The General Quasilinear Ultrahyperbolic Schrödinger Equation
نویسنده
چکیده
∂tu = − i ∂xj (ajk(x, t, u, ū,∇u,∇ū)∂xku) +~b1(x, t, u, ū,∇u,∇ū) · ∇u+~b2(x, t, u, ū,∇u,∇ū) · ∇ū + c1(x, t, u, ū)u+ c2(x, t, u, ū)ū+ f(x, t), where x ∈ R, t > 0, and A = (ajk(·))j,k=1,..,n is a real, symmetric matrix. Our aim is to study the existence, uniqueness and regularity of local solutions to the initial value problem (IVP) associated to the equation (1.1). In the case where A = (ajk(·))j,k=1,..,n is assumed to be elliptic the local solvability of the IVP associated to (1.1) was recently established in [20]. Hence, in this work we should be concerned with the case where (ajk(·))j,k=1,..,n is just a non-degenerate matrix. Equations of the form described in (1.1) with A = (ajk(·))j,k=1,..,n merely invertible arise in water wave problems, and as higher dimensions completely integrable models, see for example [1], [7], [8], [15], [27], and [30]. There are significant differences in the arguments required for the local solvability in the case where A is a non-degenerate matrix in comparison with the elliptic case treated in [20]. To illustrate them as well as to review some of the previous related results we consider first the semi-linear equation with constant coefficients (for more details and further references and comments see [19], [20], [21], and references therein)
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