Inhomogeneous solutions of partial differential equations describing the dynamics of nonlinear states in unbounded one-dimensional spatial domains, in particular, wave propagation in dispersive and dissipative media, are considered. We study the dynamic stability of these states, both nonlinear (for translationally invariant infinite-dimensional Hamiltonian systems) and spectral (linear). Nonlinear dynamic stability is studied by constructing a Lyapunov function (functional) in Hilbert spaces. Spectral stability is studied by constructing the Evans function – an analytic function in the right complex half-plane of a spectral parameter whose zeros coincide with unstable eigenvalues. Examples from hydrodynamics and elasticity theory are demonstrated.

Financial support. The course is supported by the Ministry of Science and Higher Education of the Russian Federation (the grant to the Steklov International Mathematical Center, Agreement no. 075-15-2019-1614).

Lecturer
Il'ichev Andrej Teimurazovich

Organizations
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Steklov International Mathematical Center