Search Results (20)

In a recent paper, denoted by MG24 in this text, we used a modified Korteweg–de Vries (KdV) equation to describe the evolution of wind-driven water wave packets in shallow water. The modifications were several forcing/friction terms describing wave growth due to critical-level instability in the air, wave decay due to laminar friction in the water at the air–water interface, wave growth due to turbulent wave stress in the air near the interface, and wave decay due to a turbulent bottom boundary layer. The outcome was a KdV–Burgers type of equation that can be a stable or unstable model depending on the forcing/friction parameters. In most cases that we examined, many solitary waves are generated, suggesting the formation of a soliton gas. In this paper, we extend that model in the horizontal direction transverse to the wind forcing to produce a similarly modified Kadomtsev–Petviashvili equation (KPII for water waves in the absence of surface tension). A modulation theory is described for the cnoidal and solitary wave solutions of the unforced KP equation, focusing on the forcing/friction terms and the transverse dependence. Then, using similar initial conditions to those used in MG24, that is a sinusoidal wave with a slowly varying envelope, but supplemented here with a transverse sinusoidal term, we find through numerical simulations that the radiation field upstream is enhanced, but that a soliton gas still emerges downstream as in MG24. Full article

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, $G^{\prime }/G$-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations. Full article

Based on the bosonization approach, the supersymmetric Boussinesq equation is converted into a coupled bosonic system. The symmetry group and the commutation relations of the corresponding bosonic system are determined through the Lie point symmetry theory. The group invariant solutions of the coupled bosonic system are analyzed by the symmetry reduction technique. Special traveling wave solutions are generated by using the mapping and deformation method. Some novel solutions, such as multi-soliton, soliton–cnoidal interaction solutions, and lump waves, are given by utilizing the Hirota bilinear and the consistent tanh expansion methods. The methods in this paper can be effectively expanded to study rich localized waves for other supersymmetric systems. Full article

Cnoidal wave theory perfectly describes nearshore wave characteristics. However, cnoidal wave theory is not widely applied in practical engineering because the formula for the wave profile involves a complex Jacobian elliptic function. In this paper, the approximate cnoidal wave theory is presented. Based on the Biot consolidation theory and the approximate cnoidal wave theory, an analytical solution for the pore water pressure around buried pipelines caused by waves is derived. In addition, based on the principle of effective stress, a theory of soil liquefaction around pipelines is proposed. The theoretical results were virtually identical to the results obtained in a practical flume test. Thus, the analytical method proposed in this paper is feasible. Further, the theory is applied to analyze the instantaneous liquefaction of the seabed around buried pipelines and the stability of the pipeline in the Chengdao oilfield. Full article

This study is concerned with the generation and propagation of strongly nonlinear waves in shallow water. A numerical wave flume is developed where nonlinear waves of solitary and cnoidal types are generated by use of the Level I Green-Naghdi (GN) equations by a piston-type wavemaker. Waves generated by the GN theory enter the domain where the fluid motion is governed by the Navier–Stokes equations to achieve the highest accuracy for wave propagation. The computations are performed in two dimensions, and by an open source computational fluid dynamics package, namely OpenFoam. Comparisons are made between the characteristics of the waves generated in this wave tank and by use of the GN equations and the waves generated by Boussinesq equations, Laitone’s 1st and 2nd order equations, and KdV equations. We also consider a numerical wave tank where waves generated by the GN equations enter a domain in which the fluid motion is governed by the GN equations. Discussion is provided on the limitations and applicability of the GN equations in generating accurate, nonlinear, shallow-water waves. The results, including surface elevation, velocity field, and wave celerity, are compared with laboratory experiments and other theories. It is found that the nonlinear waves generated by the GN equations are highly stable and in close agreement with laboratory measurements. Full article

