performance of the GN equations in solving these problems.

Nonlinear index of a two-level model: The essential role of the population inversion

Optical wave propagation in an assembly of two-level atoms is considered. An asymptotic model of nonlinear Schr\"odinger (NLS) type is derived in a rigorous way by means of the perturbative expansion method. The results of the obtained model are then compared with both a numerical solution of the initial set of equations of the two-level model and an exact analytical cnoidal wave solution. It is seen that the evolution of the population difference is a key factor for the accurate determination of the nonlinear index, and that the NLS approximation allows one to determine it theoretically. The knowledge of the population difference allows one to correct the cnoidal wave solution, which then matches the numerical solution up to the limits of our numerical accuracy.

Cnoidal waves and dark solitons with linear third-order dispersion and self-steepening effect

The present paper investigates the conditions needed for the generation and propagation of cnoidal waves in the form of dark solitons under the influence of self-steepening effect and linear third-order dispersion. This kind of optical pulses could be obtained in single-mode waveguides due to the precise balance between higher orders of nonlinear and dispersive phenomena. As a basic equation, it is used the nonlinear amplitude equation (NAE). It is found a new analytical solution of this equation in the form of a cnoidal wave (Jacobi elliptic sine function). In the article it is presented that the solution can be reduced to dark soliton at specific values of the modulus of ellipticity κ.

On the analytical and numerical approximations to the forced damped Gardner Kawahara equation and modeling the nonlinear structures in a collisional plasma

We perform a detailed study on the completely non-integrable forced damped Gardner/Extended Kawahara equation (FDEKE). Three techniques are introduced to determine abundance approximations to the proposed equation. In the first technique, the ansatz method is carried out for deriving some general formulas for the analytical approximations. In the second and third techniques, the FDEKE is analyzed numerically using both the septic B-spline collocation method and the method of lines. As a realistic model, the obtained approximations are employed for studying the properties of the periodic forced dissipative extended Kawahara solitary and cnoidal waves in a pair-ion plasma comprised of Maxwellian electrons and two fluid positive and negative ions. Both numerical and analytical approximations are graphically compared with each other. Also, the global maximum residual error L∞ for all obtained approximations is estimated for checking the accuracy of these approximations. Moreover, the obtained approximations can be applied for studying the features of the dissipative localized and periodic higher-order structures in optical fiber, ocean, sea, different models of plasma physics, and fluid mechanics.

Consistent Riccati expansion solvability and soliton–cnoidal wave solutions of a coupled KdV system

The consistent Riccati expansion (CRE) solvability of a coupled KdV system is investigated with the aid of the Riccati equation. Differing from the complete Painlevé integrability for three branches, this coupled KdV system is shown to be CRE solvable for one branch and non-CRE solvable for two other branches. Based on the last consistent differential equation under the CRE solvable case, a kind of soliton–cnoidal wave interaction solution is derived explicitly.

Nonlinear Excitations in Ultracold Atoms Trapped in Triple Optical Lattices

Various solitary wave excitations are found for a Bose-Einstein condensate in presence of two hybrid potentials in the form of triple mixtures of optical lattices. One of these potentials comprises of a combination of two important lattice profiles, such as frustrated optical lattice and double-well super-lattice, within one. Another represents a composite lattice combination, resulting in a wider and deeper frustrated optical lattice. The dynamical equation for such a system is solved by the exact analytical method to obtain a bright solitary wave, periodic wave and cnoidal wave excitations. We also report Anderson localization, bifurcation of condensate at the center and a competition between two different types of localizations upon trap engineering. Dynamical and structural stability analyses are also carried out, which reveal the obtained solutions as extremely stable for structural noise incorporation and sufficiently stable for dynamical stability. These triple mixtures of optical lattices impart better tunability on the condensate profile, which has made this system a true quantum simulator.

Nonlinear periodic structures in magnetoplasma with nonthermal electrons and positrons

In the present study, we address the problem of cnoidal waves (CWs) in magnetized electron-positron-ion (e-p-i) plasma with nonthermal electrons and positrons. The Korteweg-de Vries equation (KdVE) is derived using the reductive perturbation technique (RPT) and its cnoidal wave (CW) solution is analyzed. The impact of relevant plasma parameters on the characteristics of the ion-acoustic (IA) cnoidal structures are discussed in detail. The application of the present investigation is discussed.

Steady periodic waves over a planar seabed: a global characterization

The problem of propagation of steady periodic waves over a planar seabed is faced through the definition of a suitable semi-analytical iterative scheme. The latter is capable of describing highly nonlinear waves in deep, intermediate and shallow water conditions. Comparisons with the existing fifth-order theories show that the proposed model is accurate in all the regimes of motion and that it does not present any of the limitations affecting the Stokes and cnoidal wave solutions. Further, it also provides a reliable approximation of the dynamics of maximum-amplitude waves. The definition of the iterative scheme is preceded by a detailed study of the geometrical wave parameters. This latter part is aimed at the definition of a global scaling for water waves valid in all the regimes of motion.

Symmetry Reductions, Cte Method and Interaction Solutions for Sharma-Tasso-Olver-Burgers Equation

In this paper, the Sharma-Tasso-Olver-Burgers (STOB) system is analyzed by the Lie point symmetry method. The hypergeometric wave solution of the STOB equation is derived by symmetry reductions. In the meantime, the consistent tanh expansion (CTE) method is applied to the STOB equation. An nonauto-Bäcklund (BT) theorem that includes the over-determined equations and the consistent condition is obtained by the CTE method. By using the nonauto-BT theorem, the interactions between one-soliton and the cnoidal wave, and between one-soliton and the multiple resonant soliton solutions, are constructed. The dynamics of these novel interaction solutions are shown both in analytical and graphical forms. The results are potentially useful for explaining ocean phenomena.

Near-critical turbulent free-surface flow over a wavy bottom

Steady plane turbulent free-surface flow over a slightly wavy bottom is considered for very large Reynolds numbers, very small bottom slopes, and Froude numbers close to the critical value 1. As in previous works, the slope and the deviation from the critical Froude number are assumed to be coupled such that turbulence modeling is not required. The amplitudes of the periodic bottom elevations, however, are assumed to be half an order of magnitude larger than in the previous case of bumps or ramps of finite length. Asymptotic expansions give a steady-state version of an extended Korteweg–deVries (KdV) equation for the surface elevation. The extension consists of a forcing term due to the unevenness of the bottom and a damping term due to friction at the bottom. Other flow quantities, such as pressure, flow velocity components, local Froude number and bottom friction force, can be expressed in terms of the surface elevation. Exact solutions of the extended KdV equation, describing stationary cnoidal waves, are obtained for bottoms of particular periodic shapes. As a limiting case, the solitary waves over a bottom ramp are re-obtained in accord with previous results.