Exact periodic solutions of the sixth-order generalized Boussinesq equation

Abstract

This paper examines a class of nonlinear sixth-order generalized Boussinesq-like equations (SGBE): utt = uxx + 3(u2)xx + uxxxx + αuxxxxxx, α ∊ R, depending on the positive parameter α. Hirota's bilinear transformation method is applied to the above class of non-integrable equations and exact periodic solutions have been obtained. The results confirmed the well-known nonlinear superposition principle.

Solitary-Wave and Periodic Solutions of the Kuramoto-Velarde Dispersive Equation

“…This circumstance contrasts with the spatial shifts of the non-integrable partial differential equations ( [9], [10], [11]). …”

“…We can draw the conclusion that for α 2 = α 3 KVE is a semi-integrable equation, since both residual equations (10) and (11) have a bidifferential structure [9], [10] and [11]. We will search for the solution of the last two equations in the form [12]:…”

“…The bilinear structure of (10) and (11) [9], [10] allows us to apply the index parity principle to (13) [10], [11], thus obtaining their compact forms…”

The superposition of algebraic solitons for the modified Korteweg–de Vries equation

“…For partially integrable or even nonintegrable equations, some forms of superposition principle still exist, e.g. the regularized long wave [12], sixth order generalized Boussinesq [13] and convective fluid [14,15] equations. For wave patterns in two or more spatial dimensions, this superposition of solitary pulses still applies, but the dynamics is more complicated [16,17].…”

Periodic solutions of the non-integrable convective fluid equation

“…19 In case that the residual equation does not have a bilinear structure, then the original nonlinear equation is non-integrable. Such are the evolution equations: CFE, KS, KE (Kawahara equation), sixth-order generalized Boussinesq equation (SGBE), 22 etc.…”