Wave propagation in nonlinear media

The behavior of nonlinear dispersive or dissipative waves is analyzed using the decomposition method.     

Wave propagation in dissipative or dispersive nonlinear media: a numerical approach

We deal with the numerical solution of a mixed problem associated with a nonlinear partial differential equation describing the asymptotic behavior of nonlinear dispersive or dissipative one-dimensional waves.     

Estimation and control with cubic nonlinearities

The nonlinearities in a dynamic system and its measurement equations are assumed to be cubic and small, i.e., all proportional to a single scalar small parameter . The optimal digital nonlinear feedback control law is carried through the first power of , taking into account the non-Gaussian character of the state conditional distribution. The optimal law involves cubic and linear terms in the state estimate, as well as higher moments of the state conditional distribution.     

Amplitude equations for SPDEs with cubic nonlinearities

For a quite general class of stochastic partial differential equations with cubic nonlinearities, we derive rigorously amplitude equations describing the essential dynamics using the natural separation of timescales near a change of stability. Typical examples are the Swift–Hohenberg equation, the Ginzburg–Landau (or Allen–Cahn) equation and some model from surface growth. We discuss the impact of degenerate noise on the dominant behaviour, and see that additive noise has the potential to stabilize the dynamics of the dominant modes. Furthermore, we discuss higher order corrections to the amplitude equation.     

Exact solutions for the fourth order nonlinear Schrodinger equations with cubic and power law nonlinearities

The tanh method and the sine–cosine method are used for solving the fourth order nonlinear Schrodinger equations with cubic and power law nonlinearities. Several exact solutions with distinct structures are formally obtained for each type of nonlinearity. The study reveals the power of the two proposed algorithms.     

On nonlinear elliptic problems with jumping nonlinearities

Summary Elliptic equations with nonlinearities, which have different derivatives at plus and minus infinity, are studied. A characterization of solvability is given by establishing the existence of nonlinear eigenvalues of a corresponding positive-homogeneous equation.     

Blow-up properties for heat equations coupled via different nonlinearities

This paper deals with heat equations coupled via exponential and power nonlinearities, subject to null Dirichlet boundary conditions. The complete and optimal classification on non-simultaneous and simultaneous blow-ups is proposed by four sufficient and necessary conditions. We find out that, in some exponent region, the blow-up properties of the solutions depend much on the choosing of initial data. Moreover, all kinds of non-simultaneous and simultaneous blow-up rates are obtained.     

Heat propagation in linear and nonlinear media with strongly nonstationary properties

We consider a first boundary-value problem in a quadrant of the plane for a second-order linear parabolic equation with two independent variables. The boundary function may increase along with the time variable. We show that the solution remains bounded at interior points when the highest-order coefficient decreases sufficiently rapidly or when the lowest order coefficients, having appropriate signs, increase sufficiently rapidly. Similar results are established for certain quasilinear nonuniformly parabolic equations. Examples are constructed showing the accuracy of the results obtained.Translated from Trudy Seminara im. I. G. Petrovskogo, No. 12, pp. 137–148, 1987.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

A continuation theorem for periodic boundary value problems with oscillatory nonlinearities

We prove a continuation theorem for the solvability of the coincidence equationLx=Nx in normed spaces. Applications are given to the periodic boundary value problem for second order ordinary differential equations. Dealing, in particular, with the periodically forced Duffing equation

Smoothing properties of nonlinear transition semigroups: case of lipschitz nonlinearities

We consider a semilinear stochastic differential equation in a Hilbert space H with a Lipschitz continuous (possibly unbounded) nonlinearity F. We prove that the associated transition semigroup {P_t, t ≥ 0}, acting on the space of bounded measurable functions from H to , transforms bounded nondifferentiable functions into Fréchet differentiable ones. Moreover we consider the associated Kolmogorov equation and we prove that it possesses a unique “strong” solution (where “strong” means limit of classical solutions) given by the semigroup {P_t, t ≥ 0} applied to the initial condition. This result is a starting point to prove existence and uniqueness of strong solutions to Hamilton - Jacobi - Bellman equations arising in control theory. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Oscillation of second-order functional differential equations with mixed nonlinearities and oscillatory potentials

In this paper, we study oscillation of second-order functional differential equations with mixed nonlinearities

where τ0, p(t)C¹[0,∞), q(t),q_i(t),e(t)C[0,∞), p(t)>0, . Without assuming that q(t), q_i(t) and e(t) are nonnegative, the results given in [Y.G. Sun, F.W. Meng, Interval criteria for oscillation of second-order differential equations with mixed nonlinearities, Appl. Math. Comput. 198 (2008) 375–381] have been extended to the aforementioned functional differential equation.     

Wave propagation in imperfectly periodic structures: A random evolution approach

After a brief review of some results pertaining to wave propagation in periodic structures and to random evolutions, we propose a random evolution model for wave propagation in a randomly disordered periodic structure. The model is investigated in the particular case of a two layer laminated composite with layers of random thicknesses. It is shown that there is at most one propagating frequency for Floquet waves of the average solution.

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Résumé Après un bref rappel de quelques résultats concernant la propagation des ondes dans les structures périodiques et les évolutions aléatoires, on propose un modèle pour la propagation des ondes dans une structure périodique avec imperfections aléatoires basé sur les évolutions aléatoires. Ce modèle est analysé dans le cas particulier d'un matériau laminé composite à deux couches d'épaisseurs variant de manière aléatoire. On démontre qu'il existe au plus une seule fréquence correspondant à la propagation d'une onde de Floquet pour la solution moyenne.

     

On the oscillatory properties of certain fourth order nonlinear difference equations

Some new criteria for the oscillation of fourth order nonlinear difference equations of the form

     

A new perturbation solution for systems with strong quadratic and cubic nonlinearities

The new perturbation algorithm combining the method of multiple scales (MS) and Lindstedt–Poincare techniques is applied to an equation with quadratic and cubic nonlinearities. Approximate analytical solutions are found using the classical MS method and the new method. Both solutions are contrasted with the direct numerical solutions of the original equation. For the case of strong nonlinearities, solutions of the new method are in good agreement with the numerical results, whereas the amplitude and frequency estimations of classical MS yield high errors. For strongly nonlinear systems, exact periods match well with the new technique while there are large discrepancies between the exact and classical MS periods. Copyright © 2009 John Wiley & Sons, Ltd.     

On methods for continuous systems with quadratic,cubic and quantic nonlinearities

Methods for study of weakly nonlinear continuous systems are discussed. The method of multiple scales is used to analyze the nonlinear response of a relief valve under combined static and dynamic loadings. We determine a second-order approximation to the response of the system for the case of primary resonance. Second, we derive a second-order nonlinear ordinary differential equation that describes the time evolution of a single-mode, the so-called single-mode discretization. Then, we use the multiple scales method to determine second-order approximate solutions of this equation, thereby obtaining the equations describe the modulations of the amplitude and phase of the response. We show that the results of the second approach are erroneous.