The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line

Abstract

We study the local and global wellposedness of the initial-boundary value problem for the biharmonic Schrödinger equation on the half-line with inhomogeneous Dirichlet-Neumann boundary data. First, we obtain a representation formula for the solution of the linear nonhomogenenous problem by using the Fokas method (also known as the unified transform method). We use this representation formula to prove space and time estimates on the solutions of the linear model in fractional Sobolev spaces by using Fourier analysis. Secondly, we consider the nonlinear model with a power type nonlinearity and prove the local wellposedness by means of a classical contraction argument. We obtain Strichartz estimates to treat the low regularity case by using the oscillatory integral theory directly on the representation formula provided by the Fokas method. Global wellposedness of the defocusing model is established up to cubic nonlinearities by using the multiplier technique and proving hidden trace regularities.

The Modified Helmholtz Equation on a Regular Hexagon—The Symmetric Dirichlet Problem

“…and G(k) is known and given in (18). Following the same stems used in Section 3 we derive the analogue of (28), which yields the following formula for C 2 :…”

“…Furthermore, it yields new integral representations for the solution of linear elliptic PDEs in polygonal domains [11], which in the case of simple domains can be used to obtain the analytical solution of several problems which apparently cannot be solved by the standard methods [12,13]. Recently, researchers utilised the integral representations provided by the Fokas method for the local and global wellposedness analysis of Korteweg-de Vries and nonlinear Schrödinger type PDEs [14][15][16][17][18], as well as for studying problems from control theory [19].…”

The Korteweg-de Vries equation on an interval

“…Also, for a thorough introduction to the Fokas method, we refer the reader to the monograph 19 and the review article. 26 Finally, we mentioned that recently Özsar and Yolcu 43 have used the unified transform method for proving global well-posedness results for the biharmonic nonlinear Schrödinger equation on the half-line.…”

Solved and Unsolved Problems

“…On the other hand, this space for the biharmonic NLS subject to Dirichlet-Neumann b.c. u| x=0 = g, u x | x=0 = h becomes [7]. In the two dimensional case, the spaces for boundary data turn out to be of Bourgain type [1], [5].…”

“…Researchers have used quite technical tools in order to prove these estimates for inhomogeneous initial boundary value problems even in low dimensional settings. The second author has recently shown, in connection with the biharmonic NLS [7], that the kernel of the integral formula obtained by the Fokas method representing the solution has a nice space-time structure for applying the elementary tools of harmonic analysis such as Van der Corput lemma to prove decay properties in the time variable, which eventually yields necessary Strichartz estimates. The time decay of the kernel in Fokas's integral formula for the solution of the boundary value problem can also be used to prove Strichartz estimates for a wide range of dispersive PDEs, at least in the half-space case.…”