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Abstract. A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schrödinger equation (Manakov system) is developed. The allied spectral variety is a trigonal Riemann surface, which is described explicitly and the solutions of the equations are given in terms of θ-functions of the surface. The final formulae are effective in that sense that all entries like transcendental constants in exponentials, winding vectors etc. are expressed in terms of prime-form of the curve and well algorithmized operations on them. That made the result available for direct calculations in applied problems implementing the Manakov system. The simplest solutions in Jacobian ϑ-functions are given as particular case of general formulae and discussed in details.

We present a detailed bifurcation analysis for the travelling-wave solutions of the Kuramoto-Sivashinsky equation, with an emphasis on periodic solutions. The solutions are described by a I-parameter, reversible third-order ODE. In two previous papers we described new aspects in the observed bifurcations: the 'noose' bifurcation, and a novel kind of 'Shil'nikov' behaviour. This paper brings everything together, and considers the one remaining new aspect, the connected set of period-multiplying k-bifurcations. We offer a possible explanation for this set by considering a 2-parameter, reversible fourth-order ODE that contains the travelling-wave ODE in a particular limit. It is conjectured that the connected set arises from I : n resonances of the eigenvalues of a fixed paint.

It is emphasized that the dispersion relation for surface waves at a vacuum—plasma interface can be obtained by setting the reflexion coefficient to zero The relations for warm, collisional and hot, collisionless plasmas are derived in this way, and shown to be inagreement with previous results. The former relation is expressed explicitly as a cubic equation for y = 2c2k12/ω2p, as a function of x = 2ω2/ω2p − 1, where k1 is thecomponent of the wave vector along the interface, a is the frequency and w the plasma frequency; and the nature of the three branches of the y(x) curve is analyzed in detail. The rotation of the vacuum wave vector as the frequency runs from 0 to ∞ is displayed, and it is established that one and only one branch corresponds, in the collisionlesscase, to pure surface waves.

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