The Cauchy problem for a semi-infinite Volterra chain with an asymptotically periodic initial condition
The author studied the following problem for the Volterra chain: \[ \dot{a}_{n}=\tfrac{1}{2}a_{n}(a_{n-1}^{2}-a_{n+1}^{2}), \quad n=1,2,\dots \tag{1} \] \[ a_{n}(0)-\hat{a}\to 0, \quad\text{as}\quad n \to\infty, \tag{2} \] \[ a_{0}=0, \tag{3} \] where \(\hat{a}_{n+N}=\hat{a}_{n}>0\), \(n=1,2,\dots,N \) is a positive integer, and the dot above a symbol stands for derivative with respect to \( t\). Looking for solution \( a_{n}(t)\) such that \(a_{n}(0)-\hat{a}\) is rapidly decreasing, i.e. for every \(T>0\) \[ \bigg\|\sum^\infty_{n=1} n|a_n(t)-\hat{a}|\bigg\|_{C[0,T]}<\infty. \tag{4} \] Justified the method of inverse scattering problem and thereby the global solvability of (1--3) in (4) for the initial data with finite third moment, also provse a theorem about the absence of solutions.
scientific article; zbMATH DE number 5816249
scientific article