Combined Korteweg-de Vries Research Articles

Sensitivity and wave propagation analysis of the time-fractional (3+1)-dimensional shallow water waves model

This study investigates the exact solutions of the time-fractional (3+1)-dimensional combined Korteweg–de Vries Benjamin–Bona–Mahony (KdV-BBM) equation. The considered model describes the long surface gravity waves of small amplitude, which portrays the two-way propagation of waves. The modified generalized Kudryashov method and the exp(-φ(ξ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi (\\xi $$\\end{document}))-expansion methods are employed to resolve the aforementioned issue because of their effectiveness and simplicity. For generality, the time fractional version is studied; more advanced solutions that do not exist in the literature were obtained. As a result, a variety of the exact wave solutions of the conformable (3+1)-dimensional KdV-BBM equation are obtained. The dynamical behaviors of some obtained solutions are represented with the proper parameter values. The used methods yield noteworthy results in obtaining the analytical solutions of fractional differential equations under various conditions. Besides, the sensitivity of regarding dynamical system is assessed to show the numerical stability effects.

The variation of the electric field along optic fiber for null Cartan and pseudo-null frames

In this paper, we introduce new frame and new transformation associated with combined Korteweg–de Vries–modified Korteweg–de Vries (KdV–mKdV) equation, the mKdV and the KdV equations with respect to the null Cartan frame in Minkowski 3-space. Direction of the state of polarization of monochromatic linear polarized light wave traveling in the optic fiber is given by the direction of electric field vector. We show that the electric field vector [Formula: see text] can be expressed as a linear combination of null Cartan frame apparatus using this new transformation in Minkowski 3-space. Later, we study rotation of the polarization plane along optic fiber according to null Cartan frame and Frenet frame of pseudo-null curve in Minkowski 3-space. Finally, we obtain Lorentz force equations via null and pseudo null electromagnetic curves.

General improved Kudryashov method for exact solutions of nonlinear evolution equations in mathematical physics

This paper is devoted to improving the general Kudryashov method by a new and general auxiliary equation. So, a new method is introduced, which we call ‘the general improved Kudryashov method’, to produce exact solutions for nonlinear evolution equations arising in mathematical physics. As application examples, exact traveling wave solutions for the combined Korteweg–de Vries and modified Korteweg–de Vries (KdV–mKdV) equation and the (2+1)-dimensional Zakharov–Kuznetsov equation are obtained. These solutions can be classified as solitary and periodic wave solutions. Some of the obtained solutions are graphically sketched.

Symmetry classification of the Gardner equation with time-dependent coefficients arising in stratified fluids

We perform symmetry classification of a variable-coefficient combined KdV–mKdV equation. That is, the equation combining the Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations, or also known as the Gardner equation. The direct method of group classification is utilized to specify the forms of these time-dependent coefficients.

Combined equations of the Burgers hierarchy: multiple kink solutions and multiple singular kink solutions

Combined equations of the Burgers hierarchy are constructed using the sense of the combined Korteweg–de Vries (KdV)–modified KdV (mKdV) equation. The Cole–Hopf transformation method is used to study the resulting equations. Multiple kink solutions and multiple singular kink solutions are formally established for each combined equation. The kink solutions of any combination differ only in the dispersion relation.

Exact solutions of some nonlinear partial differential equations

We implemented a generalized tanh function method for approximating the solution of the combined Korteweg–de Vries (CKdV) equation and Sharma–Tasso–Olver equation. By using this scheme, we found some exact solutions of the CKdV equation and Sharma–Tasso–Olver equation. For the further investigation of the method, some exact solutions of the Boussinesq equation and the complexly coupled KdV equations have been obtained.

New travelling wave solutions for combined KdV-mKdV equation and (2+1)-dimensional Broer-Kaup-Kupershmidt system

Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg–de Vries (KdV) and modified KdV (mKdV) equation and (2+1)-dimensional Broer–Kaup–Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.

A numerical application of the decomposition method for the combined KdV–MKdV equation

We consider solitary-wave solutions of the combined Korteweg de Vries–Modified Korteweg de Vries (KdV–MKdV) equation. We prove the convergence of Adomian decomposition method (ADM) applied to the combined KdV–MKdV equation. Then we obtain the exact solitary-wave solutions and numerical solutions of the combined KdV–MKdV equation for the initial conditions. The numerical solutions are compared with the known analytical solutions. Their remarkable accuracy are finally demonstrated for the combined KdV–MKdV equation.

