Absolute and convective instabilities of temporally oscillating flows
This paper presents a theory for the classification of linear instabilities of partial differential equations on unbounded domains with time periodic coefficients. In particular, it extends the well-developed theory of absolute and convective instabilities for equations with constant coefficients by employing ideas similar to those used by the same authors in extending the theory to partial differential equations with spatially periodic coefficients [the authors, Philos. Trans. R. Soc. Lond., Ser. A 354, No. 1710, 1027-1064 (1996; Zbl 0871.76032)]. The authors first consider the linear stability problem defined by \(u_t=u_{xx}+a(t)u_x+b(t)u+g(x,t)\), \(t>0\), \(u(x,0)=u_0(x)\), \(\lim_{x\to\pm\infty}\| u(x,t)\|=0\), \(\forall t>0\), where \(u\in\mathbb{R}^n\), \(x\in\mathbb{R}\) and \(a(t)\) and \(b(t)\) are periodic in time. An application of a Fourier transform in the spatial variable \(x\) followed by a Floquet decomposition in time, and finally a Laplace transform in time leads to a formal solution to the problem in terms of an inverse Laplace-Fourier integral and to a generalised dispersion relation for the problem in terms of the temporal Floquet exponents. Then the authors present an asymptotic analysis of wave packets and of spatially amplifying waves in a spatially homogeneous oscillating case. This results in a collision criterion for absolute instability in the former case and a causality condition for the latter in terms of the temporal Floquet exponents. They also show that the oscillatory part of the asymptotics in both cases is generally quasi-periodic in time. Finally, the authors illustrate the application of their theory with two specific examples. One is the scalar problem defined by above equation, and the other is the instability of time-periodic states of the nonlinear Schrödinger equation. In concluding the authors suggest that the theory presented in this paper is relevant to open-flow systems in fluid mechanics with time-oscillatory coefficients.
classification of linear instabilities of partial differential equations
scientific article; zbMATH DE number 1077138