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The nonlinear frequency shift is derived in a transparent asymptotic form forintense Langmuir waves in general collisionless plasma. The formula describesboth fluid and kinetic effects simultaneously. The fluid nonlinearity isexpressed, for the first time, through the plasma dielectric function, and thekinetic nonlinearity accounts for both smooth distributions andtrapped-particle beams. Various known limiting scalings are reproduced asspecial cases. The calculation avoids differential equations and can beextended straightforwardly to other nonlinear plasma waves.

Nonlinear frequency shift of electrostatic waves in general collisionless plasma: unifying theory of fluid and kinetic nonlinearities