Two improved displacement shallow water equations and their solitary wave solutions

Abstract

Two improved displacement shallow water equation (DSWE) are constructed by applying the Hamilton variational principle under the shallow water approximation. The first one is the extended DSWE (EDSWE), which contains a higher order nonlinear term omitted in the original DSWE; and the second one is the fully nonlinear DSWE (FNDSWE) which contains all the nonlinear terms. By using the exp-function method, the EDSWE is analyzed and different types of waves, including the regular solitary wave, the loop solitary wave, the solitary wave with a sharp peak, and the periodic wave, are obtained. The exact solitary wave solution of the FNDSWE is also obtained. It is proved that under the shallow water approximation, the particle trajectory of the solitary wave is a parabolic curve.