On the inverse scattering problem and the low regularity solutions for the Dullin–Gottwald–Holm equation

We present an algorithm for the inverse scattering problem associated to the Dullin–Gottwald–Holm equation, arising in the study of the unidirectional propagation of waves in shallow water. In addition, a sufficient condition which guarantees the existence of the low regularity solutions for the generalized Dullin–Gottwald–Holm equation is studied by the method of energy estimation.