Time periodic solutions of nonlinear wave equations

Existence and regularity of solutions of $$(1)u_{tt} - u_{xx} = \varepsilon K(x,t,u,u_t )0< x< \pi ,0 \leqslant t \leqslant 2\pi $$ together with the periodicity and boundary conditions $$(2)u(x,t + 2\pi ) = u(x,t),u(0,t) = 0 = u(\pi ,t)$$ is studied both with an without the dissipation u_t. A solution is a pair (χ, u). A main feature of interest here is an infinite dimensional biofurcation problem. Under appropriate conditions on K, global existence results are obtained by a combination of analytical and topological methods.     

Global and periodic solutions for nonlinear wave equations with some localized nonlinear dissipation

We discuss the existence of global or periodic solutions to the nonlinear wave equation with the boundary condition , where Ω is a bounded domain in R^N,ρ(x,v) is a function like ρ(x,v)=a(x)g(v) with g′(v)?0 and β(x,u) is a source term of power nonlinearity. a(x) is assumed to be positive only in a neighborhood of a part of the boundary ∂Ω and the stability property is very delicate, which makes the problem interesting.     

New periodic wave solutions for nonlinear evolution equations with variable coefficients via mapping method

An extended mapping method with a computerized symbolic computation is used for constructing a new exact travelling wave solutions for nonlinear evolution equations arising in physics, namely, generalized Zakharov Kuznetsov equation with variable coefficients. As a result, many exact travelling wave solutions are obtained which include new periodic wave solution, trigonometric function solutions and rational solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations with variable coefficients arising in mathematical physics. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

The existence of periodic solutions of nonlinear wave equations on Sn

We are concerned with the existence of time periodic solutions of nonlinear wave equations on n-dimensional spheres. The existence of nontrivial periodic solution in case 0 is a solution is proved. the existence of multiple, especially infinitely many, time periodic solutions is established for several classes f nonlinear terms which satisfy some symmetry properties such as time translation invariance or oddness. Furhermore, this paper also studies the effect of perturbations which are not small and which destroy the symmetry, and shows how multiple solutions persist despite the nonsymmetric perturbations if the growth of nonlinear term at infinity is suitable controlled.     

Multiple periodic solutions for a class of nonlinear nonautonomous wave equations

In this paper, by means of ourZ _p index theory developed recently we study the existence of multiple periodic solutions for asymptotically linear nonautonomous wave equations. All previous known results rely either on the oddness of nonlinear terms, or on autonomous systems, and the best result for the general case is the existence of two nontrivial periodic solutions (Amann, Zehnder, K. C. Chang, S. P. Wu, S. J. Li, Z. Q. Wang). In this paper, under the assumption that the nonlinear term isT/p periodic we discuss multiple periodic solutions of nonautonomous systems and generalize a series of previous results.This research was supported in part by the National Postdoctoral Science Fund.     

Newton's method based on bifurcation for solving multiple solutions of nonlinear elliptic equations

On the basis of bifurcation theory, we use Newton's method to compute and visualize the multiple solutions to a series of typical semilinear elliptic boundary value problems with a homogeneous Dirichlet boundary condition in . We present three algorithms on the basis of the bifurcation method to solving these multiple solutions. We will compute and visualize the profiles of such multiple solutions, thereby exhibiting the geometrical effects of the domains on the multiplicity. The domains include the square, disk, symmetric or nonsymmetric annuli and dumbbell. The nonlinear partial differential equations include the Lane–Emden equation, concave–convex nonlinearities, Henon equation, and generalized Lane–Emden system. Copyright © 2013 John Wiley & Sons, Ltd.     

New periodic solutions for nonlinear evolution equations using Exp-function method

The Exp-function method is used to obtain generalized solitonary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics using symbolic computation. The method is straightforward and concise, and its applications are promising.     

Riemann theta functions periodic wave solutions and rational characteristics for the nonlinear equations

In this paper, based on a multidimensional Riemann theta function, a lucid and straightforward generalization of the Hirota-Riemann method is presented to explicitly construct multiperiodic Riemann theta functions periodic wave solutions for nonlinear equations such as the Caudrey-Dodd-Gibbon-Sawada-Kotera equation and (2+1)-dimensional breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves. The two-periodic waves are a direct generalization of one-periodic waves, their surface pattern is two-dimensional so that they have two independent spatial periods in two independent horizontal directions. A limiting procedure is presented to analyze in detail, asymptotic behavior of the multiperiodic waves and the relations between the periodic wave solutions and soliton solutions are rigorously established. This generalized Hirota-Riemann method can also be demonstrated on a class variety of nonlinear difference equations such as Toeplitz lattice equation.     

Exp-function method for traveling wave solutions of nonlinear evolution equations

In this paper, we apply the exp-function method to construct generalized solitary and periodic solutions of nonlinear evolution equations. The proposed technique is tested on the modified Zakharov-Kuznetsov (ZK) and Zakharov-Kuznetsov-Modified-Equal-Width (ZK-MEW) equations. These equations play a very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Numerical results clearly indicate the reliability and efficiency of the proposed exp-function method.     

非线性微分方程的正周期解

§ 1　 IntroductionIn[1 ] ,Saker and Agarwal studied the existence and uniqueness of positive periodicsolutions of the nonlinear differential equationN′(t) =-δ(t) N(t) + p(t) N(t) e- a N(t) ,(1 )whereδ(t) and p(t) are positive T-periodic functions.They proved that if p* >δ* ,then(1 ) has a unique T-periodic positive solution,wherep* =min0≤ t≤ Tp(t) ,δ* =max0≤ t≤ Tδ(t) .　　In view ofthe papermentioned above,whatcan be said aboutequation(1 ) when p* ≤δ* ?In this paper,we conside…     

Newton's method for nonlinear ordinary and partial differential equations

A unified approach is presented for proving the local, uniform and quadratic convergence of the approximate solutions and a-posteriori error bounds obtained by Newton's method for systems of nonlinear ordinary or partial differential equations satisfying an inverse-positive property. An important step is to show that, at each iteration, the linearized problem is inverse-positive. Many classes of problems are shown to satisfy this property. The convergence proofs depend crucially on an error bound derived previously by Rosen and the author for quasilinear elliptic, parabolic and hyperbolic problems.     

非线性发展方程新的周期解

In this paper,some new periodic solutions of nonlinear evolution equations and corresponding travelling wave solutions are obtained by using the double function method and Jacobi elliptic functions.     

Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

Multiple solutions for resonant nonlinear periodic equations

We consider a nonlinear periodic problem driven by the scalar p-Laplacian, with an asymptotically (p?1)-linear nonlinearity. We permit resonance with respect to the second positive eigenvalue of the negative periodic scalar p-Laplacian and we assume nonuniform nonresonance with respect to the first positive eigenvalue. Using a combination of variational methods, with truncation techniques and Morse theory, we show that the problem has at least three nontrivial solutions.