Abstract:

In this paper, we justify rigorously the long-wavelength limit for the two-fluid Euler-Poisson matrix. Firstly, under a long wave scaling, we establish the formal derivation of the Korteweg-de Vries (KdV) matrix from the two-fluid Euler-Poisson matrix by using a singular perturbation method. Then, with the aid of deep analysis of the complicated coupling structure of the two-fluid Euler-Poisson system and delicate energy estimates depending on such a structure, we prove the convergence of solutions of the Euler-Poisson system to that of the KdV matrix mathematically rigorously when $m_{i}/m_{e}\neq T_{i}/T_{e}$ in a time interval on which the KdV dynamics can be seen.

Key words: Two-fluid Euler-Poisson matrix, Long-wavelength limit, Singular perturbation method