On a Korteweg-de Vries-like equation with higher degree non-linearity

Abstract We deal with a non-linear partial differential equation which has been widely investigated owing to its applications in quantum field theory, as well as plasma and solid-state physics. It is the matter of a third order KdV-like equation with higher degree non-linearity in the coefficient of the transport term; it can be derived from a Lagrangian or an Hamiltonian density. In the current literature specific attention has been devoted to the search for traveling-wave solutions, depending upon a positive parameter v, which assesses the speed of the solitary wave. The velocity v is always assumed to be constant, as its dependence on the wave-amplitude is neglected in the mathematical model. In this context, Coffey [On series expansions giving closed-form solutions of Korteweg–de Vries-like equations, SIAM J. Appl. Math. 50 (6) (1990) 1580–1592] exploits an algebraic recursive technique to obtain these solutions in closed form for particular values of v. The aim of this paper is to extend these results by showing that closed-form solutions are achievable for every value of v: to this purpose we supply a proper mathematical framework for these issues by taking into account a suitable special function, namely an elliptic function in the sense of Weierstraß. Furthermore we obtain two classes of the so-called kink solutions, see [M.W. Coffey, On series expansions giving closed-form solutions of Korteweg–de Vries-like equations, SIAM J. Appl. Math. 50 (6) (1990) 1580–1592; B. Dey, Domain wall solutions of KdV-like equations with higher order nonlinearity, J. Phys. A 19 (1) (1986) L9–L12], as well as an exponential development of the general solution, for which we prove the convergence. Eventually we show how to implement the resulting functions by means of a symbolic manipulation program.