New Types Of Exact Solutions Research Articles

New types of exact solutions of high-frequency waves model in the relaxation medium

In this article, based on the extended fan-expansion method, novel soliton wave solutions of the Vakhnenko-Parkes equation are constructed. The stable property of the obtained analytical solutions is tested by implementing the Hamiltonian system's characterizations. The applied method is effective and applicable for many problems of non-linear PDE in mathematical physics.

Dynamical structures of solitons and some new types of exact solutions for the (2+1)-dimensional DJKM equation using Lie symmetry analysis

This paper is devoted to obtaining some new types of exact solutions of the (2+1)-dimensional Date–Jimbo–Kashiwara–Miwa (DJKM) equation utilizing the Lie symmetry method. All the Lie symmetries, infinitesimal generators, and possible geometric vector fields have been obtained by using the invariance condition of the group-theoretic method. Meanwhile, the Lie symmetry reductions and explicit exact solutions are obtained by a one-dimensional (1D) optimal system. All the obtained exact solutions are absolutely new and completely different from the earlier established results in the literature. Moreover, the dynamical behavior of obtained solitons like doubly solitons, dark solitons, kink wave, curved shaped multi-solitons, parabolic waves, solitary waves, and annihilation of elastic multi-soliton profiles is depicted graphically via interesting 3D-shapes. That will be widely used to provide many more attractive complex physical phenomena in the fields of plasma physics, statistical physics, fiber optics, fluid dynamics, condensed matter physics, and so on. Finally, we have verified all the achieved soliton solutions through symbolic computations with Mathematica.

Some new types of exact solutions for the Kac–Wakimoto equation associated with

Recently, it is shown that the Kac–Wakimoto equation associated with is not integrable since it does not pass Painlevé test and does not have three-soliton solution, even it has one- and two-soliton solutions. Thus in this paper, we investigate some new types of exact solutions for this equation based on its bilinear form. As a result, the rational solutions, kink-type breather solution and degenerate three-solitary wave solutions of this equation are found. The properties and space structures of these exact solutions are analyzed by displaying their profiles in (x, y)-directions. Furthermore, the Lie symmetry analysis is done to present the one-parameter group of symmetries for the KW equation.

Integrability aspects and localized wave solutions for a new $$\mathbf (4+1) $$-dimensional Boiti–Leon–Manna–Pempinelli equation

In this paper, we introduce a new integrable nonlinear evolution equation in $$4+1$$ dimensions, which is an extension of Boiti–Leon–Manna–Pempinelli equation. We prove that this new equation has the Painleve property. By using the Bell polynomial approach, we obtain the bilinear representation, bilinear Backlund transformation, Lax pair and infinite conservation laws. Furthermore, several types of new exact solutions are also constructed based on the Hirota bilinear method, including the N-soliton solutions, periodic soliton solutions and mixed lump–kink wave solutions. The dynamics and interactions of localized wave solutions are illustrated by some graphs.

New abundant wave solutions of the conformable space–time fractional (4+1)-dimensional Fokas equation in water waves

A nonlinear fractional model arising in water waves theory, namely the new conformable space–time fractional (4+1)-dimensional Fokas equation, is explored via some recently developed approaches, namely, the functional variable, the generalized Kudryashov, the Jacobi elliptic function expansion and the generalized Riccati equation mapping methods. As a result, several (abundant) distinct types of new exact solutions are constructed with the aid of symbolic computation via Mathematica 9 in terms of solitary wave, periodic wave, Jacobi elliptic function and other solutions. Furthermore, some graphical representations (numerical simulations) have been provided to illuminate the physical behavior of the resultant solutions that may be very important in real world problems and applications with fractional order.

Applications of IBSOM and ETEM for solving a discrete electrical lattice

This paper presents a number of new solutions obtained for solving the nonlinear electrical transmission lines by using the improved Bernoulli sub-ODE method and the extended trial equation method. The proposed solutions are kink soliton solution, hyperbolic solution, trigonometric solution and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and concise and can be also applied to other nonlinear partial differential equations. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to construct many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in telecommunication systems where solitons are used for codify data.

