Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

Qualitative analysis and solutions of bounded traveling wave for B-BBM equation

In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.     

Shape Analysis of Bounded Traveling Wave Solutions and Solution to the Generalized Whitham-Broer-Kaup Equation with Dissipation Terms

This paper deals with the problem of the bounded traveling wave solutions'shape and the solution to the generalized Whitham-Broer-Kaup equation with the dissipation terms which can be called WBK equati...     

耗散对广义的WBK型耗散方程行波解的影响

运用平面动力系统理论对广义的WBK型耗散方程所对应的动力系统作了定性分析,给出了其在不同参数条件下的全局相图.研究了该方程行波解的性态与耗散系数r之间的关系,得到当耗散作用较大时行波表现为扭状孤波,当耗散作用较小时行波表现为衰减振荡解的结论.     

Exact travelling wave solutions of the dissipative coupled Korteweg-de Vries equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative coupled Korteweg-de Vries equation. The possible kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful.     

Qualitative analysis and approximate damped oscillatory solutions for a kind of nonlinear dispersive-dissipative equation

This paper makes qualitative analysis to the bounded traveling wave solutions for a kind of nonlinear dispersive-dissipative equation, and considers its solving problem. The relation is investigated between behavior of its solution and the dissipation coefficient. Further, all approximate damped oscillatory solutions when dissipation coefficient is small are presented by utilizing the method of undetermined coefficients according to the theory of rotated vector field in planar dynamical systems. Finally, error estimate is given by establishing the integral equation which reflects the relation between approximate and exact damped oscillatory solutions applying the idea of homogenization principle.     

New explicit solutions for (2 + 1)-dimensional soliton equation

In this Letter, we study (2 + 1)-dimensional soliton equation by using the bifurcation theory of planar dynamical systems. Following a dynamical system approach, in different parameter regions, we depict phase portraits of a travelling wave system. Bell profile solitary wave solutions, kink profile solitary wave solutions and periodic travelling wave solutions are given. Further, we present the relations between the bounded travelling wave solutions and the energy level h. Through discussing the energy level h, we obtain all explicit formulas of solitary wave solutions and periodic wave solutions.     

New explicit solutions for the Klein-Gordon equation with cubic nonlinearity

In this paper, we investigate Klein-Gordon equation with cubic nonlinearity. All explicit expressions of the bounded travelling wave solutions for the equation are obtained by using the bifurcation method and qualitative theory of dynamical systems. These solutions contain bell-shaped solitary wave solutions, kink-shaped solitary wave solutions and Jacobi elliptic function periodic solutions. Moreover, we point out the region which these periodic wave solutions lie in. We present the relation between the bounded travelling wave solution and the energy level h. We find that these periodic wave solutions tend to the corresponding solitary wave solutions as h increases or decreases. Finally, for some special selections of the energy level h, it is shown that the exact periodic solutions evolute into solitary wave solution.     

Capillarity driven spreading of power-law fluids

We investigate the spreading of thin liquid films of power-law rheology. We construct an explicit travelling wave solution and source-type similarity solutions. We show that when the nonlinearity exponent λ for the rheology is larger than one, the governing dimensionless equation h_t + (h^λ+2|h_xxx|^λ−1h_xxx)_x=0 admits solutions with compact support and moving fronts. We also show that the solutions have bounded energy dissipation rate.     

Construction of a series of travelling wave solutions to nonlinear equations

In this paper, based on new auxiliary ordinary differential equation with a sixth-degree nonlinear term, we study the (1 + 1)-dimensional combined KdV–MKdV equation, Hirota equation and (2 + 1)-dimensional Davey–Stewartson equation. Then, a series of new types of travelling wave solutions are obtained which include new bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions. The method used here can be also extended to many other nonlinear partial differential equations.     

Exponential decay of solutions of a nonlinearly damped wave equation

The issue of stablity of solutions to nonlinear wave equations has been addressed by many authors. So many results concerning energy decay have been established. Here in this paper we consider the following nonlinearly damped wave equation a, b > 0, in a bounded domain and show that, for suitably chosen initial data, the energy of the solution decays exponentially even if m > 2.     

一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.     

Large time behavior and L−L estimate of solutions of 2-dimensional nonlinear damped wave equations

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the L^p−L^q type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the L^p−L^q estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|^αu. Our result covers the whole super critical case α>1, where the α=1 is well known as the Fujita exponent when n=2.     

Energy decay rates for solutions of the wave equation with boundary damping and source term

In this paper, we consider the wave equation

强非线性广义 Boussinesq 方程孤波解的波形分析及求解

上海理工大学理学院\quad 上海 200093该文建立了强非线性广义 Boussinesq 方程的耗散项、波速、渐进值与波形函数的导数之间的关系.利用适当变换和待定假设方法,作者求出了上述广义 Boussinesq 方程的扭状或钟状孤波解,还求出了以前文献中未曾提到过的余弦函数的周期波解.进一步给出了波速对波形影响的结论,即:``好'广义 Boussinesq 方程的行波当波速由小变大时,波形由钟状孤波变成余弦函数周期波解;``坏'广义 Boussinesq 方程的行波当波速由小变大时,波形由余弦函数周期波解变成钟状孤波.     

Travelling wave solutions of the Fornberg-Whitham equation

Zhou and Tian [J.B. Zhou, L.X. Tian, A type of bounded travelling wave solutions for the Fornberg-Whitham equation, J. Math. Anal. Appl. 346 (2008) 255-261] successfully found a type of bounded travelling wave solutions of the Fornberg-Whitham equation. In this paper, we improve the previous result by using the phase portrait analytical technology. Moreover, some smooth periodic wave, smooth solitary wave, periodic cusp wave and loop-soliton solutions are given, and the numerical simulation is made. The results show that our theoretical analysis agrees with the numerical simulation.     

Blow Up in a Nonlinearly Damped Wave Equation

In this paper we consider the nonlinearly damped semilinear wave equation u_tt – Δu + au_t |u_t|^{m – 2} = bu|u|^{p – 2} associated with initial and Dirichlet boundary conditions. We prove that any strong solution, with negative initial energy, blows up in finite time if p > m. This result improves an earlier one in [2].     

Long time existence for a slightly perturbed vortex sheet

Consider a flat two-dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ? and is analytic in a strip |??m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p, if ? is sufficiently small, with κ → 1 as ? → 0. For the particular case of sinusoidal data of wave length π and amplitude e, Moore's analysis and independent numerical results show singularity development at time t^c = |log ?| + O(log|log ?|. Our results prove existence for t < κ|log ?|, if ? is sufficiently small, with k κ → 1 as ? → 0. Thus our existence results are nearly optimal.