We consider the cubic nonlinear fourth-order Schrödinger equation \begin{document}$ i \partial_t u - \Delta^2 + \mu \Delta = \pm |u|^2 u, \quad \geq 0 $\end{document} on $ \mathbb R^N, N\geq 5 with random initial data. prove almost sure local well-posedness below scaling critical regularity. also probabilistic small data global and scattering. Finally, we scattering a large probability for rand...

Some oscillatory criteria for fourth order difference and differential equations are generalized to arbitrary time scales.

In this paper we study the existence of infinitely many nontrivial solutions of the following problem, −∆2u = u in Ω, − ∂∆u ∂ν = f(x, u) on ∂Ω, and either ∂u ∂ν = 0 or ∆u = 0 on ∂Ω. We assume that f(x, u) is superlinear and either subcritical or a sublinear perturbation of the critical case. For the proof in the critical case we apply the concentration compactness method.

In this article we show the existence of solutions to the nonlinear difference equation xn = xn−3xn−4 xn−1(an + bnxn−2xn−3xn−4) , n ∈ N0, where the sequences (an)n∈N0 and (bn)n∈N0 , and initial the values x−j , j = 1, 4, are real numbers. Also we find the set of initial values for which solutions are undefinable when an 6= 0 and bn 6= 0 for every n ∈ N0. When these two sequences are constant, w...

Radial basis function-Pseudospectral method and Fourier Pseudospectral (FPS) method are extended for stiff nonlinear partial differential equations with a particular emphasis on the comparison of the two methods. Fourth-order Runge-Kutta scheme is applied for temporal discretization. The numerical results indicate that RBF-PS method can be more accurate than standard Fourier pseudospectral meth...

We establish the existence and multiplicity of solutions for a class of fourth-order superlinear elliptic problems under Navier conditions on the boundary. Here we do not use the Ambrosetti-Rabinowitz condition; instead we assume that the nonlinear term is a nonlinear function which is nonquadratic at infinity.

We present new results on the existence of multiple positive solutions of a fourth-order differential equation subject to nonlocal and nonlinear boundary conditions that models a particular stationary state of an elastic beam with nonlinear controllers. Our results are based on classical fixed point index theory. We improve and complement previous results in the literature. This is illustrated ...

Modulation instability conditions for the generation of femtosecond pulses represented by perturbative nonlinear Schrödinger equation with higher order dispersion and nonlinear effects is investigated. It is observed that modulation instability occurs even in the normal dispersion regime due to the presence of fourth order dispersion.