Exact solutions for Fitzhugh–Nagumo model of nerve excitation via Kudryashov method

This paper presents a number of new solutions obtained for solving the Fitzhugh–Nagumo model via the Kudryashov method. The merit of the presented method is finding the further solutions of the considering problems including soliton, periodic, kink, kink-singular wave solutions. Comparing our new results with other results show that our results give the further solutions. The proposed method also allow us to establish many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in engineering and physics sciences.

Applications of IBSOM and ETEM for solving the nonlinear chains of atoms with long-range interactions

This paper presents a number of new solutions obtained for solving a complex nonlinear equation describing dynamics of nonlinear chains of atoms via the improved Bernoulli sub-ODE method (IBSOM) and the extended trial equation method (ETEM). The proposed solutions are kink solitons, anti-kink solitons, soliton solutions, hyperbolic solutions, trigonometric solutions, and bellshaped soliton solutions. Then our new results are compared with the well-known results. The methods used here are very simple and succinct and can be also applied to other nonlinear models. The balance number of these methods is not constant contrary to other methods. The proposed methods also allow us to establish many new types of exact solutions. By utilizing the Maple software package, we show that all obtained solutions satisfy the conditions of the studied model. More importantly, the solutions found in this work can have significant applications in Hamilton’s equations and generalized momentum where solitons are used for long-range interactions.

Analytical solutions with the improved (G’/G)-expansion method for nonlinear evolution equations

To seek the exact solutions of nonlinear partial differential equations (NPDEs), the improved (G'/G)-expansion method is proposed in the present work. With the aid of symbolic computation, this effective method is applied to construct exact solutions of the (1+1)-dimensional nonlinear dispersive modified Benjamin-Bona-Mahony equation and (3+1)- dimensional Kudryashov-Sinelshchikov equation. As a result, new types of exact solutions are obtained.

Extended trial equation method for nonlinear coupled Schrodinger Boussinesq partial differential equations

Abstract In this paper, we improve the extended trial equation method to construct the exact solutions for nonlinear coupled system of partial differential equations in mathematical physics. We use the extended trial equation method to find some different types of exact solutions such as the Jacobi elliptic function solutions, soliton solutions, trigonometric function solutions and rational, exact solutions to the nonlinear coupled Schrodinger Boussinesq equations when the balance number is a positive integer. The performance of this method is reliable, effective and powerful for solving more complicated nonlinear partial differential equations in mathematical physics. The balance number of this method is not constant as we have in other methods. This method allows us to construct many new types of exact solutions. By using the Maple software package we show that all obtained solutions satisfy the original partial differential equations.

Exact solution of Helmholtz equation for the case of non-paraxial Gaussian beams

A new type of exact solutions of the full 3 dimensional spatial Helmholtz equation for the case of non-paraxial Gaussian beams is presented here.We consider appropriate representation of the solution for Gaussian beams in a spherical coordinate system by substituting it to the full 3 dimensional spatial Helmholtz equation.Analyzing the structure of the final equation, we obtain that governing equations for the components of our solution are represented by the proper Riccati equations of complex value, which has no analytical solution in general case.But we find one of the possible exact solutions which is proved to satisfy to such equations for Gaussian beams.

Quasi-periodic Solutions of a Spiral Type for Photogravitational Restricted Three-Body Problem

A new type of exact solutions for photogravitational restricted three-body problem (a case of spiral motion) is presented here. A key point is that we obtain the appropriate specific case of spiral motions from the Jacobian-type integral of motion for photogravitational restricted three-body problem (when orbit of small 3-rd body is assumed to be like a spiral). Besides, we should especially note that there is a proper restriction to the type of spiral orbital motion of small 3-rd body, which could be possible for choosing as the exact solution of equations for photogravitational restricted three-body problem. The main result, which should be outlined, is that in a case of quasi-planar orbital motion (of the small 3-rd body) the asymptotic expression for component z of motion is proved to be given by the proper elliptical integral.

Comment on “New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”

In this comment we analyze the paper [Abdelhalim Ebaid, S.M. Khaled, New types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity, J. Comput. Appl. Math. 235 (2011) 1984–1992]. Using the traveling wave, Ebaid and Khaled have found “new types of exact solutions for nonlinear Schrodinger equation with cubic nonlinearity”. We demonstrate that the authors studied the well-known nonlinear ordinary differential equation with the well-known general solution. We illustrate that Ebaid and Khaled have looked for some exact solution for the reduction of the nonlinear Schrodinger equation taking the general solution of the same equation into account.

