Zakharov system

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

[math]\displaystyle{ \begin{align} i \partial_t^{} u + \nabla^2 u &= un\\ \Box n &= -\nabla^2 (|u|^2_{})\end{align} }[/math]

where [math]\displaystyle{ \Box }[/math] 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 .