An application of the inverse scattering transform to the modified intermediate long wave equation

The modified intermediate long wave (MILW) equation is a ( 1 + 1 ) -dimensional nonlinear singular integro-differential equation that possesses soliton solutions. In an appropriate limit the MILW equation reduces to the well-known modified Korteweg-de Vries equation. In this paper we solve the initial value problem for the MILW equation through a suitable implementation of the inverse scattering transform and use of the Miura-type transformation that maps solutions of the MILW equation into solutions of a complexified version of the standard intermediate long wave (ILW) equation. The initial value used for the MILW equation is assumed to be real valued, sufficiently smooth, and decaying to zero as the absolute value of the spatial variable approaches large values. An interesting feature of the procedure we develop is that soliton solutions for the ILW and MILW equations can be derived by appropriate specializations of a master set of equations.