Periodic solutions of the non-integrable convective fluid equation

Abstract

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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…”

Well-posedness result for the Kuramoto–Velarde equation

“…In [1,24], the existence of the solitons is proven, while in [25], the existence of traveling wave solutions for (1.1) is analyzed. In [26], the author analyzes the existence of the periodic solution for (1.1), under appropriate assumptions on κ, ν, δ, β, γ , α. The well-posedness of the Cauchy problem for (1.1) is proven in [27], using the energy space technique and assuming κ = 0, and in [28], through a priori estimates together with an application of the Cauchy-Kovalevskaya and choosing γ = 2α.…”

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].…”