A simple-looking relative of the Novikov, Hirota-Satsuma and Sawada-Kotera equations

Abstract

We study the simple-looking scalar integrable equation f xxt − 3( f x f t − 1) = 0, which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).

Computation of generating symmetries

“…The integrability properties of this equation were presented in [29]. The point symmetries were found there, as well:…”

“…That is, when expanded in a suitable series of powers of the spectral parameter, the components are the standard symmetries. In [25] GSs were found for KdV, associated Camassa-Holm and sine-Gordon equations and later this result was extended for other equations [27,26,28,29]. A major advance in the research of GSs was achieved in [30].…”

“…The generating symmetries in [25,27,26,28,29] were obtained via a limit process from a superposition principle. This approach has the following disadvantages: a) it is necessary to know the superposition principle (if it exists); b) not all superposition principles allow this procedure; c) not all GSs can be found.…”

Backlund transformation of the Geng-Xue system

“…6. Appendix: proofs of the identities (35) and (37) In this section, we will give proofs of the identities (34), (35) and (37). Actually, the (34) is the direct result of the Jacobi identity.…”

A view of the peakon world through the lens of approximation theory

“…The literature surrounding the Novikov equation is not yet quite as overwhelming as for the CH and DP equations, but there is no shortage of articles about PDE-analytic questions, in addition to the one just mentioned [253,290,305,176,147,314,306,135,150,195,315,196,307,53,304,143,34,149,328,302,63,228,278,207]. Stability of peakons has been considered by several researchers [215,299,257,258,54,55], and likewise solitons [237,206,260,259,301,229,326] and integrability aspects [281,30,186,29,273]. However, the numerical analysis community has not yet jumped on the bandwagon; we are only aware of two (rather similar) studies [57,58].…”