A Note on the Two-Dimensional Ginzburg–Landau Equation for Directional, Nearly Monochromatic Waves

Abstract

In this paper we deal with the two-dimensional Ginzburg–Landauequation. First we simply expand the one-dimensional Ginzburg–Landauequation to the two-dimensional one. Then the concept of thedirectionality is imported into the two-dimensional Ginzburg–Landauequation. Directional, nearly monochromatic waves have a fixedwavenumber but spread over some propagation area in propagatingdirections. Moreover, most of the energy of waves is concentrated in asingle propagating direction. In slightly unstable, directional, nearlymonochromatic waves, the fact that the envelope surface created by theamplitude modulation is presented by the product of the solution of theSchrodinger–Nohara equation and the time function is shown. In thenonlinear case, the time function depends on space.