Backlund transformations and exact solutions for a nonlinear elliptic equation modelling isothermal magnetostatic atmosphere

The equations of magnetostatic equilibria for a plasma in agravitational field are investigated analytically. For equilibriawith an ignorable spatial coordinate, the equations reduce toa single nonlinear elliptic equation for the magnetic potentialu known as the Grad-Shafranov equation. By specifying the arbitraryfunctions in this equation, a Liouville equation is obtained.Bäcklund transformations are described and applied to obtainexact solutions for the Liouville equation modelling an isothermalmagnetostatic atmosphere, in which the current density J isproportional to the exponential of the magnetic potential andmoveover falls off exponentially with distance vertical to thebase with an e-folding distance equal to the gravitational scaleheight.     

Singular limit for a nonlinear parabolic equation with terms concentrating on the boundary

We analyze the asymptotic behavior of the attractors of a parabolic problem when some reaction and potential terms are concentrated in a neighborhood of a portion Γ of the boundary and this neighborhood shrinks to Γ as a parameter ε goes to zero. We prove that the family of attractors is upper continuous at the ε=0.     

Asymptotic stability and decay rate of solutions for a nonlinear fourth order wave equation

This paper is concerned with investigating the global asymptotic behavior of the zero solution of the initial-boundary value problem for a nonlinear fourth order wave equation. Moreover an estimate of the rate of decay of the solutions is obtained.     

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.     

Asymptotic stability and decay rate of solutions for a nonlinear wave equation with variable coefficients

This paper is concerned with investigating the global asymptotic behavior of the solution to a nonlinear wave equation with variable coefficients. Moreover an estimate of the rate of decay of the solution is obtained.     

A unique solution to a nonlinear elliptic equation

The purpose of this paper is to prove the existence of a unique, classical solution to the nonlinear elliptic partial differential equation −∇⋅(a(u(x))∇u(x))=f(x) under periodic boundary conditions, where u(x₀)=u₀ at x₀∈Ω, with Ω=TN, the N-dimensional torus, and N=2,3. The function a is assumed to be smooth, and a(u(x))>0 for , where G⊂R is a bounded interval. We prove that if the functions f and a satisfy certain conditions, then a unique classical solution u exists. The range of the solution u is a subset of a specified interval . Applications of this work include stationary heat/diffusion problems with a source/sink, where the value of the solution is known at a spatial location x₀.     

A nondegeneracy result for a nonlinear elliptic equation

Let Ω be a smooth bounded domain of with N ≥ 5. In this paper we prove, for ɛ > 0 small, the nondegeneracy of the solution of the problem

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.     

Singular solutions of Hessian fully nonlinear elliptic equations

We study Hessian fully nonlinear uniformly elliptic equations and show that the second derivatives of viscosity solutions of those equations (in 12 or more dimensions) can blow up in an interior point of the domain. We prove that the optimal interior regularity of such solutions is no more than C1+?, showing the optimality of the known interior regularity result. The same is proven for Isaacs equations. We prove the existence of non-smooth solutions to fully nonlinear Hessian uniformly elliptic equations in 11 dimensions. We study also the possible singularity of solutions of Hessian equations defined in a neighborhood of a point and prove that a homogeneous order 0<α<1 solution of a Hessian uniformly elliptic equation in a punctured ball should be radial.     

Singular limit problem of the Camassa-Holm type equation

We consider a shallow water equation of Camassa-Holm type, containing nonlinear dispersive effects as well as fourth order dissipative effects. We prove the strong convergence and establish the condition under which, as diffusion and dispersion parameters tend to zero, smooth solutions of the shallow water equation converge to the entropy solution of a scalar conservation law using methodology developed by Hwang and Tzavaras [S. Hwang, A.E. Tzavaras, Kinetic decomposition of approximate solutions to conservation laws: Applications to relaxation and diffusion-dispersion approximations, Comm. Partial Differential Equations 27 (2002) 1229-1254]. The proof relies on the kinetic formulation of conservation laws and the averaging lemma.     

Singular viscosity solutions to fully nonlinear elliptic equations

We prove the existence of a viscosity solution of a fully nonlinear elliptic equation in 24 dimensions with blowing up second derivative.     

Combining linear and fast diffusion in a nonlinear elliptic equation

In this paper we analyse an elliptic equation that combines linear and nonlinear fast diffusion with a logistic type reaction function. We prove existence and non-existence results of positive solutions using bifurcation theory and sub-supersolution method. Moreover, we apply variational methods to obtain a pair of ordered positive solutions.     

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.     

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.     

The decay rate for a fractional differential equation

We consider the fractional differential equation

     

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.