Local well-posedness for higher order nonlinear dispersive systems (Q1370723)

The authors study the local well-posedness of an initial value problem for a nonlinear dispersive system of the form: \[ c\partial_tu_k+\partial _xu_k^{2j+1}+P_k(u_1,\cdots,\partial_x u_1^{2j},\cdots,\partial_x u_n^{2j})=0,\quad u_k(x,0)=u_k^0(x),\;x,t\in \mathbb{R},\tag{1} \] with \(P_k:\mathbb{C}^{n(2j+1)}\to\mathbb{C}\) a polynomial having no constant or linear terms, i.e. \[ P_k(z)=\sum_{|\alpha|\leq\rho}^Na_\alpha^k z^\alpha\text{ with }\rho\geq 2 \] for \(z=(z_1^0,\cdots,z_n^0,\cdots,z_1^{2j},\cdots,z_n^{2j})\). The system described in (1) generalizes several models, for example, the KdV, higher models in water wave problems and in elastic media. The authors prove a local well-posedness if the initial function is in a suitable function space mainly in the case \(j=1\) and obtain a decay estimate in time without smallness assumption on the initial function.