Existence of standing and traveling waves in quantum hydrodynamics with viscosity

Abstract

We prove existence of standing waves for two quantum hydrodynamics systems with linear and nonlinear viscosity. Moreover, global existence of traveling waves is proved for the former without restrictions on the viscosity and dispersion parameters, thanks to a suitable Lyapunov function. This is an improvement with respect to the global existence result in [20], where it was required that the viscosity is sufficiently strong.

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

“…Monotonicity. As it was previously mentioned, the existence of profiles satisfying (2.9) with end states P ± is studied in detail in [28,36]. In particular, it is well known that if the amplitude |P + − P − | is sufficiently small, then a heteroclinic solution exists (cfr.…”

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

“…Monotonicity. As it was previously mentioned, the existence of profiles satisfying (2.9) with end states P ± is studied in detail in [28,36]. In particular, it is well known that if the amplitude |P + − P − | is sufficiently small, then a heteroclinic solution exists (cf.…”

“…It has been proved in [36] that the solution J − 1 should not be considered, because the corresponding shock (P ± , J ± 1 , s) is not an admissible Lax shock for (2.1), while (P ± , J ± 2 , s) defines a Lax 2-shock for that system. As a consequence, the definition (2.8) yields…”

“…We invoke the existence theory from previous works (cf. [28,36]) and prove some additional useful features which will be needed in the stability analysis, such as monotonicity and exponential decay.…”

“…In the context of QHD models, purely dispersive shocks were first studied in [17,34] (see also [12,20,19] for later developments). Motivated by classical fluid theory, a new approach focuses on the interaction between dispersion and viscosity (see, e.g., [8,16,36]), where viscous-dispersive shocks play a preponderant role.…”