On existence and uniqueness of solution of Cauchy problem for equations of discrete manydimensional chiral fields assuming their values on unit sphere

A discrete model of classical field theory defined by the action
$$ S(\varphi)=\frac12\int_{-\infty}^{\infty}dt\sum_{k\in\mathbb Z^d}\biggl(|\dot{\varphi}_k|^2-\sum_{i=1}^d|\varphi_{k+e_i}-\varphi)_k|^2\biggr) $$
and constraints $|\varphi_k|^2=1$ is considered. Here $e_i$ are the basic vectors of $d$-dimensional integer lattice $\mathbb Z^d$, the functions $\varphi_k$ assume their values in $\mathbb R^\nu$. It is proved that the Cauchy problem for the equations of motion of the model with an arbitrary initial data consistent with constraints has at least one $C^\infty$-solution. The unlquness of the solution is established under the condition of uniform boundness of $\dot{\varphi}_k(0)$. In the case $\nu=2,3,4$ the uniqueness theorem is proved without this restriction.