Exact Periodic Solutions of the Nonintegrable Kawahara Equation

Abstract

In the present paper, we have obtained an exact biperiodic, one-phase solution of the Kawahara evolution equation. Two classes of real periodic waves generated by the biperiodic solution have been analyzed. A modification of the bilinear-transformation method has been applied allowing to provide a single solution of the residual equation derived from the bidifferential reduction of the considered nonintegrable equation. It is shown that the spatial displacements are individual for each separate harmonic of the real periodic solutions.

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…”