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

B. I. Shubov

Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part VI, Tome 97 (1980), pp. 217-224

Résumé

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.