Integrability Test and Travelling-Wave Solutions of Higher-Order Shallow- Water Type Equations

The derivation of exact solutions of physicallyinteresting partial differential equations (PDEs) is a topic that has long been of interest, and there are many techniques available in order to realize this aim, for example, the use of Lie symmetries [1 – 3], the variational iteration method [4, 5], and the homotopy perturbation method [6, 7]. Also of great interest, over the last thirty years or so, has been the connection between the integrability of PDEs and analytical properties of their solutions, and in particular the Weiss-Tabor-Carnevale (WTC) Painleve test [8]. Techniques arising within this context can also be used to derive exact solutions via various so-called truncation procedures [8 – 13]. In the present paper we will be considering the application of the WTC Painleve test and truncation to the higher-order shallow-water type equations discussed in [14, 15], ut −uxxt + u(2n+1)x−u(2n+3)x +3uux −2uxuxx −uuxxx = 0, n = 1,2,3, . . . . (1)