Comparison between model equations for long waves and blow-up phenomena

Recently, to describe the unidirectional propagation of water waves, Bona et al. [7] introduced a fifth order KdV–BBM type modelequation(0.1)ηt+ηx−16ηxxt+δ1ηxxxxt+δ2ηxxxxx+34(η2)x+γ(η2)xxx−112(ηx2)x−14(η3)x=0, where η=η(x,t)η=η(x,t) is a real-valued function, and δ1>0δ1>0, δ2,γ∈Rδ2,γ∈R. In this work, we plan to compare solution of the initial value problem (IVP) associated to the fifth-order KDV–BBM type model (0.1) to that of the IVP associated to the fifth-order KdV modelequation(0.2)ut+δ3uxxxxx+c1uxuxx+c2uuxxx+c3u2ux=0,ut+δ3uxxxxx+c1uxuxx+c2uuxxx+c3u2ux=0, where u=u(x,t)u=u(x,t) is a real-valued function and δ3δ3, c1c1, c2c2 and c3c3 are real constants with δ3≠0δ3≠0. This later model (0.2) was proposed by Benney in [4] to describe the interaction of long and short waves. Also, we will study the possibility of blow-up phenomenon of the fifth-order KDV–BBM type model under certain restrictions on the coefficients.