Some Remarks About The Cubic Schrodinger Equation On The Line

L. VEGA* Universidad del País Vasco Apdo. 644, 48080, Bilbao, Spain. E-mail: [email protected]

1. Introduction

We will consider the equations idtu + dlu ± \u\2u = 0 i£l teR, (1)

and more in particular the case when u(x, 0) = aSx=o + b vp -. (2)

In this note we will survey the motivations to study that particular inital value problem, the analytical questions that naturally arise trying to solve it and some recent results about it.

Also let us recall a classical result by H. Brezis and A. Friedman [1] which states that f dtu -d2u + u3 = 0 (3)

has no solution. Here the notion of solution is more restricted whether general functions or just positive are considered. Recently, J. Aguirre [2]

proved that if in (3) u(x, 0) = bvp — then there exists al least one solution for all b. As far as I know this is just an existence result and nothing has been proved about uniqueness or continuous dependence.

'Partially supported by a MECyT grant.

2. Some Physical Models

As it is well known cubic NLS is a canonical or universal model [3] which describes dispersion and nonlinear interaction. In this case the dispersion relation is quadratic and the nonlinearity can be focussing or defocussing depending on the choice of the sign in (1). In that spirit NLS appears in nonlinear optics where the temporal variable denotes a privileged direction in the physical variables. Take for example a standing wave W

¿^j W(®, y, t) - AXiWW(x, y, t) = 0, (x, y) e M2, that has y as the preferent direction of propagation. Also assume that the velocity c is a small perturbation of a constant Co > 0. Therefore we fix the ansatz

W (x,y,t) = eik(cat±y)u(x,y), to obtain that as a first approximation u has to be a solution of k2c2

Finally if we assume that the variation of u in the ¿/-variable is much smaller than k (i.e., supp u(x, •) is contained in a ball of radius small w.r.t. k) we can neglect d2u to obtain the free Schrodinger differential operators

Then the cubic term in (1) appears when one considers that the variation of the speed c around Co depends on the intensity \u\2. In this model the key dimension is two (x € R2) and explicit blow-up solutions of (1) with the positive sign can be easily given.

However our main interest comes from the geometric flow given by

Here X(s,t) denotes a curve in R3, s is the arc length parameter, c is the curvature and b the binormal.

Making the identification of the normal plane at a given point of the curve with a copy of the complex numbers at that point, we can always write b = in so that (4) becomes

Xt = icn, and therefore the relation to Schrodinger equation is clearly established. Moreover using Hasimoto transformation

we get [4] that idtV + d2V + l (|tf|2 + A(i))* = 0. (5)

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