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The understanding of astrophysical phenomena requires to deal with robust numerical tools in order to handle realistic scales in terms of energy, characteristic lengths and Mach number that cannot be easily reproduced by means of laboratory experiments. In this paper, we present the 2D numerical code HADES for the simulation of realistic astrophysical phenomena in various contexts, first taking into account radiative losses. The version of HADES including a multigroup modeling of radiative transfer will be presented in a forthcoming study. Validation of HADES is performed using several benchmark tests and some realistic applications are discussed. Optically thin radiative loss is modeled by a cooling function in the conservation law of energy. Numerical methods involve the MUSCL-Hancock finite volume scheme as well as HLLC and HLLE Riemann solvers, coupled with a second-order ODE solver by means of Strang splitting algorithm that handles source terms arising from geometrical or radiative contributions, for cartesian or axisymmetric configurations. A good agreement has been observed for all benchmark tests, either in hydrodynamic cases or in radiative cases. Furthermore, an overview of the main astrophysical studies driven with this code is proposed. First, simulations of radiative shocks in accretion columns and supernova remnant dynamics at large timescales including Vishniac instability have improved the understanding of these phenomena. Finally, astrophysical jets are investigated and the influence of the cooling effect on the jet morphology is numerically demonstrated. It is also found that periodic source enables to recover pulsating jets that mimic the structure of Herbig–Haro objects. HADES code has revealed its robustness, especially for the wall-shock test and for the so-called implosion test which turns out to be a severe one since the hydrodynamic variables are self-similar and become infinite at finite time. The simulations have proved the efficiency of HADES code and the usefulness of this tool for astrophysical applications.

In this paper, we investigate the properties of solitonic structures arising in quadratic media. First, we recall the derivation of systems governing the interaction process for waves propagating in such media and we check the local and global well-posedness of the corresponding Cauchy problem. Then, we look for stationary states in the context of normal or anomalous dispersion regimes, that lead us to either elliptic or non-elliptic systems and we address the problem of orbital stability. Finally, some numerical experiments are carried out in order to compute localized states for several regimes and to study dynamic stability as well as long-time asymptotics.