Symmetries and solutions of a third order equation

Abstract

In this paper we study a new third order evolution equation discovered a couple of years ago using a genetic programming. We show that the Lie symmetries of this equation corresponds to space and time translations, as well as a dilation on the space of independent variables and another one with respect to the depend variable. From its symmetries, explicit solutions of the equation are obtained, some of them expressed in terms of the solutions of the Airy equation and Abel equation of the second kind. Additionally, by using the direct method we establish three conservation laws for the equation, one of them new.

A family of wave equations with some remarkable properties

“…The vector fields are the evolutionary forms (10) of the point symmetries of equation (1). The first two conservation laws were first obtained in [42] and later in [37]. In this last reference it was found the third conservation law, using the direct method [5,6,7].…”

“…For instance, in [36,37] it was proved that the (finite group of) point symmetries of (1) are generated by the continuous transformations (x, t, u) → (x + s, t, u), (x, t, u) → (x, t + s, u), (x, t, u) → (x, t, e s u), and (x, t, u) → (e s x, e 3s t, u), where s is a continuous real parameter. In particular, their corresponding evolutionary generators are…”

“…In [37] it was proved that (7) is a solution of (1) provided that k 2 < 0, where k 2 is that in (6). Otherwise, if k 2 > 0, one would have an oscillatory solution, given by u(x, t) = sin c/a( − 2)(x − ct) + φ 0 , where φ 0 is an initial phase.…”

“…Additionally, if k 2 < 0, then we have exponential solutions given by (see [36,37]) u ± (x, t) = e ± √ c a(2− ) (x−ct) .…”

“…For instance, in [42] the authors found two conservation laws for (1) for arbitrary values of . Later, in [37], two of us have established a new conservation law for = −2. In the present paper we find new conservation laws for = 0 and = −2/3 obtained from the point symmetries of (1) and the results proved in [19,20,12].…”