Construction of new wave structures and stability analysis for the nonlinear Klein-Gordon equation

Under investigation in this paper is a nonlinear conformable time-fractional Boussinesq equations as an important class of fractional differential equations in mathematical physics. The extended trial equation method, the $$\exp (-\Omega (\eta ))$$ -expansion method and the $$\tan (\phi (\eta )/2)$$ -expansion method are used in examining the analytical solution of the nonlinear fractional equations. The proposed methods are based on the integration method and a wave transformation. The fractional derivative in the sense of conformable time-fractional derivative is defined. Fractional complex transform is implemented to change fractional differential equations into ordinary differential equations in this paper. In addition, explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of nonlinear conformable time-fractional Boussinesq equations.

In this article, the extended multiple Riccati equations expansion method has been used to construct a series of double soliton- like solutions, double triangular function solutions and complexiton soliton solutions for some nonlinear partial differential equations in mathematical physics via the (2+1) dimensional breaking soliton equations, (2+1) dimensional painleve integrable Burger's equations and (2+1) dimensional Nizhnik- Novikov- Vesselov equations. With the help of symbolic computation as Maple or Mathematica, we obtain many new types of complexiton soliton solutions, for example, various combination of trigonometric periodic function and hyperbolic function solutions, various combination of trigonometric periodic function and rational function solutions, various combination of hyperbolic function and rational function solutions. Key words: The extended multiple Riccati equations expansion method, double soliton-like solutions, double triangular function solutions, complexiton soliton solutions, the (2+1) dimensional breaking soliton equations, the (2+1) dimensional painleve integrable Burger's equations, the (2+1)-dimensional Nizhnik-Novikov- Vesselov equations.

In this study, several novel solutions of strain wave equation for micro-structured solids are obtained by using the powerful modified extended direct algebraic method. In micro-structured solids, the strain wave dynamical equation is utilized to model the wave propagation in micro-structured materials. We have profitably obtained exact traveling wave solutions, including solitons, solitary waves, kink and anti-kink solitary waves, periodic and rational solutions have been found. This approach can not only produce the same solution, but also find new solutions that we think other researchers have missed. These attained solutions help researchers for knowing the physical phenomena of this dynamical equation. The three-dimensional maps of some solutions obtained in this study are plotted through giving appropriate values of the parameters engaged in the solutions for knowing the physical interpretation. The computational work and obtained results demonstrate that the new extended technique, gives an efficient and direct mathematical tools for resolving the nonlinear models in engineering and mathematical physics.

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

Effects of dust charges on the nonlinear ion acoustic solitary waves in a fully relativistic dusty plasma for both cases of negative and positive dusts are numerically studied based on the pseudopotential method. In the presence of dusty particles, it is found that various types of nonlinear acoustic waves exist in forms which can be viewed as sequential combinations of three kinds of elementary solitary waves: bump, dip, and kink-type solitary waves. The number and the sequence of the constituent elementary solitary waves in a given nonlinear waves depend more sensitively on dust particle density than any other parameters. For negatively charged dust particles of low density, the nonlinear wave is in the shape of bumpy solitary wave. For a somewhat higher density, the wave changes into a form which can be viewed as a combination of bump and dip-type solitary waves. As the density is increased further, a more complex nonlinear wave composed of bump, kink, and dip-type solitary waves emerges. For a much higher density of dust particle, the nonlinear wave can have a shape that can be considered as a combination of bump and kink-type solitary waves. For the case of positively charged dust particles, two kinds of nonlinear waves can exist: bump-type solitary wave and a combination of bump and kink solitary waves. For both cases of negative and positive dust particles, it is found that single dip-type solitary wave does not exist. It is also found that as dust particle density increases, the signature of the elementary waves becomes less prominent.

We analyze the effect of phase-matched third-harmonic generation on the existence and stability of (1+1)-dimensional bright and dark spatial solitary waves in optical media with a cubic (or Kerr) nonlinear response. We demonstrate that parametric coupling of the fundamental beam with the third harmonic leads to the existence of two-color solitary waves resembling those in a χ(2) medium and that it can modify drastically the properties of solitary waves due to effective non-Kerr nonlinearities. In particular, we find a power threshold for the existence of two-frequency parametric bright solitons and also reveal the soliton multistability in a Kerr medium that becomes possible owing to a higher-order nonlinear phase shift caused by cascaded third-order processes. We also analyze dark solitary waves and their stabilities. We show that, in a certain parameter domain, parametric χ(3) dark solitons may become unstable owing to the modulational instability of the supporting background or to other instability mechanisms caused by the parametric coupling between the harmonics.

In this paper, we propose a new method called exp(−I•(ξ)) fractional expansion method to seek traveling wave solutions of the nonlinear fractional Sharma-Tasso-Olver equation. The result reveals that the method together with the new fractional ordinary differential equation is a very ini¬‚uential and effective tool for solving nonlinear fractional partial differential equations in mathematical physics and engineering. The obtained solutions have been articulated by the hyperbolic functions, trigonometric functions and rational functions with arbitrary constants

In this article, many new exact solutions of the (2+1)-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation and the (1+1)-dimensional nonlinear heat conduction equation are constructed using the Riccati equation mapping method. By means of this method, many new exact solutions are successfully obtained. This method can be applied to many other nonlinear evolution equations in mathematical physics. doi: 10.14456/WJST.2014.14

Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this work, the exact traveling wave solutions of the Boussinesq equation is studied by using the new generalized (G′/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G′/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations in mathematical physics and engineering.PACS05.45.Yv, 02.30.Jr, 02.30.Ik

In this paper, we investigate closed-form meromorphic solutions of the fifth-order Sawada-Kotera (fSK) equation and (3+1)-dimensional generalized shallow water (gSW) equation. The study of high-order and high-dimensional differential equations is pivotal for modeling complex nonlinear phenomena in physics and engineering, where higher-order dispersion, dissipation, and multidimensional dynamics govern system behavior. Constructing explicit solutions is of great significance for the study of these equations. The elliptic, hyperbolic, rational, and exponential function solutions for these high-order and high-dimensional differential equations are achieved by proposing the extended complex method. The planar dynamics behavior of the (3+1)-dimensional gSW equation and its phase portraits are analyzed. Using computational simulation, the chaos behaviors of the high-dimensional differential equation under noise perturbations are examined. The dynamic structures of some obtained solutions are revealed via some 2D and 3D graphs. The results show that the extended complex method is an efficient and straightforward approach to solving diverse differential equations in mathematical physics.

Optical solitary wave solutions for the higher order nonlinear Schrödinger equation with cubic-quintic non-Kerr terms

Analytical approach for space–time fractional Klein–Gordon equation

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

New solitons and periodic solutions for nonlinear physical models in mathematical physics

New analytical solutions of (2 + 1)-dimensional conformable time fractional Zoomeron equation via two distinct techniques