Exact Solitary Research Articles

Diversity of soliton solutions to the (3 + 1)-Dimensional Wazwaz-Benjamin-Bona-Mahony equations arising in mathematical physics

The current study scrutinizes exact singular solitary, kink, anti-kink, bell-shaped, and periodic traveling wave solutions for the newly implemented 3-dimensional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations family using a modified Exp-function approach. In spite of the presence of several trigonometric, hyperbolic, and rational functions, some new exact solitary, hyperbolic, and periodic traveling wave solutions to the considered equations are obtained by putting the modified Exp-function method into practice using the computer program Maple. The arrangement of the functions tanh, sech2, tan, and sec2 expresses the specific solutions produced by the defined procedure. The information was obtained to evaluate how well the suggested method could compute the exact solutions of the WBBM equations that could be applied to nonlinear water model applications in the ocean and coastal engineering. To show the physical composition and properties of the discovered solitons, contour, three-, and two-dimensional graphs are plotted using the computer program Mathematica. The acquired results imply that the proposed method can be applied to find various, enhanced, useful, and compatible solutions for other important nonlinear partial differential equations. All of the solutions given have been qualified by swapping the respective equations with those generated via the computer program Maple. Furthermore, we compared our obtained results with the solutions already put forward in the literature.

The Application of the Functional Variable Method for Solving the Loaded Non-linear Evaluation Equations

In this article, we construct exact traveling wave solutions of the loaded Korteweg-de Vries, the loaded modified Korteweg-de Vries, and the loaded Gardner equation by the functional variable method. The performance of this method is reliable and effective and gives the exact solitary and periodic wave solutions. All solutions to these equations have been examined and 3D graphics of the obtained solutions have been drawn by using the Matlab program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. The graphical representations of some obtained solutions are demonstrated to better understand their physical features, including bell-shaped solitary wave solutions, singular soliton solutions, and solitary wave solutions of kink type. Our results reveal that the method is a very effective and straightforward way of formulating the exact traveling wave solutions of non-linear wave equations arising in mathematical physics and engineering.

Drift ion-acoustic waves in a nonuniform rotating magnetoplasma with two-temperature superthermal electrons

The theory of low-frequency (in comparison with the ion cyclotron frequency), long wavelength, electrostatic drift ion-acoustic waves (IAWs) is studied in a nonuniform rotating magnetoplasma with two temperature superthermal electrons. In the linear limit, the coupling of IAWs and drift waves by the density inhomogeneity is shown to produce a new wave mode which typically depends on the density gradient, the rotational frequency and the spectral indexes of superthermal electrons. In the nonlinear regime, an evolution equation for the drift IAWs is derived by the dispersion approach, and using the Jacobi elliptic function expansion technique its exact solitary and periodic wave solutions (namely, cnoidal and dnoidal) are also obtained. The properties of these solutions are numerically examined and it is found that they are significantly modified by the effects of the background density gradient, the superthermality of electrons and the Coriolis force associated with the rotational motion of ions.

The new exact solutions of the Fifth-Order Sawada–Kotera equation using three wave method

In this paper, traveling waves with different frequencies and velocities can be constructed by three wave method. Some new exact solitary and periodic solitary solutions are obtained for the Fifth-Order Sawada–Kotera equation using three wave type method via Hiröta bilinear form. The solutions investigated by three wave method are more than solutions by others method such as homoclinic test method.

Exact solitary and periodic wave solutions for the coupled Schrödinger–Boussinesq equation

In this paper, exact solutions for the coupled Schrödinger–Boussinesq equation are studied. Under given parameter conditions, exact explicit formulas of some solitary and periodic traveling wave solutions are obtained by applying the method of dynamical systems. Furthermore, our results generalize the known results in the previous literature.

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

On the Galerkin/Finite-Element Method for the Serre Equations

A highly accurate numerical scheme is presented for the Serre system of partial differential equations, which models the propagation of dispersive shallow water waves in the fully-nonlinear regime. The fully-discrete scheme utilizes the Galerkin / finite-element method based on smooth periodic splines in space, and an explicit fourth-order Runge-Kutta method in time. Computations compared with exact solitary and cnoidal wave solutions show that the scheme achieves the optimal orders of accuracy in space and time. These computations also show that the stability of this scheme does not impose restrictive conditions on the temporal step size. In addition, solitary, cnoidal, and dispersive shock waves are studied in detail using this numerical scheme for the Serre system and compared with the 'classical' Boussinesq system for small-amplitude shallow water waves. The results show that the interaction of solitary waves in the Serre system is more inelastic. The efficacy of the numerical scheme for modeling dispersive shocks is shown by comparison with asymptotic results. These results have application to the modeling of shallow water waves of intermediate or large amplitude, such as occurs in the nearshore zone.

