Solitary waves for the delayed shallow-water wave equations

The shallow-water wave equations with different forms of delays are presented in this work, such as no delay, local delay and nonlocal delay, which are described in the form of convolutions with different kernels. These shallow-wave equations satisfy the asymptotic integrability condition and include the Korteweg–de Vries equation, Camassa–Holm equation and Degasperis–Procesi equation as particular cases. The existence and non-existence of solitary wave are established by the invariant manifold theory and geometric singular perturbation theory. It is found that different delays have various effects on the existence of solitary waves. In particular, the Melnikov functions with divergence free or not are derived for different delays to measure the separation of stable and unstable manifolds, so that the existence of solitary waves could be justified.