Related papers: Microlocal analysis of fractional wave equations

We investigate the Cauchy problem for a semilinear spatio--temporal fractional diffusion equation with a time-dependent forcing term: \[ \partial_t^\alpha u + (-\Delta)^{\mathsf{s}} u = |u|^p + t^{\sigma}\,\mathbf{w}(x), \quad (t,x) \in…

This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence…

The existence of solutions to Cauchy type problems of linear Riemann-Liouville fractional differential equations with variable coefficients is considered in a space of integrable functions. First, we consider the existence and uniqueness of…

We consider the Cauchy problem to the 3D fractional Schr\"odinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty…

In this paper, the regularity properties of Cauchy problem for linear and nonlinear nonlocal wave equations are studied.The equation involves a convolution integral operators with a general kernel operator functions whose Fourier transform…

We study the Cauchy problem for the fractional semilinear heat equation with distributional inhomogeneous terms. By introducing the Lorentz--Morrey spaces, we overcome limitations of real interpolation in the classical local Morrey spaces…

We prove the existence of global solutions to the Cauchy problem for noncommutative nonlinear wave equations in arbitrary even spatial dimensions where the noncommutativity is only in the spatial directions. We find that for existence there…

In physics, phenomena of diffusion and wave propagation have great relevance; these physical processes are governed in the simplest cases by partial differential equations of order 1 and 2 in time, respectively. By replacing the time…

We prove global existence and modified scattering for the solutions of the Cauchy problem to the fractional Korteweg-de Vries equation with cubic nonlinearity for small, smooth and localized initial data.

We obtain necessary conditions and sufficient conditions on the existence of solutions to the Cauchy problem for a fractional semilinear heat equation with an inhomogeneous term. We identify the strongest spatial singularity of the…

In this paper, we characterize the wave front sets of solutions to fractional Schr\"{o}dinger equations \(i\partial_{t}u =(-\Delta)^{\theta/2}u + V(x)u\) with $0<\theta <2$ via the wave packet transform (short-time Fourier transform). We…

In this paper we consider the nonlinear fractional logarithmic Schr\"{o}dinger equation. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. We also prove…

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schr\"{o}dinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist…

This paper introduces fractional type evolutionary equations modeling the interaction between short waves and long waves. We consider a fractional Benney type system, which is given by a fractional Schr\"odinger equation coupled with a…

Given a Hilbert space, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with discrete non-negative spectrum acting on it. We consider the cases when the time-dependent propagation speed is regular,…

We investigate the solutions for a time dependent potential by considering two scenarios for the fractional Schr\"odinger equation. The first scenario analyzes the influence of the time dependent potential in the absence of the kinetic…

In this paper, we consider the Cauchy problem for the fractional Camassa-Holm equation which models the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. Using Kato's…

In this paper we study fractional as well as semi-local Chern-Simons vortices in G = U(1) x SO(2M) and G = U(1) x USp(2M) theories. The master equations are solved numerically using appropriate Ansatze for the moduli matrix field. In the…

We consider Scr\"odinger equations with real-valued smooth Hamiltonians, and non-smooth bounded pseudo-differential potentials, whose symbols may be not even differentiable. The well-posedness of the Cauchy problem is proved in the frame of…

In this paper, we show that all non-trivial solutions of a broad class of nonlinear dispersive equations, whose linear evolution is governed by a dispersion relation under minimal regularity assumptions, cannot remain compactly supported…