Existence and uniqueness on periodic solutions of fourth-order nonlinear differential equations

In this paper, we study the problem of periodic solutions for fourth-order nonlinear differential equations. Under proper conditions, we employ a novel proof to establish some criteria to ensure the existence and uniqueness of $T$-periodic solutions. Moreover, we give two examples to illustrate the effectiveness of our main results

Exact Solitary Wave and Periodic Wave Solutions of a Class of Higher-Order Nonlinear Wave Equations

We study the exact traveling wave solutions of a general fifth-order nonlinear wave equation and a generalized sixth-order KdV equation. We find the solvable lower-order subequations of a general related fourth-order ordinary differential equation involving only even order derivatives and polynomial functions of the dependent variable. It is shown that the exact solitary wave and periodic wave solutions of some high-order nonlinear wave equations can be obtained easily by using this algorithm. As examples, we derive some solitary wave and periodic wave solutions of the Lax equation, the Ito equation, and a general sixth-order KdV equation

Ultimate boundedness and periodicity results for a certain system Of third-order nonlinear Vector delay differential equations

In the last years, there has been increasing interest in obtaining the sufficient conditions for stability, instability, boundedness, ultimately boundedness, convergence, etc. For instance, in applied sciences some practical problems concerning mechanics, engineering technique fields, economy, control theory, physical sciences and so on are associated with third, fourth and higher order nonlinear differential equations. The problem of the boundedness and stability of solutions of vector differential equations has been widely studied by many authors, who have provided many techniques especially for delay differential equations. In this work a class of third order nonlinear non-autonomous vector delay differential equations is considered by employing the direct technique of Lyapunov as basic tool, where a complete Lyapunov functional is constructed and used to obtain sufficient conditions that guarantee existence of solutions that are periodic, uniformly asymptotically stable, uniformly ultimately bounded and the behavior of solutions at infinity. In addition to being for a more general equation, the obtained results here are new even when our equation is specialized to the forms previously studied and include many recent results in the literature. Finally, an example is given to show the feasibility of our results

Lie symmetries and exact solutions for a fourth-order nonlinear diffusion equation

In this paper, we consider a fourth-order nonlinear diffusion partial differential equation, depending on two arbitrary functions. First, we perform an analysis of the symmetry reductions for this parabolic partial differential equation by applying the Lie symmetry method. The invariance property of a partial differential equation under a Lie group of transformations yields the infinitesimal generators. By using this invariance condition, we present a complete classification of the Lie point symmetries for the different forms of the functions that the partial differential equation involves. Afterwards, the optimal systems of one-dimensional subalgebras for each maximal Lie algebra are determined, by computing previously the commutation relations, with the Lie bracket operator, and the adjoint representation. Next, the reductions to ordinary differential equations are derived from the optimal systems of one-dimensional subalgebras. Furthermore, we study travelling wave reductions depending on the form of the two arbitrary functions of the original equation. Some travelling wave solutions are obtained, such as solitons, kinks and periodic waves

On higher order fully periodic boundary value problems

In this paper we present sufficient conditions for the existence of periodic solutions of some higher order fully differential equation where the nonlinear part verifies a Nagumotype growth condition. A new type of lower and upper solutions, eventually non-ordered, allows us to obtain, not only the existence, but also some qualitative properties on the solution. The last section contains two examples to stress the application to both cases of n odd and n even

On the finite space blow up of the solutions of the Swift-Hohenberg equation

The aim of this paper is to study the finite space blow up of the solutions for a class of fourth order differential equations. Our results answer a conjecture in [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations. Arch. Ration. Mech. Anal., 207(2):717--752, 2013] and they have implications on the nonexistence of beam oscillation given by traveling wave profile at low speed propagation.Comment: 24 pages, 2 figure

Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators

Dozens of exponential integration formulas have been proposed for the high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries and Ginzburg-Landau equations. We report the results of extensive comparisons in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and higher order methods, and periodic semilinear stiff PDEs with constant coefficients. Our conclusion is that it is hard to do much better than one of the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews