Exact solutions of the classical Boussinesq system

Abstract

In this paper, we study exact solutions of the classical Boussinesq (CB) system, which describes propagations of shallow water waves. By using the bilinear form, with exponential expansions, we obtain solitary wave solutions of the CB system. Based on asymptotic analysis method, we study the elastic and elastic-inelastic-coupled interactions of the obtained solitary wave solutions. With extended three-wave method, we obtain the periodic solitary solution of the CB system. And with polynomial expansions, we get the rational solutions of the CB system. These interesting exact solutions may be useful in the study of some phenomena appeared in shallow water waves. ARTICLE HISTORY

Factoring Third Order Ordinary Differential Operators over Spectral Curves

“…For short, we will call L a Boussinesq operator. Boussinesq systems have been widely studied, especially their rational solutions [3,5,20]. They generate a hierarchy of integrable equations, the Boussinesq hierarchy, one of the Gelfand and Dickii integrable hierarchies of equations associated to differential operators of any order [4].…”

“…Spectral Factorization (SpF) Input: A Boussinesq operator L as in(20) and λ 0 ∈ C. Output: A list [I(L), P 0 = (λ 0 , µ 0 , γ 0 ), L 1 ], with: I(L) the defining ideal of the spectral curve Γ of L; the coordinates of a point P 0 of Γ;L 1 = ∂ + φ(P 0 ) the right gcd of {L − λ 0 , A 1 − µ 0 , A 2 − γ 0 }, for generators A 1 , A 2 of the centralizer C K (L), that verifies (32). 1 Compute A 1 and A 2 , by means of Bsq operators in (3), such that C K (L) = C[L, A 1 , A 2 ].…”

A new convolution variational boundary element technique for design sensitivity analysis and topology optimization of anisotropic thermo-poroelastic structures

“…Exact (Mohyud-Din, Irshad, Ahmed, & Khan, 2017;Sun & Chen, 2018), fractional (Khan, Ellahi, Khan, & Mohyud-Din, 2017), optimal (Sikander, Khan, Ahmed, & Mohyud-Din, 2017; and numerical (Abd-Alla, El-Naggar, Abd-Alla, Fahmy, & El-Shahat, 2008;;Fahmy, 2008;Iqbal, Mohyud-Din, & Bin-Mohsin, 2016;Mohamed, 2019;Mohyud-Din, Noor, & Noor, 2009;Mohyud-Din, Yıldırım, & Sarıaydın, 2011;Shakeel & Mohyud-Din, 2015) solutions have been implemented in various physical problems. Topology optimization is a numerical design method which optimizes material distribution in a design domain under given loads, boundary conditions and constraints.…”