Two new classes of exact solutions for the KdV equation via Bäcklund transformations

The Korteweg-de Vries equation which includes nonlinear and dispersive terms quadratic in the wave amplitude is considered. The exact solutions can be obtained by the AKNS class. The technique developed relies on the construction of the wave functions which are solutions of the associated AKNS system; that is, a linear eigenvalue problem in the form of a system of first order partial differential equations. The method of characteristics is used and Bäcklund transformations (BTs) are employed to generate two new solutions from the old one.