On the exact solutions of the intermediate long-wave equation
The long-wave notion was introduced by R. I. Joseph in 1977. It is a nonlinear dispersive wave at the interface between two fluids of different densities at rest in a long, horizontal channel in the \(x\)-direction. The pseudo-differential operator equation for the unknown function \(\eta (x,t)\) is \[ \eta_t+2 \eta\eta_x- (N_H\eta)_x+ (1/H)\eta_x=0, \] where \(H>0\) is a constant and \(N_H\) is given by \(N_H\widehat\eta (k)=(k \text{coth} H)\widehat \eta(k)\) (the circumflex sign denotes the Fourier transform). The authors, based on results by R. I. Joseph (loc. cit.), study the equation \[ (N+\gamma) \Phi= \Phi^2,\;\gamma>-1 \] for which they prove the main theorem of the paper concerning existence and properties of the solution.
scientific article; zbMATH DE number 597480