Exact special solitary solutions with compact support for the nonlinear dispersive K(m, n) equations

The nonlinear dispersive K(m, n) equations, u_t−(u^m)_x−(uⁿ)_xxx = 0 which exhibit compactons: solitons with compact support, are studied. New exact solitary solutions with compact support are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. General formulas for the solutions of K(m, n) equations are established.     

New exact solitary-wave special solutions for the nonlinear dispersive K(m, n) equations

In this paper, we study the nonlinear dispersive K(m, n) equations: u_t + (u^m)_x − (uⁿ)_xxx = 0 which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the decomposition method in K(m, n) equations. The nonlinear equations K(m, n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m, n) equations are established.     

Exact solutions of generalized Zakharov and Ginzburg–Landau equations

By using the homogeneous balance principle, the exact solutions of the generalized Zakharov equations and generalized Ginzburg–Landau equation are obtained with the aid of a set of subsidiary higher-order ordinary differential equations (sub-equations for short).     

Exact and explicit solutions of Euler-Painlevé equations through generalized Cole-Hopf transformations

More general Euler-Painlevé equations are exactly linearized using generalized Cole-Hopf transform and are shown to admit exact solutions in terms of Kummer functions. The asymptotic behaviours of Euler-Painlevé equations are also derived.     

New solitary-wave special solutions with compact support for the nonlinear dispersive K(m,n) equations

The genuinely nonlinear dispersive K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons: solitons with compact support, is investigated. New solitary-wave solutions with compact support are developed. The specific cases, K(2,2) and K(3,3), are used to illustrate the pertinent features of the proposed scheme. An entirely new general formula for the solution of the K(m,n) equation is established, and the existing general formula is modified as well.     

四阶非线性发展方程的精确解和广义条件对称

讨论了允许二阶广义条件对称的四阶非线性发展方程.通过广义条件对称方法得到了其对称约化和精确解.     

Exact peakon solutions given by the generalized hyperbolic functions for some nonlinear wave equations

In 1993, Camassa and Holm drived a shallow water equation and found that this equation has a peakon solution with the form $\phi(\xi)=ce^{-|\xi|}$. In this paper, we show that three nonlinear wave systems have peakon solutions which needs to be represented as generalized hyperbolic functions. For the existence of these solutions, some constraint parameter conditions are derived.     

The periodic wave solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations

More periodic wave solutions expressed by Jacobi elliptic functions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equations are obtained by using the extended F-expansion method. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained.     

含任意次正幂项的广义五阶KdV方程的精确解

利用齐次平衡原理,通过引进含非线性辅助微分方程（sub-ODE）,获得了含任意次正幂项的广义五阶KdV方程的精确解,包括钟状孤波解,扭状孤波解和三角函数表示的周期波解.所得精确解与前人用其它方法所获得一致,并包含了以往文献未提供的部分解,扩充并完善了以往文献的相关结果.     

Exact solutions ofG-invariant chiral equations

A method is suggested for solving the chiral equations (αg,zg ⁻¹),ˉz+(αg,ˉzg ⁻¹),z=0, whereg belongs to some Lie groupG. The solution is written out in terms of harmonic maps. The method can be used even for some infinite-dimensional Lie groups. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 710–716, November, 1995. The work was supported in part by the foundation CONACyT-México.     

New exact solutions of the Broer–Kaup equations in (2 + 1)-dimensional spaces

With the aid of Maple, several new kinds of exact solutions for the Broer–Kaup equations in (2 + 1)-dimensional spaces are obtained by using a new ansätz. This approach can also be applied to other nonlinear evolution equations.     

Symbolic computation of exact solutions for the (2 + 1)-dimensional generalized stochastic KP equation

In this paper, the F-expansion method is extended and applied to construct the exact solutions of the (2 + 1)-dimensional generalized Wick-type stochastic Kadomtsev–Petviashvili equation by the aid of the symbolic computation system Maple. Some new stochastic exact solutions which include kink-shaped soliton solution, singular soliton solution and triangular periodic solutions are obtained via this method and Hermite transformation.     

