TY - JOUR

T1 - The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence

AU - Schochet, Steven H.

AU - Weinstein, Michael I.

PY - 1986/12

Y1 - 1986/12

N2 - We consider the initial value problem for the Zakharov equations {Mathematical expression} (x∈ℝk, k=2, 3, t ≧0) which model the propagation of Langmuir waves in plasmas. For suitable initial data solutions are shown to exist for a time interval independent of λ, a parameter proportional to the ion acoustic speed. For such data, solutions of (Z) converge as λ → ∞ to a solution of the cubic nonlinear Schrödinger equation (CSE)iEt+ΔE+|E|2E=0. We consider both weak and strong solutions. For the case of strong solutions the results are analogous to previous results on the incompressible limit of compressible fluids.

AB - We consider the initial value problem for the Zakharov equations {Mathematical expression} (x∈ℝk, k=2, 3, t ≧0) which model the propagation of Langmuir waves in plasmas. For suitable initial data solutions are shown to exist for a time interval independent of λ, a parameter proportional to the ion acoustic speed. For such data, solutions of (Z) converge as λ → ∞ to a solution of the cubic nonlinear Schrödinger equation (CSE)iEt+ΔE+|E|2E=0. We consider both weak and strong solutions. For the case of strong solutions the results are analogous to previous results on the incompressible limit of compressible fluids.

UR - http://www.scopus.com/inward/record.url?scp=0000014949&partnerID=8YFLogxK

U2 - 10.1007/BF01463396

DO - 10.1007/BF01463396

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AN - SCOPUS:0000014949

SN - 0010-3616

VL - 106

SP - 569

EP - 580

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 4

ER -