The integrable Boussinesq equation and it’s breather,lump and soliton solutions

The fourth-order nonlinear Boussinesq water wave equation, which explains the propagation of long waves in shallow water, is explored in this article. We used the Lie symmetry approach to analyze the Lie symmetries and vector fields. Then, by using similarity variables, we obtained the symmetry reductions and soliton wave solutions. In addition, the Kudryashov method and its modification are used to explore the bright and singular solitons while the Hirota bilinear method is effectively used to obtain a form of breather and lump wave solutions. The physical explanation of the extracted solutions was shown with the free choice of different parameters by depicting some 2-D, 3-D, and their corresponding contour plots.

     

Lax pair,conservation laws,and multi-shock wave solutions of the DJKM equation with Bell polynomials and symbolic computation

The integrability and multi-shock wave solutions of the DJKM equation are studied by means of Bell polynomials scheme, Hirota bilinear method, and symbolic computation. A more generalized bilinear system of the DJKM equation is constructed via Bell polynomials scheme. Moreover, Lax pair and infinite conservation laws of this equation are first obtained via its corresponding Bell-polynomials-type Bäcklund transformation. Furthermore, the multi-shock wave solutions are also obtained by applying standard Hirota bilinear method, and the propagation and collision of shock waves are graphically demonstrated by graphs.     

A ship motion forecasting approach based on empirical mode decomposition method hybrid deep learning network and quantum butterfly optimization algorithm

Studies of the shallow water waves are active, possessing the applications in ocean engineering, marine environment, atmospheric science, etc. In this paper, we investigate a (3+1)-dimensional shallow water wave equation with time-dependent coefficients. Hirota method and symbolic computation help us work out (1) a bilinear form, (2) N-soliton solutions with N being a positive integer, (3) the higher-order breather solutions, (4) periodic-wave solutions and (5) hybrid solutions composed of one first-order breather and one soliton/two solitons. Moreover, we provide some nonlinear phenomena described by the associated solutions. All of the obtained results are determined via the time-dependent coefficients of that equation.

     

Bilinear Bäcklund transformation,Lax pair and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics/fluid mechanics/plasma physics

In this paper, outcomes of the study on the Bäcklund transformation, Lax pair, and interactions of nonlinear waves for a generalized (2 + 1)-dimensional nonlinear wave equation in nonlinear optics, fluid mechanics, and plasma physics are presented. Via the Hirota bilinear method, a bilinear Bäcklund transformation is obtained, based on which a Lax pair is constructed. Via the symbolic computation, mixed rogue–solitary and rogue–periodic wave solutions are derived. Interactions between the rogue waves and solitary waves, and interactions between the rogue waves and periodic waves, are studied. It is found that (1) the one rogue wave appears between the two solitary waves and then merges with the two solitary waves; (2) the interaction between the one rogue wave and one periodic wave is periodic; and (3) the periodic lump waves with the amplitudes invariant are depicted. Furthermore, effects of the noise perturbations on the obtained solutions will be investigated.

     

Novel evolutionary behaviors of localized wave solutions and bilinear auto-Bäcklund transformations for the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation

Water waves are common phenomena in nature, which have attracted extensive attention of researchers. In the present paper, we first deduce five kinds of bilinear auto-Bäcklund transformations of the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation starting from the specially exchange identities of the Hirota bilinear operators; then, we construct the N-soliton solutions and several new structures of the localized wave solutions which are studied by using the long wave limit method and the complex conjugate condition technique. In addition, the propagation orbit, velocity and extremum of the first-order lump solution on (x, y)-plane are studied in detail, and seven mixed solutions are summarized. Finally, the dynamical behaviors and physical properties of different localized wave solutions are illustrated and analyzed. It is hoped that the obtained results can provide a feasibility analysis for water wave dynamics.

     

Different wave structures for the completely generalized Hirota–Satsuma–Ito equation

Under investigation is a completely generalized Hirota–Satsuma–Ito equation in (2 + 1)-dimensional. Multiple lump solutions are obtained based on three test functions, including 1-, 2- and 3-order lump solutions. Subsequently, the interaction between lump wave and solitary waves, and the interaction solution between lump wave and periodic wave are studied by using the bilinear form. Final, the stability and phase velocity are investigated. In order to analyze the dynamic behavior of these solutions, some 3D plots and contour plots are given by Mathematica.

     

Characteristics of the solitary waves and lump waves with interaction phenomena in a (2 + 1)-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada equation

In this paper, we consider a (\(2+1\))-dimensional generalized Caudrey–Dodd–Gibbon–Kotera–Sawada (gCDGKS) equation, which is a higher-order generalization of the celebrated Kadomtsev–Petviashvili (KP) equation. By considering the Hirota bilinear form of the CDGKS equation, we study a type of exact interaction waves by the way of vector notations. The interaction solutions, which possess extensive applications in the nonlinear system, are composed by lump wave parts and soliton wave parts, respectively. Under certain conditions, this kind of solutions can be transformed into the pure lump waves or the stripe solitons. Moreover, we provide the graphical analysis of such solutions in order to better understand their dynamical behavior.     