An improved generalized F-expansion method and its application to the (2 + 1)-dimensional KdV equations

An improved generalized F-expansion method is proposed to seek exact solutions of nonlinear partial differential equations. With the aid of symbolic computation, we choose the (2 + 1)-dimensional KdV equations to illustrate the validity and advantages of the proposed method. Many new and more general non-travelling wave solutions are obtained, including single and combined non-degenerate Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions, each of which contains two arbitrary functions.     

Exact solutions of equations for the Burgers hierarchy

Some classes of the rational, periodic and solitary wave solutions for the Burgers hierarchy are presented. The solutions for this hierarchy are obtained by using the generalized Cole–Hopf transformation.     

Exact solutions of the unsteady two-dimensional Navier-Stokes equations

This paper studies the two-dimensional incompressible viscous flow in which the local vorticity is proportional to the stream function perturbed by a uniform stream. It was known by Taylor and Kovasznay that the Navier-Stokes equations for flow of this kind become linear. From the general solution to the linear equations for steady flow, we show that there exist only two types of steady flow of this kind: Kovasznay downstream flow of a two-dimensional grid and Lin and Tobak reversed flow about a flat plate with suction. In the unsteady flow case, new classes of exact analytical solutions are found which include Taylor vortex array solution as a special case. It is shown that these unsteady flows are, as viewed from a frame of reference moving with the undisturbed uniform stream, pseudo-steady in the sense that the flow pattern is steady but the magnitude of motion decays, or grows, exponentially in time. All these solutions are valid for any Reynolds number.

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Résumé Dans ce travail nous étudions l'écoulement plan d'un fluide visqueux incompressible dans lequel la rotation locale est proportioneile à la fonction de courant perturbée par un courant uniforme. Conformément aux travaux de Taylor et Kovasznay les équations de Navier-Stokes pour cet écoulement deviennent linéaires. Par conséquent nous utilisons la solution générale pour démontrer que seulement deux catégories d'écoulement stationnaire peuvent exister: l'écoulement de Kovasznay en aval d'une grille plane, et l'écoulement inversé de Lin et Tobak pour une plaque plane avec aspiration. Nous étudions aussi l'écoulement non stationnaire et nous découvrons des classes nouvelles de solutions exactes qui contiennent, en particulier, le réseau de tourbillons de Taylor. Enfin nous démontrons que ces écoulements sont pseudo-stationnaires dans un système de coordonnées en mouvement avec le courant uniforme non perturbé; ce qui signifie que l'amplitude de l'écoulement stationnaire croit ou décroit exponentiellment dans le temps. Toutes ces solutions sont valides pour tous les nombres de Reynolds.

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On leave from University of Waterloo, Ontario, Canada.     

Exact elliptic compactons in generalized Korteweg–De Vries equations

Using the action principle, and assuming a solitary wave of the generic form u(x,t) = AZ(β(x + q(t)), we derive a general theorem relating the energy, momentum, and velocity of any solitary wave solution of the generalized Korteweg‐De Vries equation K*(l,p). Specifically we find that , where l,p are nonlinearity parameters. We also relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two‐parameter family of exact solutions to these equations, which include elliptic and hyper‐elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria. © 2006 Wiley Periodicals, Inc. Complexity 11: 30–34, 2006     

Exact travelling wave solutions of the generalized Bretherton equation

We search for exact travelling wave solutions of the generalized Bretherton equation for integer, greater than one, values of the exponent m of the nonlinear term by two methods: the truncated Painlevé expansion method and an algebraic method. We find periodic solutions for m=2 and m=5, to add to those already known for m=3; in all three cases these solutions exist for finite intervals of the wave velocity. We also find a “kink” shaped solitary wave for m=5 and a family of elementary unbounded solutions for arbitrary m.