Periodic Solutions Of Nonlinear Equations Research Articles

Analytical analysis of the fourth-order Boussinesq equation by traveling wave solutions

To find periodic solutions of nonlinear equations, the Jacobi elliptic function expansion method is used. This strategy is more extensive than expanding the hyperbolic tangent series. This approach can produce periodic shock wave solutions such as solitary wave solutions.

Periodic solutions of nonlinear equations generalizing logistic equations with delay

The existence of periodic solutions of nonlinear equations generalizing logistic equations with delay is studied. The existence of sets of periodic solutions of two types is established. The stability and asymptotics of periodic solutions under changes of the parameters are studied.

Periodic Solutions of Nonlinear Equations Generalizing Logistic Equations with Delay

Исследуется вопрос о существовании периодических решений семейств нелинейных уравнений, обобщающих логистическое уравнение с запаздыванием. Установлено существование множеств периодических решений двух типов. Рассмотрены вопросы устойчивости и асимптотики периодических решений при изменении параметров. Библиография: 13 названий.

Almost Periodic Solutions of Nonlinear Equations that are not Necessarily Almost Periodic in Bochner’s Sense

We introduce a new class of almost periodic operators and establish conditions for the existence of almost periodic solutions of nonlinear equations that are not necessarily almost periodic in Bochner’s sense.

Envelope periodic solutions for a discrete network with the Jacobi elliptic functions and the alternative (G′/G)-expansion method including the generalized Riccati equation

Using the Jacobi elliptic functions and the alternative (G′/G-expansion method including the generalized Riccati equation, we derive exact soliton solutions for a discrete nonlinear electrical transmission line in (2+1) dimension. More precisely, these methods are general as they lead us to diverse solutions that have not been previously obtained for the nonlinear electrical transmission lines. This study seeks to show that it is not often necessary to transform the equation of the network into a well-known differential equation before finding its solutions. The solutions obtained by the current methods are generalized periodic solutions of nonlinear equations. The shape of solutions can be well controlled by adjusting the parameters of the network. These exact solutions may have significant applications in telecommunication systems where solitons are used to codify or for the transmission of data.

Uniformly constructing soliton solutions and periodic solutions to Burgers–Fisher equation

In this paper, we propose a new algorithm to finding all forms of soliton solutions and periodic solutions of nonlinear equations. And we successfully solve Burgers–Fisher equation. The method can also be applied to other nonlinear partial differential equations.

Existence and Regularity of Periodic Solutions of Nonlinear Equations of a Suspended String

We shall consider BVP to a nonlinear equation of suspended string with a special power density of order 1/2 to which a nonlinear time-independent outer force operates. We shall show the existence and the regularity of a family of infinitely many smooth time-periodic solutions of BVP near each normal mode. By considering our BVP in the Sobolev-type function spaces with weights at the origin, we show that under the weak Poincare-type Diophantine condition, the regularity of the solutions coincides with the differentiability of the nonlinear forcing term. The set of the periods is contained in a neighborhood of each period of normal mode, and is uncountable and dense in the interval, and has the Lebesgue measure zero.

Infinitely Many Periodic Solutions of Nonlinear Equations of Suspended Strings

We shall deal with a nonlinear equation of a suspended string with a special density to which a nonlinear time-independent external force is applied. We shall show the existence of infinitely many time-periodic solutions, near time-periodic solutions which are independent of position. The set of the periods is constructed by using algebraic numbers of degree 2 and the continued fractions.

Periodic solutions for nonlinear equations with mean curvature-like operators

We give an existence result for a periodic boundary value problem involving mean curvature-like operators in the scalar case. Following [R. Manásevich, J. Mawhin, Periodic solutions for nonlinear systems with p -Laplacian-like operators, J. Differential Equations 145 (1998), 367–393], we use an approach based on the Leray–Schauder degree.

