On the stochastic Benjamin-Ono equation

We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H²(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation.     

On vortex perturbations in a free-interacting boundary layer

Large-scale structures with an inviscid, non-linear subdomain (deck) on the bottom of a boundary layer in the case of subsonic and transonic free stream velocities are considered. A class of locally inviscid perturbations with an internal line of discontinuity of the tangential velocity, which leads to the appearance of a free term on the right-hand side of the Benjamin-Ono equations, is investigated. The shape of the above-mentioned line is sought and it is determined from the solution of a system of one-dimensional non-stationary equations in which, apart from the Benjamin-Ono equation, a kinematic condition and an equation for the inviscid deck close to the wall also occur. An example of a periodic, non-linear solution is constructed and amplitude constraints which ensure its realization are formulated.     

Chaos in the beam equation

We show the existence of chaotic solutions for certain weakly damped linear beam equations with slowly periodic perturbations resting on weakly non-linear elastic bearings.     

Periodic solutions of a periodically forced and undamped beam resting on weakly elastic bearings

We study the problem of existence of periodic solutions to a partial differential equation modelling the behavior of an undamped beam subject to an external periodic force. We assume that the ordinary differential equation associated to the first two modes of vibration of the beam has a symmetric homoclinic solution. By using methods borrowed by dynamical systems theory we prove that, if the period is non resonant with the (infinitely many) internal periods of the PDE, the equation has a weak periodic solution of the same period as the external force. In particular we obtain continua of periodic solutions for the undamped beam in absence of external forces. This result may be considered as an infinite dimensional analogue of a result obtained in [16] concerning accumulation of periodic solutions to homoclinic orbits in finite dimensional reversible systems. Matteo Franca: Partially supported by G.N.A.M.P.A. – INdAM (Italy).     

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee–Infante equation is discussed.     

Exact periodic wave solutions for the hKdV equation

In this paper, by using the Hirota bilinear method and the Jacobian theta functions for the higher order KdV equation, the existence of periodic wave solutions with one and two period are obtained. The asymptotic properties of the periodic wave solutions are analyzed in detail. It is shown that the well-known soliton solutions can be reduced from the periodic wave solutions.     

Orbital stability of peakons for the Degasperis-Procesi equation with strong dispersion

In this paper, we study the orbital stability of the peakons for the Degasperis-Procesi equation with a strong dispersive term on the line. Using the method in [Z. Lin, Y. Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math. 62 (2009) 125-146], we prove that the shapes of these peakons are stable under small perturbations. Some previous results are extended.     

The complexity of wave propagation of the nonlinear Schrödinger equation with weak periodic external field

A nonlinear Schrödinger equation with external field is obtained for nonlinear optics in a non-homogeneous medium. On the basis of the Melnikov function and KAM theory, the complexity of the wave propagation of the equation is studied in the case of weak periodic external field.     

Blow-up behavior for the Klein–Gordon and other perturbed semilinear wave equations

We give blow-up results for the Klein–Gordon equation and other perturbations of the semilinear wave equations with superlinear power nonlinearity, in one space dimension or in higher dimension under radial symmetry outside the origin.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.     

Error Estimate of the Fourier Collocation Method for the Benjamin-Ono Equation

In this paper, the Fourier collocation method for solving the generalized Benjamin-Ono equation with periodic boundary conditions is analyzed. Stability of the semi-discrete scheme is proved and error estimate in H1/2-norm is obtained.     

Different physical structures of solutions for a generalized Boussinesq wave equation

A technique based on the reduction of order for solving differential equations is employed to investigate a generalized nonlinear Boussinesq wave equation. The compacton solutions, solitons, solitary pattern solutions, periodic solutions and algebraic travelling wave solutions for the equation are expressed analytically under several circumstances. The qualitative change in the physical structures of the solutions is highlighted.     

Nonlinear perturbations of a ‐Laplacian equation with critical growth in

We prove the existence of solution for a class of ‐Laplacian equations where the nonlinearity has a critical growth. Here, we consider two cases: the first case involves the situation where the variable exponents are periodic functions. The second one involves the case where the variable exponents are nonperiodic perturbations.     

Spreading speeds and traveling waves for periodic evolution systems

The theory of spreading speeds and traveling waves for monotone autonomous semiflows is extended to periodic semiflows in the monostable case. Then these abstract results are applied to a periodic system modeling man-environment-man epidemics, a periodic time-delayed and diffusive equation, and a periodic reaction-diffusion equation on a cylinder.     

Well-posedness for a higher-order Benjamin-Ono equation

In this paper we prove that the initial value problem associated to the following higher-order Benjamin-Ono equation

     

Instability of cnoidal‐peak for the NLS‐δ equation

We study the existence and stability of standing waves for the periodic cubic nonlinear Schrödinger equation with a point defect determined by the periodic Dirac distribution at the origin. We show that this model admits a smooth curve of periodic‐peak standing wave solutions with a profile determined by the Jacobi elliptic function of cnoidal type. Via a perturbation method and continuation argument, we obtain that in the repulsive defect, the cnoidal‐peak standing wave solutions are unstable in $H^1_{per}$ with respect to perturbations which have the same period as the wave itself. Global well‐posedness is verified for the Cauchy problem in $H^1_{per}$ .     

Global attractors for doubly nonlinear evolution equations with non-monotone perturbations

This paper addresses the analysis of dynamical systems generated by doubly nonlinear evolution equations governed by subdifferential operators with non-monotone perturbations in a reflexive Banach space setting. In order to construct global attractors, an approach based on the notion of generalized semiflow is employed instead of the usual semigroup approach, since solutions of the Cauchy problem for the equation might not be unique. Moreover, the preceding abstract theory is applied to a generalized Allen-Cahn equation as well as a semilinear parabolic equation with a nonlinear term involving gradients.     

Solution set splitting at low energy levels in Schrödinger equations with periodic and symmetric potential

The time-independent superlinear Schrödinger equation with spatially periodic and positive potential admits sign-changing two-bump solutions if the set of positive solutions at the minimal nontrivial energy level is the disjoint union of period translates of a compact set. Assuming a reflection symmetric potential we give a condition on the equation that ensures this splitting property for the solution set. Moreover, we provide a recipe to explicitly verify the condition, and we carry out the calculation in dimension one for a specific class of potentials.     

Multilinear estimates for periodic KdV equations, and applications

We prove an endpoint multilinear estimate for the X^s,b spaces associated to the periodic Airy equation. As a consequence we obtain sharp local well-posedness results for periodic generalized KdV equations, as well as some global well-posedness results below the energy norm.