We study the Cauchy problem for the modified KdV equation ut + uxxx + (u )x = 0, u(0) = u0 for data u0 in the space Ĥr s defined by the norm ‖u0‖Ĥr s := ‖〈ξ〉 sû0‖Lr′ ξ . Local well-posedness of this problem is established in the parameter range 2 ≥ r > 1, s ≥ 1 2 − 1 2r , so the case (s, r) = (0, 1), which is critical in view of scaling considerations, is almost reached. To show this result, we...

A set of wave equations with fractional loss operators in time and space are analyzed. The fractional Szabo equation, the power law wave equation and the causal fractional Laplacian wave equation are all found to be low-frequency approximations of the fractional Kelvin-Voigt wave equation and the more general fractional Zener wave equation. The latter two equations are based on fractional const...

In this paper, we are concerned with seeking exact solutions for fractional differential-difference equations by an extended Riccati sub-ODE method. The fractional derivative is defined in the sense of the modified Riemann-liouville derivative. By a combination of this method and a fractional complex transformation, the iterative relations from indices n to n ± 1 are established. As for applica...

In this paper, the space-time Riesz fractional partial differential equations with periodic conditions are considered. The equations are obtained from the integral partial differential equation by replacing the time derivative with a Caputo fractional derivative and the space derivative with Riesz potential. The fundamental solutions of the space Riesz fractional partial differential equation (...

In this paper, we study a change in the propagation trajectory of set autofocusing laser beams using fractional Fourier transform. Clusters displaced bounded Airy-Gaussian supplemented by phase function that deflects beam similarly to prism are considered. The shift and deviation (according carrier spatial frequencies) make it possible beams. influence parameters under consideration on properti...

We study the stochastic fractional diffusive limit of a kinetic equation involving a small parameter and perturbed by a smooth random term. Generalizing the method of perturbed test functions, under an appropriate scaling for the small parameter, and with the moment method used in the deterministic case, we show the convergence in law to a stochastic fluid limit involving a fractional Laplacian.

In this Letter we propose the concept of the evanescent Airy beam. We analyze the structure of the ideal evanescent Airy beam, which initial profile has the Airy form, while the spectral decomposition consists of only evanescent partial waves. Also, we discuss the refraction of the Airy beam through the interface and investigate the field of the transmitted evanescent Airy beam. c © 2009 Optica...

We consider dynamical systems that are described by fractional power of coordinates and momenta. The fractional powers can be considered as a convenient way to describe systems in the fractional dimension space. For the usual space the fractional systems are non-Hamiltonian. Generalized transport equation is derived from Liouville and Bogoliubov equations for fractional systems. Fractional gene...

in this paper, a technique generally known as meshless numerical scheme for solving fractional dierential equations isconsidered. we approximate the exact solution by use of radial basis function(rbf) collocation method. this techniqueplays an important role to reduce a fractional dierential equation to a system of equations. the numerical results demonstrate the accuracy and ability of this me...