Integrability, conservation laws and exact solutions for a model equation under non-canonical perturbation expansions

A Hamiltonian equation produces a variety of Painlevé integrable equations: solutions of distinct physical structures

“…Olver (1984) has studied a (1 þ 1)-dimensional unidirectional model to describe the long and small amplitude waves over a shallow water. He presented a Hamiltonian model equation (Olver, 1984;Wael et al, 2022) from the second-order perturbation expansion for unidirectional waves, given as:…”

“…Hence, we have only the weak Painlevé property; the general solution is not single-valued around ϕ = 0. Based on this, our work will be performed based on this weak Painlevé property, where integrability was emphasized in Wael et al (2022).…”

“…Multiple soliton solutions, shocks, freak waves, breathers, lattice waves, propagation of nonlinear waves in plasma, and many other scientific areas have been investigated, and useful results were achieved in Leblond and Mihalache (2013), Olver (1984), Triki and Wazwaz (2017) and the references therein. Integrable evolution equations give many insights into scientific applications such as fluid flows, turbulence flows, oceanography and many others (Wael et al, 2022;Wazwaz et al, 2023aWazwaz et al, , 2023bWeiss et al, 1983;Zhou and Zhu, 2014).…”

“…However, there is one more branch, the one with the constant dominant term u 0 ¼ À 4 3a ð Þ ; this branch describes the situation when the coefficient at u xxx goes to 0 near f ¼ 0 and the expansion of u for this branch contains (Wazwaz et al, 2023a(Wazwaz et al, , 2023bKhalique, 2013;Khuri, 2005); the fractional degrees f n 2 with positive integer n. Hence, we have only the weak Painlev e property; the general solution is not single-valued around f ¼ 0. Based on this, our work will be performed based on this weak Painlev e property, where integrability was emphasized in Wael et al (2022).…”

“…Olver (1984) has studied a (1 + 1)-dimensional unidirectional model to describe the long and small amplitude waves over a shallow water. He presented a Hamiltonian model equation (Olver, 1984; Wael et al , 2022) from the second-order perturbation expansion for unidirectional waves, given as: where the small parameters α and β represent, respectively, the ratio of wave amplitude to undisturbed fluid depth and the square of the ratio of fluid depth to wave length (Olver, 1984; Wael et al , 2022), which are assumed to be of the same order of smallness. Moreover, equation (1) includes three linear terms and three nonlinear terms (Olver, 1984; Wael et al , 2022).…”