Numerical investigations of dispersive shocks and spectral analysis for linearized quantum hydrodynamics

Abstract

The aim of this paper is to study solutions of one dimensional compressible Euler system with dissipation–dispersion terms, where the dispersive term is originated by the quantum effects described through the Bohm potential, as customary in quantum hydrodynamic models. We shall investigate numerically the sensitivity of the profiles with respect to the viscosity parameter, in particular in terms of their monotonicity properties. In addition, we shall also pinpoint numerically how the profile becomes more oscillatory as the end states approach the vacuum. The analysis of spectral properties of the linearized system around constant states is also provided, as well as the (numerical) localization of the point spectrum of the linearization along a profile. The latter investigation is carried out through the Evans function method.

Introduction

The aim of this paper is to study the following Euler system with dissipation–dispersion terms:

$$\{\begin{array}{c}{\rho }_t+m_x=0,\hfill \\ m_t+(\frac{m^2}{\rho }+{\rho }^{\gamma }){}_x=ϵ\mu m_{xx}+{ϵ}^2{k}^2\rho \left(\frac{{\left(\sqrt{\rho }\right)}_{xx}}{\sqrt{\rho }}\right){}_x,\hfill \end{array}$$

where t ≥ 0, $x\in \mathbb{R},$ $\rho =\rho (t,x)>0,$ $m=m(t,x),$ γ ≥ 1, 0 < ϵ ≪ 1, μ > 0, k > 0. The positive coefficients ϵμ and ϵ²k² stand for the viscosity and dispersive coefficients, respectively, and ρ^γ is the pressure. The particular shape of the dispersion terms is due to the Bohm potential, and the resulting system is referred to as the quantum hydrodynamics system, being used for instance in superfluidity or to model semiconductor devices. We are in particular interested in the analysis of the linearized version of (1) around special solutions, as constant states and dispersive shocks, namely solutions of (1) written as traveling waves

$$\rho =P\left(\frac{x-st}{ϵ}\right)\text{,}m=J\left(\frac{x-st}{ϵ}\right)\text{.}$$

As customary, the speed $s\in \mathbb{R}$ of the traveling wave and its limiting end states

$$\underset{y\to \pm \infty }{\lim }P\left(y\right)=P^{\pm }\text{and}\underset{y\to \pm \infty }{\lim }J\left(y\right)=J^{\pm }$$

are assumed to verify the Rankine–Hugoniot conditions:

$$J^{+}-J^{-}=s(P^{+}-P^{-}),$$

$$(\frac{J^2}{P}+P^{\gamma }){}^{+}-(\frac{J^2}{P}+P^{\gamma }){}^{-}=s(J^{+}-J^{-}).$$

The existence of such solution is studied in full details in the paper [18], where in particular the interplay between the diffusive and dispersive effects is analyzed in connections with the existence, monotonicity, and stability of such solutions. The notion and study of the effect of dispersive terms have been first considered by [13], [20], see also [12], [14], [19], [22], while a fairly complete analysis, via the Whitham modulation theory, has been investigated in [15], which also includes a wide bibliography on these topics. The first attempt to analyze the spectral theory of the linearized operator around dispersive shocks has been discussed in [16] regarding the case of p-system with real viscosity and linear capillarity, but only in the case of monotone shocks, while the mathematical theory of the quantum hydrodynamic systems has been developed in [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].

In the present paper, we shall set up the dynamical systems solved by these profiles, obtained as heteroclinic connections, and give some numerical computations of them in the next two sections. In particular we shall investigate numerically how the monotonicity properties of the profile, which appears to exist far outside the range of parameters fixed by the existence theorem proved in [18], are affected by the interplay of the viscosity and the dispersive parameters. More specifically, the profile becomes more oscillatory as the viscosity coefficient becomes smaller. In addition, we shall also underline as the profiles increases its oscillatory behavior as the end states approach the vacuum. Then, Section 4 is devoted to the study of the linearization of (1), both around a profile and around a constant state, the latter case being relevant for the subsequent analysis, which strongly relies on the linearized, constant coefficient, asymptotic operators, obtained as the parameter of the profile tends to  ± ∞. In Section 5 we treat the analysis of the spectrum of linearized operators, in particular giving a resolvent estimate for the constant coefficient case, namely, for the linearized operator about a constant state. To investigate the point spectrum of the linearized operator around a profile, one needs to supplement the resolvent estimate (available for large eigenvalues) with the study of the Evans function (see, for instance [21]); the corresponding numerical results about it are collected in Section 5.2.