Conservation laws for Volterra chain with initial step-like condition

Abstract

In the present work we study a system of equations in the Volterra chain with initial step-like condition. The solutions to the Cauchy problem are sought in the class of positive functions. The nature of the problem is in some sense close to the problem on collapse of a discontinuity for the Korteweg-de-Vries equation. We show that the solution to the Cauchy problem for the Volterra chani can be constructed as a Taylor series. For bounded initial conditions, we obtain estimates implying that the convergence series exceeds zero. We formulate a local existence and uniqueness theorem for the solution to the Cauchy problem with bounded initial conditions. We consider a special condition of the break of the Volterra chain: $b_nb_{n+1}=1$, $n\ge N\ge2$. We provide specified estimates for solutions of the break of the chain. We prove that under the break, the solutions to the chain are defined for all positive time. We also establish two conservation laws for the broken chain. One of the laws follows the break condition, while the other is implied by the Lagrange property.