Mediterr. J. Math.
Springer International Publishing AG 2017
The Time-Periodic Solutions
to the Modiﬁed Zakharov Equations
with a Quantum Correction
Xiaoxiao Zheng, Yadong Shang and Huafei Di
Abstract. This paper investigates the existence and uniqueness of time-
periodic solutions of the periodic initial value problem for the modiﬁed
Zakharov equations with a quantum correction. By combining a priori
estimates with the Galerkin method and Leray–Schauder ﬁxed point
theorem, we prove that there exist a unique strong time-periodic solution
and a unique classical time-periodic solution under some conditions on
the forcing terms f and g.
Mathematics Subject Classiﬁcation. 35B10, 35G25, 35Q55.
Keywords. modiﬁed Zakharov equations, quantum correction, time-periodic
solution, a priori estimate, Galerkin method, Leray–Schauder ﬁxed point
In the laser plasma interaction, the most general useful model for the strong
Langmuir turbulence is described by Zakharov equations. These are a pair
of coupled equations with one for the envelope of the Langmuir wave of
high-frequency electric ﬁeld and with another for the ion-acoustic wave den-
sity perturbation. By linearizing Maxwell’s equations and using the ion and
electric hydrodynamical approximations, Zakharov  derived the following
model of Langmuir turbulence in plasma:
These equations can also be obtained less rigorously but in a simpler way
using the dispersion relation for Langmuir wave and the ponderomotive force
created by the high-frequency wave propagation in plasma . Zakharov
equations play important roles in the strong turbulence theory for plasma
waves. An important eﬀect in this theory is the so-called Langmuir collapse.