Asymptotic integration of certain systems of linear partial differential equations (Q760583)

The authors study the question of the construction of a general asymptotic solution for a system of equations of the form \[ A(\sigma,\epsilon)L^ 2u+\epsilon C(\sigma,\epsilon)Lu+B(\sigma,\epsilon)u=p(\sigma,\epsilon)e^{i\theta (x,\epsilon)}, \] where u(x,\(\epsilon)\), p(\(\sigma\),\(\epsilon)\) are n- dimensional vector functions, A(\(\sigma\),\(\epsilon)\), B(\(\sigma\),\(\epsilon)\), C(\(\sigma\),\(\epsilon)\) are real (n\(\times n)\)- matrices, \(\theta\) (x,\(\epsilon)\) is a scalar function, \(x=(x_ 1,...,x_ m)\), \(\sigma =\epsilon^{p/q}x\), \(\epsilon \in (0,\epsilon_ 0]\) is a small parameter, p, q are relatively prime numbers, L is the operator \(L=\partial /\partial x_ 1+\partial /\partial x_ 2+...+\partial /\partial x_ m\). If \(A(\sigma,\epsilon)=\sum A_ k(\sigma)\epsilon^ k,\) \(B(\sigma,\epsilon)=\sum B_ k(\sigma)\epsilon^ k,\) they consider the case when the characteristic equation \(\det \| B_ 0(\sigma)-\omega A_ 0(\sigma)\| =0\) has multiple roots, to which correspond elementary divisors of the same multiplicities. The non-resonance case and the resonance case are discussed.