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. Full article

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. Full article

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. Full article

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. Full article

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. Full article

The reverse space-time nonlocal complex modified Kortewewg–de Vries (mKdV) equation is investigated by using the consistent tanh expansion (CTE) method. According to the CTE method, a nonauto-Bäcklund transformation theorem of nonlocal complex mKdV is obtained. The interactions between one kink soliton and other different nonlinear excitations are constructed via the nonauto-Bäcklund transformation theorem. By selecting cnoidal periodic waves, the interaction between one kink soliton and the cnoidal periodic waves is derived. The specific Jacobi function-type solution and graphs of its analysis are provided in this paper. Full article

This paper presents a combined analytical and numerical (CAN) model to simulate the scattering of cnoidal waves by a fixed and partially immersed box-type breakwater. A set of Boussinesq equations are solved in the outer region using the finite-difference method to model the propagation of cnoidal waves and their subsequent reflection and transmission after encountering the breakwater. The two-dimensional (2D) velocity potential in the inner region beneath the body is derived analytically by solving the equations formulated from the orthogonality of eigenfunctions and the interfacial matching conditions. Experimental measurements on the wave profiles were carried out in a wave tank to verify the model solutions. Reflected and transmitted wave elevations obtained from the present CAN model match closely with the measured data. Additionally, the calculated horizontal and vertical forces on the body using the developed CAN model are in reasonable agreement with those from a potential 2D flow-based fully nonlinear wave model (FNWM). The method and proposed CAN model, if applied to a simple parametric investigation, can provide the expected trends in terms of applied forces, wave reflection, and transmission. Full article

Traveling wave solutions, including localized and periodic structures (e.g., solitary waves, cnoidal waves, and periodic waves), to a symmetry Korteweg–de Vries equation (KdV) with integer and rational power law nonlinearity are reported using several approaches. In the case of the localized wave solutions, i.e., solitary waves, to the evolution equation, two different methods are devoted for this purpose. In the first one, new hypotheses with Cole–Hopf transformation are employed to find general solitary wave solutions. In the second one, the ansatz method with hyperbolic sech algorithm are utilized to obtain a general solitary wave solution. The obtained solutions recover the solitary wave solutions to all one-dimensional KdV equations with a power law nonlinearity, such as the KdV equation with quadratic nonlinearity, the modified KdV (mKdV) equation with cubic nonlinearity, the super KdV equation with quartic nonlinearity, and so on. Furthermore, two different approaches with two different formulas for the Weierstrass elliptic functions (WSEFs) are adopted for deriving some general periodic wave solutions to the evolution equation. Additionally, in the form of Jacobi elliptic functions (JEFs), the cnoidal wave solutions to the KdV-, mKdV-, and SKdV equations are obtained. These results help many authors to understand the mystery of several nonlinear phenomena in different branches of sciences, such as plasma physics, fluid mechanics, nonlinear optics, Bose Einstein condensates, and so on. Full article

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum. Full article

This study is aimed at numerically investigating the cnoidal wave-induced dynamics characteristics and the liquefaction process in a loosely deposited seabed floor in a shallow water environment. To achieve this goal, the integrated model FSSI-CAS 2D is taken as the computational platform, and the advanced soil model Pastor–Zienkiewicz Mark III is utilized to describe the complicated mechanical behavior of loose seabed soil. The computational results show that a significant lateral spreading and vertical subsidence could be observed in the loosely deposited seabed floor due to the gradual loss of soil skeleton stiffness caused by the accumulation of pore pressure. The accumulation of pore pressure in the loose seabed is not infinite but limited by the liquefaction resistance line. The seabed soil at some locations could be reached to the full liquefaction state, becoming a type of heavy fluid with great viscosity. Residual liquefaction is a progressive process that is initiated at the upper part of the seabed floor and then enlarges downward. For waves with great height in shallow water, the depth of the liquefaction zone will be greatly overestimated if the Stokes wave theory is used. This study can enhance the understanding of the characteristics of the liquefaction process in a loosely deposited seabed under coastal shallow water and provide a reference for engineering activities. Full article