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The Cauchy problem for quasi-linear Schrödinger equations. (English) Zbl 1177.35221

The authors study the local in time solvability of the nonlinear Schrödinger initial value problem \[ \begin{split} \partial_t u= ia_{jk}(x, t,u,\overline u,\nabla u,\nabla\overline u)\partial^2_{x_jx_k} u+\vec b_1(x, t,u,\overline u,\nabla u,\nabla\overline u)\nabla u+\\ \vec b_2(x,t,u,\overline u,\nabla u,\nabla\overline u)\nabla\overline u+ C_1(x,t,u,\overline u)u+ C_2(x,t,u,\overline u)\overline u+ f(x,t):= L(u)u+ f(x,t),\end{split}\tag{1} \]
\[ u(x,0)= u_0(x)\tag{2} \] for \(x\in\mathbb R^n\), \(t\in\mathbb R\) and \(\nabla= (\partial_{x_1},\dots, \partial_{x_n})\), where \(\overline u\) is the conjugate complex function to \(u\).
Under the hypotheses of ellipticity, asymptotic flatness, growth of the first-order coefficients \(b_l\), regularity of equation (1) with the given functions \(u_0\), \(f\) and \(b_e= (b_{e_1},\dots, b_{e_n})\), \(l= 1,2\) and assuming that the Hamiltonian flow \(H_{h(u_0)}\) is non-trapping (for details see the reviewed paper), they prove the following main result. Theorem A: There exists a constant \(T_0> 0\) and an unique smooth solution \(u\) of the initial value problem (1), (2) on the finite time interval \([-T_0, T_0]\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
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