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Variable coefficient Schrödinger flows for ultrahyperbolic operators. (English) Zbl 1088.35065
The authors consider the nonlinear Schrödinger equation \[ \begin{cases} \partial_tu=iL(x,D)u+b_1(x)\cdot\nabla_xu+ b_2(x)\cdot\nabla_x\bar{u}+c_1(x)u+c_2(x)\bar{u}+ P(u,\bar{u},\nabla_xu,\nabla_x\bar{u}),\\ u(x,0)=u_0(x),\end{cases}\tag{1} \] where \(x\in{\mathbb R}^n,\) \(t>0,\) \(L(x,D)=-\sum_{j,k=1}^n\partial_{x_j}( a_{jk}(x)\partial_{x_k}\cdot),\) \(A(x)=(a_{jk}(x))_{j,k=1,\ldots,n}\) is a real, symmetric and nondegenerate matrix, and \(P\) is a polynomial with no linear or constant terms. Equations such as (1) with \(A(x)\) only invertible (the ultrahyperbolic case) as opposed to the positive definite case, arise in connection with water wave problems, and in higher dimensions as completely integrable models.
The authors study here the existence, uniqueness and regularity of the local solutions to problem (1). Appropriate assumptions are imposed on the smoothness and the decay of the coefficients \(a_{jk},\) \(b_1,\) \(b_2,\) \(c_1\) and \(c_2\) and on the initial datum \(u_0(x),\) as well as on the asymptotic behavior of \(a_{jk}(x)\) as \(| x| \to+\infty\). In addition, it is necessary to measure the regularity of solutions in weighted Sobolev spaces of high order. Unless \(b_1\) is real and \(P\) has special symmetries, the nonlinear term incurs the so-called “loss of derivatives”, which the authors overcome by constructing an integrating factor using some nonstandard pseudodifferential symbols. By a careful study of the bicharacteristic flow associated with \(L,\) they at first establish the local smoothing effect for the linear equation (obtained by replacing the nonlinear term \(P\) by a function \(f(x,t)\)), which is then used to treat the general nonlinear equation by a fixed-point argument.

35Q55 NLS equations (nonlinear Schrödinger equations)
35S05 Pseudodifferential operators as generalizations of partial differential operators
35B45 A priori estimates in context of PDEs
35L82 Pseudohyperbolic equations
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