Exact traveling wave solutions to 2D-generalized Benney-Luke equation

By using the dynamical system method to study the 2D-generalized Benney- Luke equation, the existence of kink wave solutions and uncountably infinite many smooth periodic wave solutions is shown. Explicit exact parametric representations for solutions of kink wave, periodic wave and unbounded traveling wave are obtained.     

Bifurcations of travelling wave solutions for a coupled nonlinear wave system

Introduction FangShaomeiandGuoBoling[1]consideredthefollowingtimeperiodicproblemof dampedcouplednonlinearwaveequations:ut f(u)x-αuxx βuxxx 2vvx=G1(u,v) h1(x),vt-γvxx 2(uv)x g(v)x=G2(u,v) h2(x),(1)whereα,β,γareconstants,andγ>0,β≠0.Undertheperiodicboundaryconditions,the authorsobtainedtheuniqueexistenceofstrongsolutionsfortheabovesystem.InthispaperweshallconsiderbifurcationbehaviorofthetravellingwavesolutionsofEq.(1)inthecaseGi(u,v)≡0,hi(u,v)≡0(i=1,2).Letξ=x-ct,u=u(x-ct),where cis…     

Similarity analysis and exact solutions for a general discrete two-velocity model of Boltzmann equation

Summary This paper concerns with the similarity analysis for a general discrete two-velocity model of the Boltzmann equation introduced by Illner [12]. We find the general groups of invariance and we get some exact solutions, recovering general results which contain either solutions extensively described in the literature or undiscovered ones.

---

Sommario In questa nota si applica l'analisi dei gruppi infinitesimi di trasformazione ad un modello generale discreto a due velocità dell'equazione di Boltzmann introdotto da Illner [12]. Si trovano i più generali gruppi di invarianza e si ottengono alcune soluzioni esatte, ritrovando risultati generali che contengono sia soluzioni ampiamente descritte in letteratura che nuove soluzioni.

---

---

Work supported by the C.N.R. through the G.N.F.M.     

Nonlinear evolution of probability vectors of interest in discrete kinetic theory

In the framework of the discrete Boltzmann equation, a suitable space-time discretization of the one-dimensional fourteen discrete velocity model by Cabannes, leads in a bounded domain to the nonlinear Markovian evolution of a probability vector, whose moments represent the macroscopic quantities of the gas. Convergence of the probability vector towards the equilibrium steady state is proven when the walls are at a temperature compatible with the equilibrium itself. A physical application is subsequently dealt with. The classical problem of heat transfer between two parallel plates at different temperatures is formulated and solved, and the properties of the final steady state are discussed.     

A new auxiliary equation method for finding travelling wave solutions to KdV equation

In this paper, a new auxiliary equation method is used to find exact travelling wave solutions to the （1＋1）-dimensional KdV equation. Some exact travelling wave solu- tions with parameters have been obtained, which cover the existing solutions. Compared to other methods, the presented method is more direct, more concise, more effective, and easier for calculations. In addition, it can be used to solve other nonlinear evolution equations in mathematical physics.     

Exact traveling wave solutions for an integrable nonlinear evolution equation given by M. Wadati

By using the method of dynamical systems, the travelling wave solutions of for an integrable nonlinear evolution equation is studied. Exact explicit parametric representations of kink and anti-kink wave, periodic wave solutions and uncountably infinite many smooth solitary wave solutions are given.     

Bifurcations of travelling wave solutions for Jaulent-Miodek equations

By using the theory of bifurcations of planar dynamic systems to the coupled Jaulent-Miodek equations,the existence of smooth solitary travelling wave solutions and uncountably infinite many smooth periodic travelling wave solutions is studied and the bifurcation parametric sets are shown.Under the given parametric conditions,all possible representations of explicit exact solitary wave solutions and periodic wave solutions are obtained.     