Generalized index for Hamiltonian systems with applications

Some classes of linear Hamiltonian equations with periodic coefficients (nondegenerate, strongly stable, completely unstable) are determined by the disposition of the Floquet multipliers. Periodic solutions of nonlinear equations (e.g. elliptic or hyperbolic solutions) are also defined by the multipliers of the corresponding variational equation. In this paper, we consider a general set M of linear Hamiltonian equations with multipliers satisfying some arbitrary conditions and a specific condition on the multiplier lying at some point of the unit circle (all known sets admit such a definition). We show that the set M consists of a finite number of subsets Mi which comprise a countable number of domains within which any two Hamiltonians can be continuously deformed into each other. The corresponding integer index q is expressed through the eigenvalues of some self-adjoint problem. It is shown that this index (and, therefore, the known indices relating to specific sets) increases on increasing the Hamiltonian. Using the obtained results, some known and new sets are studied from the unified point of view. It is shown that for the sets of nondegenerate and completely unstable equations, the domains are directionally convex; for strongly stable equations, necessary and sufficient conditions for directional convexity are found. The results are applied to problems of existence and stability of periodic solutions of nonlinear Hamiltonian equations.

The new extended Jacobian elliptic function expansion algorithm and its applications in nonlinear mathematical physics equations

More recently we have presented the extended Jacobian elliptic function expansion method and its algorithm to seek more types of doubly periodic solutions. Based on the idea of the method, by studying more relations among all twelve kinds of Jacobian elliptic functions. we further extend the method to be a more general method, which is still called the extended Jacobian elliptic function expansion method for convenience. The new method is more powerful to construct more new exact doubly periodic solutions of nonlinear equations. We choose the (2+1)-dimensional dispersive long-wave system to illustrate our algorithm. As a result, twenty-four families of new doubly periodic solutions are obtained. When the modulus m→1 or 0, these doubly periodic solutions degenerate as soliton solutions and trigonometric function solutions. This algorithm can be also applied to other nonlinear equations.

Periodic solutions of nonlinear equations obtained by linear superposition

We show that a type of linear superposition principle works for several nonlinear differential equations. Using this approach, we find periodic solutions of the Kadomtsev-Petviashvili (KP) equation, the nonlinear Schrodinger (NLS) equation, the $\lambda \phi^4$ model, the sine-Gordon equation and the Boussinesq equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure for generating solutions is successful as a consequence of some powerful, recently discovered, cyclic identities satisfied by the Jacobi elliptic functions.

Periodic Solutions for Non-linear Equations of Structured Populations

We show the existence of non-trivial periodic solutions for a class of non-linear equations, model of age-structured populations. To this aim we use the theory of Centre Manifold for a class of abstract differential equations introduced by Desch and Schappacher, and show that a Hopf Bifurcation Theorem can be proved for this class of equations. Finally, we consider some special cases, and present an algorithm for the determination of the type of bifurcation, and show that in the considered cases the bifurcation is always supercritical.

Hopf Bifurcation in Gas Journal Bearings

This paper reviews a numerical investigation of the problem of small self-excited vibrations in gas journal bearings. The method of analysis is based on the Hopf bifurcation theory, in which the approximate periodic solutions of nonlinear equations of motion are computed using the Fredholm alternative. This theory enables us to construct the bifurcating periodic solutions and to determine their stability. The equations of motion of the investigated gas journal bearing have been formulated after estimating the damping and stiffness coefficients of a gas film. For this purpose, a new method of identification has been proposed.

Non-linear wave and Schrödinger equations

We investigate stability of periodic and quasiperiodic solutions of linear wave and Schrodinger equations under non-linear perturbations. We show in the case of the wave equations that such solutions are unstable for generic perturbations. For the Schrodinger equations periodic solutions are stable while the quasiperiodic ones are not. We extend these results to periodic solutions of non-linear equations.

Stability of Periodic Solutions of Nonlinear Integrodifferential Systems

We apply the two-timing perturbation procedure to problems of bifurcation of periodic solutions of nonlinear integrodifferential equations of a type arising in population dynamics. This yields also the stability of the bifurcating branch. Next a Floquet theory for linear integrodifferential equations with periodic coefficients is stated along with three theorems characterizing the stability of periodic solutions of nonlinear equations of the kind referred to above. This theory is shown to confirm the stability results of the perturbation procedure.

On the existence of periodic solutions in the nonlinear theory of oscillations of nonshallow reissner shells of revolution, accounting for decay

The problem of nonlinear oscillations of a homogeneous, isotropic nonshallow Reissner shell of revolution of constant thickness, with decay and periodicity of the application of the external load, is considered. The proof of existence of a generalized periodic solution and of convergence of the Bubnov-Galerkin method is given. The problem of existence of periodic solutions of nonlinear equations of the theory of plates and shallow shells with decay, were studied in /1,2/.