Bifurcation of an equilibrium point in a system of nonlinear parabolic equations with transformed argument (Q2735859)

Denoting \(u_\Delta(t,x)= u(t,x-\Delta)\), where \(\Delta\) is a transformation of the argument, one considers the quasilinear parabolic system NEWLINE\[NEWLINE\frac{\partial u}{\partial t}= D(t,\varepsilon) \frac{\partial^2u} {\partial x^2}+ A(t,\varepsilon)u+ B(t,\varepsilon) u_\Delta+ f(t,u, u_\Delta, \varepsilon) \tag{1}NEWLINE\]NEWLINE with periodicity condition NEWLINE\[NEWLINEu(t,x+2\pi)= u(t,x). \tag{2}NEWLINE\]NEWLINE This system arose in modelling of nonlinear effects in optics. One supposed, that the vector-function \(f(t,u,v, \varepsilon)\), the matrices \(D(t,\varepsilon)\), \(A(t, \varepsilon)\) and \(B(t, \varepsilon)\) are periodic with respect to \(t\) and sufficiently smooth with respect to all variables, in addition \(f(t,u,v, \varepsilon)= O(|u|^2+ |v|^2)\) as \(|u|+ |v|\to 0\). Earlier, some results had been known about the asymptotic behaviour of the autonomous equation with transformed argument. NEWLINENEWLINENEWLINEIn this paper, using representation of solution of (1), (2) in the form of complex Fourier series, the original problem is reduced to a countable system of ordinary nonlinear differential equations with parameter \(\Delta\). For this system the author investigates questions of existence of integral manifolds, central manifolds, invariant torus and its qualitative properties. In particular, one obtains estimates of solutions on invariant manifolds, exponential stability of invariant manifold and quasiperiodicity of solutions on invariant torus.