稀薄流非线性模型方程离散速度坐标法有限差分解

从一般非线性Bo ltzm ann方程出发,发展并实现了一套适于大范围K nudsen数稀薄流问题数值模拟的统一算法。采用BGK模型和Shakov模型近似碰撞项,进而引入两个二速度无量纲简化分布函数,通过关于分子速度第三分量取矩积分,将三速度单一模型方程变换为二速度微分方程组。基于G auss-H erm ite积分公式和正交多项式G auss积分公式,借助离散速度坐标法消除简化模型方程对分子速度空间的连续依赖性,从相空间到物理空间得到一组带源项双曲守恒离散方程,并给出其显式和隐式二阶迎风TVD有限差分解。以二维圆柱A r气体超声速绕流算例,验证了数值算法的有效性,比较分析了漫反射和镜面反射两种气体分子壁面反射模型的计算结果。     

A lattice Boltzmann model for two‐dimensional sound wave in the small perturbation compressible flows

A multi‐entropy‐level lattice Boltzmann model for two‐dimensional sound wave equation in the small perturbation flows is presented. In this model, we used higher‐order moment method, multi‐scale technique and the Chapman–Enskog expansion, and multi‐entropy‐level to obtain sound wave equation with isentropic equation. As numerical examples, the Doppler effects in the sound wave propagation, the sound scattering from circular cylinder are simulated. The numerical results show that this model can be used to simulate sound wave propagation. Copyright © 2010 John Wiley & Sons, Ltd.     

A higher‐order accuracy lattice Boltzmann model for the wave equation

A lattice Boltzmann model with higher‐order accuracy for the wave motion is proposed. The new model is based on the technique of the higher‐order moment of equilibrium distribution functions and a series of lattice Boltzmann equations in different time scales. The forms of moments are derived from the binary wave equation by designing the higher‐order dissipation and dispersion terms. The numerical results agree well with classical ones. Copyright © 2008 John Wiley & Sons, Ltd.     

New Explicit and Exact Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ ,ｔｈｅｓｔｕｄｙｋｅｒｎｅｌｏｆｍｏｄｅｒｎｓｃｉｅｎｃｅｉｓｃｈａｎｇｅｄｆｒｏｍｌｉｎｅａｒｔｏｎｏｎｌｉｎｅａｒｓｔｅｐｂｙｓｔｅｐ .Ｍａｎｙｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｐｒｏｂｌｅｍｓｃａｎｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｅｄｅｓｃｒｉｂｅｄｂｙｕｓｉｎｇｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎ .Ｕｐｔｏｎｏｗ ,ｍａｎｙｉｍｐｏｒ…     

New groups of solutions to the Whitham-Broer-Kaup equation

The Whitham-Broer-Kaup model is widely used to study the tsunami waves. The classical Whitham-Broer-Kaup equations are re-investigated in detail by the generalized projective Riccati-equation method. 20 sets of solutions are obtained of which, to the best of the authors’ knowledge, some have not been reported in literature.Bifurcation analysis of the planar dynamical systems is then used to show different phase portraits of the traveling wave solutions under various parametric conditions.     

ON CAUCHY PROBLEMS FOR THE RLW EQUATION IN TWO SPACE DIMENSIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｓｙｓｔｅｍｏｆｔｈｅｎｏｎｌｉｎｅａｒｄｉｓｐｅｒｓｉｖｅ ,ｔｈｅｐａｐｅｒｓ [1 ]ａｎｄ [2 ]ｈａｖｅｓｔｕｄｉｅｄｔｈｅｍｏｄｅｌｅｑｕａｔｉｏｎｏｆｓｉｍｐｌｅｗａｖｅｓｐｒｏｐａｇａｔｉｏｎｏｆｌｏｎｇｗａｖｅ ,ｕｔ+αｕｘ+ βｕｕｘ-γｕｘｘｔ=0　　ｘ∈Ｒ ,ｔ≥ 0ｉｓｃａｌｌｅｄｔｈｅＲＬＷｅｑｕａｔｉｏｎ .ＳＨＡＮＧ[3]ｏｂｔａｉｎｅｄｔｈｅｅｘａｃｔｓｏｌｉｔａｒｙｗａｖｅｓｏｌｕｔｉｏｎｓｏｆｔｈｅｔｗｏｄｉｍｅｎｓｉｏｎａｌＲＬＷｅｑｕａｔｉｏｎ .Ｉｎ…     

