Exact traveling wave solutions and dynamical behavior for the (n+1)-dimensional multiple sine-Gordon equation

Using the methods of dynamical systems for the (n 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions axe obtained. For the double sine-Gordon equation, the exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined.     

Traveling Waves and Capillarity Driven Spreading of Shear-Thinning Fluids

We study capillary spreadings of thin films of liquids of power-law rheology. These satisfy ut＋（u^λ＋2｜uxxx｜^λ-1uxxx）x=0,where u （x, t） represents the thickness of the one-dimensional liquid and λ 〉 1. We look for traveling wave solutions so that u（x,t） =g（x＋ct） and thus g satisfies g＇＇＇=｜g-ε｜^1/λ/g^1＋2/λ sgn（g-ε） We show that for each ε 〉 0 there is an infinitely oscillating solution, gε, such that limt→∞ gε=ε and that gε→ g0 as ε → O, where g0≡ 0 for t ≥ 0 and g0=cλ｜t｜3λ/2λ＋1 for t〈0 for some constant cλ.     

Multiple rogue wave solutions for a (3+1)-dimensional Hirota bilinear equation

In this paper, we considered the multiple rogue wave solutions of a (3+1)-dimensional Hirota bilinear equation by using a symbolic computation approach. Based on the bilinear form of this equation, the first-order rogue waves, the second-order rogue waves and the third-order rogue waves are presented. Moreover, some basic properties of multiple rogue waves and their collision structures are explained by drawing the three dimensional plot.     

Research on traveling wave solutions for a class of (3+1)-dimensional nonlinear equation

Nonlinear wave phenomena are of great importance in the nature, and have became for a long time a challenging research topic for both pure and applied mathematicians. In this paper the solitary wave, kink (anti-kink) wave and periodic wave solutions for a class of (3+1)-dimensional nonlinear equation were obtained by some effective methods from the dynamical systems theory.     

Numerical similarity reductions of the (1+3)-dimensional Burgers equation

We consider the (1+3)-dimensional Burgers equation u_t = u_xx + u_yy + u_zz + uu_x which has considerable interest in mathematical physics. Lie symmetries are used to reduce it to certain ordinary differential equations. We employ numerical methods to solve a number of these ordinary differential equations.     

Exact traveling wave solutions and dynamical behavior for the (<Emphasis Type="Italic">n</Emphasis> + 1)-dimensional multiple sine-Gordon equation

Using the methods of dynamical systems for the (n + 1)-dimensional multiple sine-Gordon equation, the existences of uncountably infinite many periodic wave solutions and breaking bounded wave solutions are obtained. For the double sine-Gordon equation, the exact explicit parametric representations of the bounded traveling solutions are given. To guarantee the existence of the above solutions, all parameter conditions are determined. This work was supported by the National Natural Science Foundation of China (Grant No. 11671179) and the Natural Science Foundation of Yunnan Province (Grant No. 2005A0092M).     

Interaction solutions and abundant exact solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation in fluid mechanics

In this work, we present the interaction solutions and abundant exact solutions for the new (3+1)-dimensional generalized Kadomtsev-Petviashvili equation based on the Hirota''s bilinear form and a direct function. The obtained interaction solutions contain the interaction between the rational function and the $\tanh$ function and the interaction between the rational function and the $\cos$ function. The dynamical properties of these resulting solutions are analyzed and shown in three-dimensional plots, corresponding contour graphs and plane figures.     

On integrability of the extended (3+1)-dimensional Jimbo-Miwa equation

By applying the binary bell polynomial scheme, the bilinear form, Bäcklund Transformations, and lax pairs of an extended (3+1)-dimensional Jimbo-miwa (JM) equation are constructed. Next, periodic wave-type solutions can also be obtained to the extended (3+1)-dimensional JM equation through the three-wave method with the help of maple. Finally, a test function of the sech-function method is utilized to get solitary waves of this study problem. These new results can help us better understand interesting physical phenomena and mechanism.     