Analysis of lumps,single-stripe,breather-wave,and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method

In this paper, we implement the Hirota’s bilinear method to extract diverse wave profiles to the generalized perturbed-KdV equation when the test function approaches are taken into consideration. Several novel solutions such as lump-soliton, lump-periodic, single-stripe soliton, breather waves, and two-wave solutions are obtained to the proposed model. We conduct some graphical analysis including 2D and 3D plots to show the physical structures of the recovery solutions. On the other hand, this work contains a correction of previous published results for a special case of the perturbed KdV. Moreover, we investigate the significance of the nonlinearity, perturbation, and dispersion parameters being acting on the propagation of the perturbed KdV. Finally, our obtained solutions are verified by inserting them into the governing equation.

     

A meshless numerical wave tank for simulation of nonlinear irregular waves in shallow water

Time domain simulation of the interaction between offshore structures and irregular waves in shallow water becomes a focus due to significant increase of liquefied natural gas (LNG) terminals. To obtain the time series of irregular waves in shallow water, a numerical wave tank is developed by using the meshless method for simulation of 2D nonlinear irregular waves propagating from deep water to shallow water. Using the fundamental solution of Laplace equation as the radial basis function (RBF) and locating the source points outside the computational domain, the problem of water wave propagation is solved by collocation of boundary points. In order to improve the computation stability, both the incident wave elevation and velocity potential are applied to the wave generation. A sponge damping layer combined with the Sommerfeld radiation condition is used on the radiation boundary. The present model is applied to simulate the propagation of regular and irregular waves. The numerical results are validated by analytical solutions and experimental data and good agreements are observed. Copyright © 2008 John Wiley & Sons, Ltd.     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.     

The structure and asymptotic behaviour of compression waves

In the study of weak solutions to nonlinear hyperbolic partial differential equations both rarefaction waves and compression waves arise. Although the behavior of rarefaction waves is known for all time, the characteristics that determine a compression wave intersect and hence the development of the wave is not easily determined. The purpose of this paper is to study compression waves. As a first step we consider the Cauchy problem for the nonlinear wave equation. We show that if the data outside some finite interval consist of constant states, then after finite time the solution involves the same states as does the solution to the Riemann problem determined by these constant states. This result is then applied to compression waves to obtain information on the shock that arises and on the steady-state solution. The region of interaction is also described. This information is obtained via a constructive procedure.     

Characteristics of rogue waves on a periodic background for the Hirota equation

We consider the rogue dn-periodic waves (the rogue wave solutions on the dn-periodic waves background) for the Hirota equation by using Darboux transformation. We take Jacobian elliptic function dn as a seed solution, which is modulationally unstable as regards long wave perturbations. Through nonlinearization of the Lax pair for Hirota equation, the corresponding periodic eigenfunctions are successfully obtained. Based on these periodic eigenfunctions, we further construct the solutions of the Lax pair equations with dn-periodic wave seed solutions. In addition, numerical simulations are presented to reveal the phenomena of these solutions under different parameters choices.     

Lump,soliton, and interaction solutions to a generalized two-mode higher-order nonlinear evolution equation in plasma physics

This article investigates a nonlinear fifth-order partial differential equation (PDE) in two-mode waves. The equation generalizes two-mode Sawada-Kotera (tmSK), two-mode Lax (tmLax), and two-mode Caudrey–Dodd–Gibbon (tmCDG) equations. In 2017, Wazwaz [1] presented three two-mode fifth-order evolutions equations as tmSK, tmLax, and tmCDG equations for the integrable two-mode KdV equation and established solitons up to three-soliton solutions. In light of the research above, we examine a generalized two-mode evolution equation using a logarithmic transformation concerning the equation’s dispersion. It utilizes the simplified technique of the Hirota method to obtain the multiple solitons as a single soliton, two solitons, and three solitons with their interactions. Also, we construct one-lump solutions and their interaction with a soliton and depict the dynamical structures of the obtained solutions for solitons, lump, and their interactions. We show the 3D graphics with their contour plots for the obtained solutions by taking suitable values of the parameters presented in the solutions. These equations simultaneously study the propagation of two-mode waves in the identical direction with different phase velocities, dispersion parameters, and nonlinearity. These equations have applications in several real-life examples, such as gravity-affected waves or gravity-capillary waves, waves in shallow water, propagating waves in fast-mode and the slow-mode with their phase velocity in a strong and weak magnetic field, known as magneto-sound propagation in plasmas.

