New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations

In this work, we establish new solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations. Unlike the standard Degasperis–Procesi and Camassa–Holm equations, where multi-peakon solutions arise, the modified forms cause a change in the characteristic of these solutions and change it to bell-shaped solitons, periodic, and complex solutions. The extended tanh method, the rational hyperbolic functions method, and the rational exponential functions method are employed to reveal these new solutions.