Analytical and Multishaped Solitary Wave Solutions for Extended Reduced Ostrovsky Equation

We present the analytical and multishaped solitary wave solutions for extended reduced Ostrovsky equation (EX-ROE). The exact solitary (traveling) wave solutions are expressed by three types of functions which are hyperbolic function solution, trigonometric function solution, and rational solution. These results generalized the previous results. Multishape solitary wave solutions such as loop-shaped, cusp-shaped, and hump-shaped can be obtained as well when the special values of the parameters are taken. The -expansion method presents a wide applicability for handling nonlinear partial differential equations.

Exact Peakon, Compacton, Solitary Wave, and Periodic Wave Solutions for a Generalized KdV Equation

We employ the approaches of both dynamical system and numerical simulation to investigate a generalized KdV equation, which is presented by Yin (2012). Some peakon, compacton, solitary wave, smooth periodic wave, and periodic cusp wave solutions are obtained, and the planar graphs of the compactons and the periodic cusp waves are simulated.

A Variety of Exact Periodic Wave and Solitary Wave Solutions for the Coupled Higgs Equation

In this work, the coupled Higgs field equation is studied. The extended Jacobi elliptic function expansion methods are efficiently employed to construct the exact periodic solutions of this model. As a result, many exact travelling wave solutions are obtained which include new shock wave solutions or kink-shaped soliton solutions, solitary wave solutions or bell-shaped soliton solutions, and combined solitary wave solutions are formally obtained.

Exact periodic, cusp, solitary and loop wave solutions of the EX-ROE

The extended reduced Ostrovsky equation (EX-ROE) is investigated by using the dynamical system theory. The bifurcation phase portraits are drawn in different regions of parameter plane. The bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakon, solitary loop waves and periodic loop waves are obtained. The dynamic characters of these solutions are investigated.

Finding the one‐loop soliton solution of the short‐pulse equation by means of the homotopy analysis method

AbstractThe homotopy analysis method is applied to the short‐pulse equation in order to find an analytic approximation to the known exact solitary upright‐loop solution. It is demonstrated that the approximate solution agrees well with the exact solution. This provides further evidence that the homotopy analysis method is a powerful tool for finding excellent approximations to nonlinear solitary waves. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Exact solitary- and periodic-wave modes in coupled equations with saturable nonlinearity

We demonstrate that the known method, based on the Hirota bilinear operator, generates classes of exact solutions to a system of coupled nonlinear Schrödinger (CNLS) equations with nonpolynomial nonlinearity, either rational or algebraic (the latter involves a square root). The choice of the CNLS equations is suggested by known models for photorefractive media and Bose–Einstein condensation. The solutions are, generally, periodic, and they form families expressed in terms of the Jacobi elliptic functions, the elliptic modulus k being a free parameter of the family. In the limit case corresponding to the infinite period ( k = 1 ), the solutions amount to solitons. In some cases, the exact solutions may feature patterns with two peaks per period. Exact solutions are also found in a single NLS equation with a rational saturable nonlinearity and periodic potential in the form of squared Jacobi sine.

Simplest equation method to look for exact solutions of nonlinear differential equations

New method is presented to look for exact solutions of nonlinear differential equations . Two basic ideas are at the heart of our approach. One of them is to use the general solutions of the simplest nonlinear differential equations. Another idea is to take into consideration all possible singularities of equation studied. Application of our approach to search exact solutions of nonlinear differential equations is discussed in detail. The method is used to look for exact solutions of the Kuramoto–Sivashinsky equation and the equation for description of nonlinear waves in a convective fluid. New exact solitary and periodic waves of these equations are given.

Exact Periodic and New Solitary Wave Solutions to the Generalization of Integrable (2+1)-dimensional Dispersive Long Wave Equations

A general solution involving three arbitrary functions is first obtained for the generalization of integrable (2+1)-dimensional dispersive long wave equations by means of WTC truncation method. Exact periodic wave solutions are then expressed as rational functions of the Jacobi elliptic functions. For the first time the interaction of Jacobi elliptic waves is studied and found to be nonelastic! Limit cases are studied and some interesting, new solitary structures are revealed. The interactions of between two dromions, between dromion and solitoff and between y -periodic solitons are all nonelastic, and x -periodic solitons can propagate steadily. It is shown that the Jacobi elliptic wave solutions can be viewed as the generalization of dromion, dromion-solitoff and periodic solitons.

Exact electron-acoustic solitary waves

Nonlinear propagation of an electron-acoustic wave in a magnetoplasma is considered. Accounting for the exact electron and ion nonlinearities, we show that the equations governing the dynamics of these waves can be written in terms of the energy integral of a classical particle. A careful analysis of the potential reveals the existence of exact planar solitary waves. Application of our investigation to space plasmas is pointed out.