On the theory of long waves in an inclined channel (Q1078012)

A method resembling the asymptotic small-parameter method, is used to study the long steady waves in an inclined channel, with the waves degenerating into solitons as their length tends to infinity. By analogy with the theory of stability of elastic rods, the process of transition from one-dimensional steady flow to two-dimensional flow, can be represented as instantaneous, with the result that all rectilinear stream lines becomes curved, but the values of the Froude and Reynolds numbers remain the same. It is shown that solutions of this type can exist, provided that the velocity of wave propagation and the value of the Reynolds number are nearly critical. Simple formulas are obtained for the wave profile, and the dependence of the wave propagation on the amplitude. If the Reynolds number is small and the angle of inclination of the channel is nearly \(\pi\) /2, the same formulas hold even without the assumption that the Reynolds number is nearly critical. Technical difficulties arise in connection with the estimates for Green's function for the biharmonic operator.