An exact, periodic solution of the Kaup-Newell equation

Abstract

An exact, periodic solution of a special derivative nonlinear Schrödinger model, the Kaup–Newell equation incorporating cubic nonlinearity, is derived. The polar, or Madelung, representation is employed. The amplitude and the phase are expressed in terms of elliptic function and elliptic integral of the third kind respectively. Insisting on strict periodicity for both the amplitude and phase yields a ‘quantized’, or ‘eigenvalue’, condition for the modulus of the elliptic functions. Plane wave and solitary pulses are recovered in the appropriate limiting regimes. For intermediate values of the modulus, numerical results are presented.