A Higher-Order Water-Wave Equation and the Method for Solving It

Abstract

By a new technique, we have found another nonlinear evolution equation which can be solved exactly by inverse scattering techniques. This equation has a cubic nonlinearity added to the Boussinesq equation and can also be derived from the water-wave equations. This eigenvalue problem differs from any studied before, but in some aspects it is similar to the sine-Gordon eigenvalue problem in laboratory coordinates. Also, the solution to the inverse scattering problem is given.