New types of exact solutions for the fourth-order dispersive cubic–quintic nonlinear Schrödinger equation

In this study, we use two direct algebraic methods to solve a fourth-order dispersive cubic–quintic nonlinear Schrödinger equation, which is used to describe the propagation of optical pulse in a medium exhibiting a parabolic nonlinearity law. By using complex envelope ansatz method, we first obtain a new dark soliton and bright soliton, which may approach nonzero when the time variable approaches infinity. Then a series of analytical exact solutions are constructed by means of F-expansion method. These solutions include solitary wave solutions of the bell shape, solitary wave solutions of the kink shape, and periodic wave solutions of Jacobian elliptic function.

New types of exact solutions for nonlinear Schrödinger equation with cubic nonlinearity

Although, many exact solutions were obtained for the cubic Schrödinger equation by many researchers, we obtained in this research not only more exact solutions but also new types of exact solutions in terms of Jacobi-elliptic functions and Weierstrass-elliptic function.

Travelling wave solutions to some important equations of mathematical physics

Exact travelling wave solutions of some nonlinear evolution equations of mathematical physics are obtained by using the mapping method and the extended F -expansion method. It is well known that different types of exact solutions of a given auxiliary ordinary differential equation produce new types of exact solutions to nonlinear evolution equations. Many new exact travelling wave solutions of Zakharov-Kuzentsov (ZK), modified Kadomtsev-Petviashvilli (KP) with square root nonlinearity and modified fifth-order Korteweg-de Vries (KdV) equations are constructed by using mapping method. We also apply the extended F -expansion method to the long-short-wave interaction system and the coupled modified KdV equations. The solutions obtained in this paper include single and combined Jacobi elliptic function solutions, rational solutions and hyperbolic function solutions. In the limiting case, the solitary wave solutions of such equations and systems are also studied.

A new note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev–Petviashvili equation

Exact solutions of the (2+1)-dimensional Kadomtsev–Petviashvili by Zhang [Huiqun Zhang, A note on exact complex travelling wave solutions for (2+1)-dimensional B-type Kadomtsev–Petviashvili equation, Appl. Math. Comput. 216 (2010) 2771–2777] are considered. To look for “new types of exact solutions travelling wave solutions” of equation Zhang has used the G′/ G-expansion method. We demonstrate that there is the general solution for the reduction by Zhang from the (2+1)-dimensional Kadomtsev–Petviashvili equation and all solutions by Zhang are found as partial cases from the general solution.

Exact Solutions for Coupled mKdV Equations by a New Symbolic Computation Method

By introducing (G′/G)-expansion method and symbolic computation software MAPLE, two types of new exact solutions are constructed for coupled mKdV equations. The solutions included trigonometric function solutions and hyperbolic function solutions. The procedure is concise and straightforward, and the method is also helpful to find exact solutions for other nonlinear evolution equations.

The new tri-function method to multiple exact solutions of nonlinear wave equations

In this paper, based on a system of the first order differential equations with three nonlinear ordinary differential equations (ODEs), a new tri-function method is presented to investigate exact solutions of a wide class of nonlinear wave equations. The method is constructive and can be carried out in a computer with the aid of symbolic computation. In particular, we apply the tri-function method to the (3+1)-dimensional Kadomtsev–Petviashvili (KP) equation and the (2+1)-dimensional nonlinear Schrödinger (NLS) equation such that many types of new exact solutions are obtained, which contain doubly periodic solutions and solitary wave solutions.

Exactly fractional solutions of the (2+1)-dimensional modified KP equation via some fractional transformations

In this paper, the (2+1)-dimensional modified Kadomtsev–Petviashvil (KP) equation is investigated with the aid of symbolic computation. We use some fractional transformations to obtain many types of new exact solutions of (2+1)-dimensional modified KP equation. These solutions include rational solutions, periodic wave solutions, solitary wave solutions and doubly periodic wave solutions. These transformations can be also extended to other nonlinear wave equations.