Functional and differential equations in three problems of nonlinear mathematical physics (Q2738429)

Suppose we have a \(2n\) dimensional Euclidean space with coordinates \(y_i\), \(\overline y_i\). Assume the existence of two commutative generators \(D= \partial/(\partial y_n)+\sum u^\mu \partial/(\partial y_\mu)\) and \(\overline D=\partial/(\partial \overline y_n)+\sum v^\nu \partial/(\partial\overline y_\nu)\) satisfying \(D\overline y=\overline D y=0\). If we demand commutativity \([D,\overline D]=0\) then we have \(Dv^\nu\equiv v^\nu_{y_n}+\sum u^\mu v^\nu_{y_\mu}=0\), \(\overline Du^\mu\equiv u^\mu_{\overline y_n}+\sum v^\nu u^\mu_{\overline y_\nu}=0\). The last system is called the \(uv\) system, and the corresponding manifold, the \(UV\) manifold. Nontrivial multidimensional solutions are constructed for this \(uv\) system. In some cases, their general solutions are presented in explicit form. The cases of complex Monge-Ampère equation and complex Bateman equation are related to \(uv\) systems and discussed in detail.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00039].