     

Generalized Darboux transformation and localized waves in coupled Hirota equations

In this paper, we construct a generalized Darboux transformation to the coupled Hirota equations with high-order nonlinear effects like the third dispersion, self-steepening and inelastic Raman scattering terms. As application, an N$N$th-order localized wave solution on the plane backgrounds with the same spectral parameter is derived through the direct iterative rule. In particular, some semi-rational, multi-parametric localized wave solutions are obtained: (1) vector generalization of the first- and the second-order rogue wave solutions; (2) interactional solutions between a dark–bright soliton and a rogue wave, two dark–bright solitons and a second-order rogue wave; (3) interactional solutions between a breather and a rogue wave, two breathers and a second-order rogue wave. The results further reveal the striking dynamic structures of localized waves in complex coupled systems.     

An extended Korteweg–de Vries equation: multi-soliton solutions and conservation laws

In this paper, we consider an extended KdV equation, which arises in the analysis of several problems in soliton theory. First, we converted the underlying equation into the Hirota bilinear form. Then, using the novel test function method, abundant multi-soliton solutions were obtained. Second, we have performed some distinct methods to extended KdV equation for getting some exact wave solutions. In this regard, Kudryashov’s simplest equation methods were examined. Third, the local conservation laws are deduced by multiplier/homotopy methods. Finally, the graphical simulations of the exact solutions are depicted.     

Amplification,reshaping, fission and annihilation of optical solitons in dispersion-decreasing fiber

Amplification, reshaping, fission and annihilation of optical solitons can be applied in fiber lasers, all-optical switching devices and optical communications. In this paper, for the variable coefficient high-order nonlinear Schrödinger equation, analytic two- and three-soliton solutions are derived by the Hirota bilinear method. Optical solitons propagation in the dispersion-decreasing fibers is investigated theoretically. The influence of corresponding parameters is discussed based on obtained solutions. By choosing properly parameters, optical solitons are amplified and reshaped stably in a long distance. Besides, the number of amplified solitons can be chosen as required. Moreover, a novel phenomenon that three solitons can split into four solitons or merge into two solitons has been proposed. Results may be helpful to realize the amplification, reshaping, fission and annihilation of solitons, and will be valuable to the applications of optical amplifier, all-optical switching and optical self-routing.     

Nonlinear tunneling for controllable rogue waves in two dimensional graded-index waveguides

With the help of the similarity transformation connected the variable-coefficient nonlinear Schrödinger equation with the standard nonlinear Schrödinger equation, we firstly obtain first-order and second-order rogue wave solutions in two dimensional graded-index waveguides. Then, we investigate the nonlinear tunneling of controllable rogue waves when they pass through nonlinear barrier and nonlinear well. Our results indicate that the propagation behaviors of rogue waves, such as postpone, sustainment and restraint, can be manipulated by choosing the relation between the maximum value of the effective propagation distance Z _m and the effective propagation distance corresponding to maximum amplitude of rogue waves Z ₀. Postponed, sustained and restrained rogue waves can tunnel through the nonlinear barrier or well with increasing, unchanged and decreasing amplitudes by modulating the ratio of the amplitudes of rogue waves to barrier or well height.     

Sixth order solitary wave equations

In this work, we present higher order solitary wave equations, in particular sixth order. We show how these equations can be derived using fundamental physics laws, such as the Ohm’s law. We use the Taylor series expansion and in some cases the Hirota’s bilinear operator to obtain these model equations. The sixth order solitary wave equations model different physical problems such as problems in the electrical domain and the propagation of dispersive water waves.     

浅水区海底管道周围海床渗透压力计算研究

我国海上油田开采起步较晚,大部分油田处于浅水区,因此,在设计管道时,应充分考虑由浅水区波浪引起的管道周围海床渗流力。根据浅水波相关假设,考虑自由水面非线性影响,推导出椭圆余弦波的波面方程,在此基础上进一步得到一个关于速度势的表达式,并根据该表达式得出作用于海床表面的波压公式。考虑海床土的压缩性,推导出一阶近似椭圆余弦波作用下浅水区埋置管道周围海床的渗流压力解析解,最后将计算结果与大型水槽试验及以往研究成果作对比。结果表明,在椭圆余弦波的作用下,由一阶椭圆余弦波理论得到的计算结果与试验结果规律基本一致,与相似工况下的现有理论成果数值基本相同,具有一定的可行性和工程价值。     

Canonical equations for almost periodic,weakly nonlinear gravity waves

A set of stable canonical equations of second order is derived, which describe the propagation of almost periodic waves in the horizontal plane, including weakly nonlinear interactions. The derivation is based on the Hamiltonian theory of surface waves, using an extension of the Ritz variational method. For waves of infinitesimal amplitude the well-known linear refraction-diffraction model (the mild-slope equation) is recovered. In deep water the nonlinear dispersion relation for Stokes waves is found. In shallow water the equations reduce to Airy's nonlinear shallow-water equations for very long waves. Periodic solutions with steady profile show the occurrence of a singularity at the crest, at a critical wave height.