Class Of Dispersive Equations Research Articles

SOLITARY WAVE SOLUTIONS FOR A CLASS OF DISPERSIVE EQUATIONS

The focus of the present work is the one-dimensional nonlinear equation ut − uxxt + ux + uxxx + αuux = λ(uuxxx + 2uxuxx), (1) modeling the wave breaking phenomenon in the shallow water regime. When α, λ > 0, using a variational approximation, we show that (1) admits solitary wave solutions which propagate in the negative x−direction. AMS Subject Classification: 35Q35, 35Q51, 35A15

Symmetry analysis for a class of nonlinear dispersive equations

A class of dispersive equations is studied within the framework of group analysis of differential equations. The enhanced Lie group classification is achieved. The complete list of equivalence transformations is presented. It is shown that certain equations from the class admit nonclassical reductions. Potential and potential nonclassical symmetries are also considered.

Radial Strichartz Estimates with Application to the 2-D Dirac-Klein-Gordon System

We prove Strichartz estimates for data with spherical symmetry for a large class of dispersive equations. Our method is based on weighted space-time estimates and does not rely on the dispersive estimate. The dual inhomogeneous Strichartz estimates are derived too and it is shown that in the context of the wave and the Klein-Gordon equation the results that we obtain are essentially optimal. Furthermore, we extend the inhomogeneous estimates to the range . We also give an application to the global well-posedness of the Dirac-Klein-Gordon system in two spatial dimensions. For data that is spherically symmetric we improve the regularity assumptions of the recent result of Grünrock and Pecher [13].

Resolvent Estimates Related with a Class of Dispersive Equations

We present a simple proof of the resolvent estimates of elliptic Fourier multipliers on the Euclidean space, and apply them to the analysis of time-global and spatially-local smoothing estimates of a class of dispersive equations. For this purpose we study in detail the properties of the restriction of Fourier transform on the unit cotangent sphere associated with the symbols of multipliers.

Pointwise Estimates for Oscillatory Integrals and Related Lp–Lq Estimates II: Multidimensional Case

In this article we first establish some pointwise estimates for a class of multidimensional oscillatory integrals, and then use such estimates to establish Lp – Lq estimates for a class of dispersive equations of the form Dtu − P(Dx)u = 0, where Dt = −i∂t, Dx = −i∇x, and P(Dx) is a partial differential operator whose symbol P(ξ) (ξ ∈ Rn) is an inhomogeneous real nondegenerate elliptic polynomial. These estimates not only improve, but also extend some known results in the related topics. Tools used to obtain such results are the Van der Corput lemma, integration by parts, Young’s inequality and the interpolation theory.