New exact solutions of a nonlinear integrable equation

Analytic solutions of the partial differential equations are needed to explain many phenomena seen in thermodynamics, aerodynamics, plasma physics, and other fields. In this paper, variational principle is analyzed of the integrable nonlinear Korteweg–de Vries (KdV) typed equation. In addition, exact solutions of this equation are obtained by using various methods such as direct integration, homogeneous balance method, Exp-function method, and Kudryashov method.     

Symbolic computation of exact solutions for a nonlinear evolution equation

In this paper, by means of the Jacobi elliptic function method, exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are presented. It can be shown that not only the obtained solitary wave solutions have the property of loop-shaped, cusp-shaped and hump-shaped for different values of parameters, but also different types of double periodic wave solutions are possible, namely periodic loop-shaped wave solutions, periodic hump-shaped wave solutions or periodic cusp-shaped wave solutions. Furthermore, periodic loop-shaped wave solutions will be degenerated to loop-shaped solitary wave solutions for the same values of parameters. So do cusp-shaped solutions and hump-shaped solutions. All these solutions are new and first reported here.     

Bifurcations and exact solutions for a nonlinear surface wind waves equation

By using the method of dynamical systems, for the nonlinear surface wind waves equation, which is given by Manna, we study its dynamical behavior to determine all exact explicit traveling wave solutions. To guarantee the existence of the aforementioned solutions, all parameter conditions are determined. Our procedure shows that the nonlinear surface wind waves equation has no peakon solution. Copyright © 2015 John Wiley & Sons, Ltd.     

Partial regularity for weak solutions of a nonlinear elliptic equation

For scalar non-linear elliptic equations, stationary solutions are defined to be critical points of a functional with respect to the variations of the domain. We consideru a weak positive solution of −Δu=u ^α in -Δu=u ^α in Ω ⊂ ℝ ⁿ , which is stationary. We prove that the Hausdorff dimension of the singular set ofu is less thann−2α+1/α−1, if α≥n+2/n−2.     

Non-Lie ansatzen and exact solutions of a nonlinear spinor equation

Using the conditional symmetry of the nonlinear Dirac equation new ansatzen are obtained for a spinor field which reduce this equation to ordinary differential equations. A new class of exact solutions of the nonlinear Dirac equation, which contains three arbitrary functions, is constructed.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 958–962, July, 1990.     

New exact Jacobian elliptic function solutions for Wick-type stochastic KdV equation

Wick-type stochastic KdV equations are researched. By the means of Hermite transformation and mapping method, many new exact Jacobi function solutions of stochastic KdV equations are derived.     

The exact analytic solutions of a nonlinear differential iterative equation

This paper is concerned with a second-order nonlinear iterated differential equation of the form _c0x^″(z)+_c1x^′(z)+_c2x(z)=x(p(z)+bx(z))+h(z)$c_0{x}^{″}\left(z\right)+c_1{x}^{\prime }\left(z\right)+c_{2}x\left(z\right)=x(p\left(z\right)+bx\left(z\right))+h\left(z\right)$. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained. We discuss not only the general case |β|≠1$\left|\beta \right|\ne 1$, but also the critical case |β|=1$\left|\beta \right|=1$, especially when β$\beta$ is a root of unity. Furthermore, the exact and explicit analytic solution of the original equation is investigated for the first time. Such equations are important in both applications and the theory of iterations.     

New exact solutions for the nonlinear wave equation with fifth-order nonlinear term

The purpose of this paper is to reveal the dynamical behavior of the nonlinear wave equation with fifth-order nonlinear term, and provides its bounded traveling wave solutions. Applying the bifurcation theory of planar dynamical systems, we depict phase portraits of the traveling wave system corresponding to this equation under various parameter conditions. Through discussing the bifurcation of phase portraits, we obtain all explicit expressions of solitary wave solutions and kink wave solutions. Further, we investigate the relation between the bounded orbit of the traveling wave system and the energy level h. By analyzing the energy level constant h, we get all possible periodic wave solutions.     

非线性偏微分方程的约化和精确解

§ 1　IntroductionSeeking the exact solutions of the nonlinear partial differential equation is one of thevery importantsubjectin PDE research.Up to now,many methods offinding the exact so-lutions for NLPDE are constructed,such as inverse scattering transformation(IST) [1 ] ,Liepoint symmetry and similar reductions[2 ,3] ,B cklund[4— 6] and Cole-Hofe transformations,Hirota s bilinear method[7] ,the homogeneous balance method[8,9] ,tanh function method[1 0 ]and so on.In this paper,we giv…     

Conditional invariance and exact solutions of the nonlinear equation

The conditional invariance of the nonlinear heat equation is studied. Conditionalinvariance operators are applied for reducing the original equation to ordinary differential equations, and also for finding its exact solutions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1370–1376, October, 1990.     

New exact solutions for some power law nonlinear diffusion equation

An extended auxiliary equation method for exact traveling wave solutions of constant coefficient nonlinear partial differential equations of evolution is proposed. This, together with a convenient characterization, affords new exact traveling wave solutions of some classes of nonlinear power law diffusion equations to be obtained.     

Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions

In this paper, existence of nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions is considered by using the monotone operator principle, Mountain pass lemma, Linking theorem and some results in Morse theory. The obtained results are also valid and new for the corresponding difference periodic boundary value problem.     

Partial regularity for solutions of a nonlinear elliptic equation with singular nonlinearity

We consider the following nonlinear elliptic equation with singular nonlinearity:

where α>β>1, a>0, and Ω is an open subset of , n2. Let uH¹(Ω) with and be a nonnegative stationary solution. If we denote the zero set of u by

we shall prove that the Hausdorff dimension of Σ is less than or equal to .     

Backlund transformations and Painleve analysis: Exact solutions for a Grad--Shafranov-type magnetohydrodynamic equilibrium

The problem of plasma equilibrium in a gravitational field isinvestigated analytically. For the two-dimensional problem,the system of ideal magnetohydrodynamic equations is reducedto a single nonlinear elliptic equation of the magnetic potentialas a Grad-Shafranov-type equation. By specifying the arbitraryfunctions in this equation, the sinh-Poisson equation can beobtained. Using the Bäcklund-transformation technique andPainlevé analysis, a set of exact solutions are obtainedwhich adequately describe force-free models for solar flaresand plane-parallel filaments of a diffuse magnetized plasmasuspended horizontally in equilibrium in a uniform gravitationalfield.     

Large solutions for quasilinear elliptic equation with nonlinear gradient term

We study the existence, asymptotic behavior near the boundary and uniqueness of large solutions for a class of quasilinear elliptic equation with a nonlinear gradient term. By constructing the suitable blow-up upper and lower solutions, we obtain the existence and the asymptotic behavior of radial large solutions of the problem in balls and then derive the existence of solutions in a general domain by a comparison argument. By using a perturbation method and constructing comparison functions, we show the exact asymptotic behavior of any nonnegative solution of it near the boundary. The uniqueness is shown by a standard argument.     

Symmetry reduction and some exact solutions of a nonlinear five-dimensional wave equation

By using decomposable subgroups of the generalized Poincaré group P(1,4), we perform a symmetry reduction of a nonlinear five-dimensional wave equation to differential equations with a smaller number of independent variables. On the basis of solutions of the reduced equations, we construct some classes of exact solutions of the equation under consideration.