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Summary: ON THE INITIAL VALUE PROBLEM FOR THE ONE
DIMENSIONAL QUASILINEAR SCHR
ODINGER EQUATIONS
WEE KEONG LIM y AND GUSTAVO PONCE z
Abstract. We study the local in time solvability of the initial value problem of the one di-
mensional fully nonlinear Schrodinger equation. Under appropriate assumptions on the nonlinearity
(regularity and ellipticity) and on the initial data (regularity and decay at in nity), we establish the
existence and uniqueness of solutions of the initial value problem in weighted Sobolev spaces. The
equation can be reduced to its quasilinear version by taking space derivative. The desired results are
obtained by combining a change of variables, energy estimates and the arti cial viscosity method.
Key words. quasilinear Schrodinger equations, a priori estimates
AMS subject classi cations. Primary 35Q55, 35B45; Secondary 35A07, 35J10
1. Introduction. This paper is concerned with the initial value problem (IVP)
for the general quasilinear Schrodinger equation in one space dimension
8 > <
> :
@ t u = ia(u;
u; @ x u; @ x
u)@ 2
x u + ib(u; u; @ x u; @ x
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