Lump solutions to the generalized (2+1)-dimensional B-type Kadomtsev-Petviashvili equation

Through symbolic computation with Maple, the (2+1)-dimensional B-type Kadomtsev-Petviashvili(BKP) equation is considered. The generalized bilinear form not the Hirota bilinear method is the starting point in the computation process in this paper. The resulting lump solutions contain six free parameters, four of which satisfy two determinant conditions to guarantee the analyticity and rational localization of the solutions, while the others are arbitrary. Finally, the dynamic properties of these solutions are shown in figures by choosing the values of the parameters.     

Traveling Waves in Degenerate Diffusion Equations

In this survey, we present a literature review on the study of traveling waves in degenerate diffusion equations by illustrating the interesting and singular wave behavior caused by degeneracy. The main results on wave existence and stability are presented for the typical degenerate equations, including porous medium equations, flux limited diffusion equations, delayed degenerate diffusion equations, and other strong degenerate diffusion equations.     

Traveling wave solutions of the Degasperis-Procesi equation

We classify all weak traveling wave solutions of the Degasperis-Procesi equation. In addition to smooth and peaked solutions, the equation is shown to admit more exotic traveling waves such as cuspons, stumpons, and composite waves.     

一类具有扩散项的流行性传染病模型中的行波解

讨论了一类具有扩散项的流行性传染病模型中的行波解的存在性.首先,将对该模型所对应的反应扩散系统的行波解的讨论转化为对二阶常微分系统的上下解的讨论;然后,通过上下解方法建立了这个具有扩散项的传染病模型中行波解的存在性条件,并进一步讨论了扩散因素对行波解的波速的影响,得到被感染人群的流动对病毒的传播有一定的影响.     

广义KdV方程孤子解的存在参数区

§ 1　IntroductionThe classic Kd V equation has the formut=32gh uux +32 αux +13σuxxx . (1 )Where g,h,α,andσ are physical parameters.After some transformations,various ver-sions of the modified Kd V equation may appear.The Kd V equation is a classic soliton e-quation and is well discussed.Different methods have been created to find the precise orapproximate solitons or solitary waves.The following Kd V equation is the mostcommonone:ut-6 uux +uxxx =0 . (2 )Considering traveling waves an…     

Asymptotic behavior of the periodic wave solution for the (3+1)-dimensional Kadomtsev-Petviashvili equation

In this paper, the one- and two-periodic wave solutions for the (3+1)-dimensional Kadomtsev-Petviashvili equation are presented by means of the Hirota’s bilinear method and the Riemann theta function. The rigorous proofs on asymptotic behaviors of these two solutions are given that soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.     

An analytical approach to solve the fractional-order (2 + 1)-dimensional Wu–Zhang equation

This paper considers time-fractional $(2+1)$-dimensional Wu–Zhang nonlinear system of partial differential equation describing a long dispersive wave. An approximate analytical solution of the dispersion relation of the long wave has been obtained by the fractional reduced differential transform method (FRDTM). The effect of fractional-order $\alpha$ on the wave profile of the solution is discussed graphically and comparing the exact solution of Wu–Zhang equation when $\alpha =1$. The result shows that the present method reveals the effectiveness, efficiency, and reliability of computed mathematical results to easily solve the fractional-order Wu–Zhang (WZ) system of differential equations.     

Extend three-wave method for the (1+2)-dimensional Ito equation

In this work, Extend three-wave method (ETM) is used to construct the novel multi-wave solutions of the (1+2)-dimensional Ito equation. As a result, three-soliton solution, doubly periodic solitary wave solutions, periodic two solitary wave solutions are obtained. It is shown that the Extend three-wave method may provide us with a straightforward and effective mathematical tool for seeking multi-wave solutions of higher dimensional nonlinear evolution equations.     

On the integrability of the (1+1)-dimensional and (2+1)-dimensional Ito equations

Under investigation in this paper are the (1+1)-dimensional and (2+1)-dimensional Ito equations. With the help of the Bell polynomials method, Hirota bilinear method and symbolic computation, the bilinear representations, N-soliton solutions, bilinear Bäcklund transformations and Lax pairs of these two equations are obtained, respectively. In particular, we obtain a new bilinear form and N-soliton solutions of the (2+1)-dimensional Ito equation. The bilinear Bäcklund transformation and Lax pair of the (2+1)-dimensional Ito equation are also obtained for the first time. Copyright © 2014 John Wiley & Sons, Ltd.