A class of multidimensional nonlinear Langmuir waves

Abstract

Nonlinear Langmuir waves in a plasma governed by the dimensionless equations i∂E/∂t=−∇2E+nE, ∂2n/∂t2 =∇2[n+g (‖E‖2)] are studied, where E is the complex amplitude of the high‐frequency electric field, n is the low frequency perturbation in the ion density from its constant equilibrium value, and g is a given function of ‖E‖2. General conditions for the existence or nonexistence of a class of multidimensional solitary‐wave and nonlinear periodic travelling‐wave solutions in the form E(t,x) =h(k⋅x−vt) and n (t,x) =s (k⋅x −vt) are established. The results are applied to the special cases: (i) g (‖E‖2) =‖E‖2 corresponding to the usual pondermotive force, and (ii) g (‖E‖2) =K[1−exp(−‖E‖2)], K is a positive constant, representing ion density saturation.