In mathematics, the Zakharov system is a system of non-linear partial differential equations, introduced by Vladimir Zakharov in 1972 to describe the propagation of Langmuir waves in an ionized plasma. The system consists of a complex field u and a real field n satisfying the equations
i ∂ t
u + ∇ 2 u = u n ◻ n = − ∇ 2 ( | u |
2 ) {\displaystyle {\begin{aligned}i\partial _{t}^{}u+\nabla ^{2}u&=un\\\Box n&=-\nabla ^{2}(|u|_{}^{2})\end{aligned}}}
where ◻ {\displaystyle \Box } is the d'Alembert operator.
See also Resonant interaction; the Zakharov equation describes non-linear resonant interactions.
References Zakharov, V. E. (1968). Stability of periodic waves of finite amplitude on the surface of a deep fluid. Journal of Applied Mechanics and Technical Physics, 9(2), 190-194. Zakharov, V. E. (1972), "Collapse of Langmuir waves", Soviet Journal of Experimental and Theoretical Physics, 35: 908–914, Bibcode:1972JETP...35..908Z.