We investigate the L p asymptotic behavior (1 ≤ p ≤ ∞) of a perturbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equations can be strictly parabolic, or have real viscosity matrix (partially parabolic, e.g., compressible Navier-Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corresponding to shift in shock position or shape, a part which is the sum of diffusion waves, i.e., the solutions to a viscous Burger's equation, conserving the initial mass and convecting away from the shock profile in outgoing modes, and another part which is more rapidly decaying in L p .
Generalizing similar results for viscous shock and relaxation waves, we establish sharp pointwise Green function bounds and linearized and nonlinear stability for traveling wave solutions of an abstract viscous combustion model including both Majda's model and the full reacting compressible Navier-Stokes equations with artificial viscosity with general multi-species reaction and reaction-dependent equation of state, under the necessary conditions of strong spectral stability, i.e., stable point spectrum of the linearized operator about the wave, transversality of the profile as a connection in the traveling-wave ODE, and hyperbolic stability of the associated Chapman-Jouguet (square-wave) approximation. Notably, our results apply to combustion waves of any type: weak or strong, detonations or deflagrations, reducing the study of stability to verification of a readily numerically checkable Evans function condition. Together with spectral results of Lyng and Zumbrun, this gives immediately stability of small-amplitude strong detonations in the small heat-release (i.e., fluid-dynamical) limit, simplifying and greatly extending previous results obtained by energy methods by Liu-Ying and Tesei-Tan for Majda's model and the reactive Navier-Stokes equations, respectively.
Extending to systems of hyperbolic-parabolic conservation laws results of Howard and Zumbrun for strictly parabolic systems, we show for viscous shock profiles of arbitrary amplitude and type that necessary spectral (Evans function) conditions for linearized stability established by Mascia and Zumbrun are also sufficient for linearized and nonlinear phase-asymptotic stability, yielding detailed pointwise estimates and sharp rates of convergence in L p , 1 p ∞.
Refining previous work of Zumbrun, Mascia-Zumbrun, Raoofi, HowardZumbrun and Howard-Raoofi, we derive sharp pointwise bounds on behavior of perturbed viscous shock profiles for large-amplitude Lax or overcompressive type shocks and physical viscosity. These extend well-known results of Liu obtained by somewhat different techniques for small-amplitude Lax type shocks and artificial viscosity, completing a program initiated by Zumbrun and Howard. As pointed out by Liu, the key to obtaining sharp bounds is to take account of cancellation associated with the property that the solution decays faster along characteristic than in other directions. Thus, we must here estimate characteristic derivatives for the entire nonlinear perturbation, rather than judicially chosen parts as in the work of Raoofi and Howard-Raoofi, a requirement that greatly complicates the analysis.
Intertwined Hamiltonian formalism originally has its roots in quantum field theory and nonrelativistic quantum mechanics. In this work, we develop the non-relativistic two dimensional intertwined Hamiltonian formalism in classical optics. We obtain the properties of the intertwined media in detail and show that the differential part of intertwining operator is a series in Euclidean algebra generators. Also, we investigate quadratic gradient-index medium as an example of this structure, and obtain the intertwining operator and intertwined medium refractive index. Moreover, we study the preservation of quantum properties in the intertwined medium. For this, we consider superposition preservation as the most important property of quantum characters. We show that when a Schrödinger cat state is generated in gradient-index medium, we can construct another Schrödinger cat state in the intertwined one.
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