On cauchy problems for the RLW equation in two space dimensions

A cnoidal wave solution of the two dimensional RLW equation of are obtained by elliptic integral method, and the some estimations the uniqueness and the stability of the periodic solution with both x, y to the Cauchy problem are proved by the priori estimations. Biography: Huang Zheng-hong (1956-)     

A multi‐energy‐level lattice Boltzmann model for two‐dimensional wave equation

A lattice Boltzmann model for two‐dimensional wave equation is presented. In this model, we used higher‐order moment method, multi‐scale technique and Chapman–Enskog expansion, and multi‐energy‐level to obtain wave equation and energy conservation equation. As numerical examples, the interference and diffraction of wave are simulated. The numerical results show this model can be used to simulate two‐dimensional wave propagation. Copyright © 2009 John Wiley & Sons, Ltd.     

Wronskian and Grammian solutions for the (2+1)-dimensional BKP equation

The (2 + 1)-dimensional BKP equation in the Hirota bilinear form is studied during this work. Wronskian and Grammian techniques are applied to the construction of Wronskian and Grammian solutions of this equation, respectively. It is shown that these solutions can be expressed as not only Pfaffians but also Wronskians and Grammians.     

Two types of traveling wave solutions to burgers-KdV equations

ＩｎｔｒｏｄｕｃｔｉｏｎＳｏｃａｌｌｅｄＢｕｒｇｅｒｓ＿ＫｄＶｅｑｕａｔｉｏｎｓａｒｅｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｅｖｏｌｕｔｉｏｎｅｑｕａｔｉｏｎｓｕｔ ｕｕｘ-αｕｘｘ βｕｘｘｘ =0 ,(1 )ｗｈｉｃｈｗａｓｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｉｎｔｒｏｄｕｃｅｄｗｈｅｎｏｎｅｓｔｕｄｉｅｄｓｏｍｅｐｈｙｓｉｃａｌｐｒｏｂｌｅｍｓｓｕｃｈａｓｌｉｑｕｉｄｆｌｏｗｓｗｉｔｈａｉｒｂｕｂｂｌｅｓ,ｔｈｅｆｌｕｉｄｉｎｔｈｅｅｌａｓｔｉｃｔｕｂｅｓａｎｄｓｏｏｎ ,ｗｈｅｒｅαａｎｄβｓｔａｎ…     

Bifurcations of traveling wave solutions and exact solutions to generalized Zakharov equation and Ginzburg-Landau equation

This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.     

Exact traveling wave solutions in the coupled plane-base rotator model of DNA

The internal dynamics of the DNA base pairs is studied starting from the generalized coupled plane base-rotator model of DNA, obtained by Yomosa and later improved by Takeno and Homma. We conceive the double-stranded DNA as an anisotropically coupled spin chain simple model. The generalized Hamiltonian expressed in terms of quasi-spin operators is averaged over the generalized coherent states in the Perelomov sense, in order to obtain the classical non-linear evolution equations of this molecular system where the inhomogeneity has not been considered. This approach provides the equations of motion, which could be reduced to a nonlinear Schrödinger equation with a saturable nonlinearity. This non-linear equation, under certain restrictions in the parametric space, supports traveling periodic triangular, bell, bubble and kink like solutions.     

Exact solutions to the problem of deep-bed filtration with retardation of a jump in concentration within the framework of the nonlinear two-velocity model

Exact solutions with plane and cylindricalwaves are obtained for one-dimensional problems of injection of a suspension into a porous reservoir when lagging of the suspended particles behind the carrier fluid is taken into account in the case of large change in the porosity. It is shown that taking lagging of the particles behind the fluid into account can lead to slowing-down the motion of jump in concentration. This is in agreement with the results of a series of experiments. It is also noted that, in principle, models in which the particles pass in average ahead of the carrier fluid are possible in the problems of deep bed filtration.