Special exact soliton solutions for the K(2, 2) equation with non-zero constant pedestal

Abstract

Special exact solutions of the K(2, 2) equation, u t + ( u 2) x + ( u 2) xxx = 0, are investigated by employing the qualitative theory of differential equations. Our procedure shows that the K(2, 2) equation either has loop soliton, cusped soliton and smooth soliton solutions when sitting on the non-zero constant pedestal lim x→±∞ u = A ≠ 0, or possesses compacton solutions only when lim x→±∞ u = 0. Mathematical analysis and numerical simulations are provided for these soliton solutions of the K(2, 2) equation.