Transformation of Short Waves in a Nonuniform Flow Field on the Ocean Surface. The Effect of Wind Growth Rate Modulation

We develop a model of transformation of the short surface wave spectrum in the presence of a nonuniform flow on a water surface, in which the modulation of wind-wave growth rate is taken into account. The model of a turbulent near-water atmospheric layer is used to calculate the modulated growth rate. In this model, turbulent stresses in the wind are described using a gradient approximation with model eddy viscosity specified with allowance for the known laboratory experiments. The examples of short-wave modulation in the presence of nonuniform flows on a water surface, originating from ripples and intense internal waves, are considered. It is shown that deformations of the wind-velocity profile and its long-wavelength perturbation due to the nonlinear interaction between the wind surface waves and the wind has a significant effects on the short-wave growth rate and its modulation. In the case of ripples, this deformation reduces to an increase in the roughness parameter of the wind-velocity profile and to a velocity-profile modulation with ripple period. The modulated growth rate is calculated within the framework of a quasi-linear model of surface-wave generation by a turbulent wind, in which the hypothesis of random phases of the wind-wave field is used. The amplitude and phase of the hydrodynamical modulation transfer function are calculated within the framework of the relaxation model. The calculation results are in reasonable agreement with the available experimental data. A model described by the combined Korteweg–de Vries equation is used to study a surface flow field generated by intense internal waves. The internal-wave parameters are takes from the results of the COPE experiment. We calculate the wind growth-rate dependences on the wave-train phase for the cases of downwind and upwind propagation of an internal wave. The calculation results agree qualitatively with experimental data.

Lie Group of Transformations for a KdV–Boussinesq Equation

We consider a combined Korteweg–deVries and Boussinesq equation governing long surface waves in shallow water. Considering traveling wave solutions, the basic equations will be reduced to a second order ordinary differential equation. Using the Lie group of transformations we reduce it to a first order ordinary differential equation and employ a direct method to derive its periodic solutions in terms of Jacobian elliptic functions and their corresponding solitary wave and explode decay mode solutions.

Soliton Perturbations for a Combined KdV-MKdV Equation

The soliton perturbations for the combined Korteweg de Vries and modified Korteweg de Vries (KdV-MKdV) equation are studied. The first-order effects of perturbation on a soliton, namely both the slow time-dependence of the soliton parameters and the first-order correction are derived through constructing the appropriate Green's function.

On the theory of internal surges in shallow basins

On the basis of well-known solutions, we analyse an internal surge, i.e. a sharp step-like variation of the pycnocline depth in a shallow basin. According to results obtained, in those cases where the pycnocline is situated approximately at mid-depth of the basin, the surge may have the form of a steadily moving non-linear wave (kink) governed by the combined Korteweg-de Vries equation with quadratic and cubic non-linearities. The vertical velocity component in such a wave has the form of a pulse and represents a soliton. Estimates of surge parameters obtained from real hydrologic data are in good agreement with the experimentally measured data.

On Series Expansions Giving Closed-Form Solutions of Korteweg–de Vries-Like Equations

Korteweg–deVries (KdV)-like equations with higher-degree nonlinearity are solved by a direct algebraic technique due to Hereman et al. [J. Phys. A, 19 (1986), pp. 607–628]. For two KdV-like equations, one with fifth-degree nonlinearity, the other a combined KdV and mKdV equation, for particular choices of the coefficients of the nonlinear terms, the kink and antikink solutions found by Dey are recovered. Furthermore, soliton solutions of the combined KdV and mKdV equation are found for all values of the coefficients. Closed-form solutions for the Calogero–Degasperis–Fokas modified mKdV equation are also developed. Applications of the solutions of these equations in quantum field theory, plasma physics, and solid-state physics are mentioned. The Hereman et al. method is illustrated and slightly extended and this direct series method is briefly compared to Hirota’s method.

Effects of viscosity on water waves

The paper deals with the behaviour of a viscous liquid whose flow preserves the structure of the material columns. A balance law for energy is established which accounts for Navier-Stokes dissipation; on appealing to the invariance of such a low under rigid motions, balance equations for mass and linear momentum are derived. It is an outstanding consequence of the theory that the simultaneous occurence of viscosity and inertia terms makes long gravity waves be governed by the combined Korteweg-de Vries and Burgers equation.