Rational solutions from Pad\'e approximants for the generalized Hunter-Saxton equation

Exact rational solutions of the generalized Hunter-Saxton equation are obtained using Pad\'e approximant approach for the traveling-wave and self-similarity reduction. A larger class of algebraic solutions are also obtained by extending a range of parameters within the solutions obtained by this approach.


Introduction
The generalized Hunter-Saxton (HS) equation was originally proposed in [1] as This model reproduces the original HS equation when k = 1/2, which is a model for wave propagation in a director field of nematic liquid crystals [2]. Besides, equation (1) appears in different applications, such as the study of the Einstein-Weyl space for k = −1/3 [5]. Solutions of (1) in a non-closed form were obtained in ref. [1] by a hodographic transformation, however the task of expressing these solutions in a closed form of the original variables may be complicated and indirect.
It is known that the Padé approximant usually gives a good representation of the solution near a regular point (see [3] for more details) and it has widely been used to construct approximative solutions of differential equations with already determined initial conditions [7,8]. Moreover, an exact solution was found for the Boussinesq equation in [8] for a predetermined initial condition. In this note we employ the inverse path for study the generalized HS equation. We start with a rational expression given by the Padé approximant and seek for a set parameters or initial conditions that allows the ansatz to become an exact solution. Although there is no guarantee that a given ODE has rational or polynomial solutions, this approach provides a useful tool for searching for them.
The Padé approximant consists of a ratio of two polynomials that can be expressed as Thus, in order to determine the Padé approximant of a Taylor expansion at x = 0 given by f = j=0 a j x j , we need to solve a system of L + M equations, namely, m+n=j a m q n − p j = 0, j = 0, 1, ..., L + M.
Consider now an arbitrary ordinary differential equation (ODE): and let v = v(z; S 0 × S 1 ) be the general solution of this equation, where S 1 is the Cartesian product of the sets of initial conditions. Let us assume the origin has a regular local representation, i.e. v = ∞ j=0 v j z j . Substituting in the differential equation (2), we have where n is the order of the equation. The above expression determines the coefficients v j such that Calculating the Padé approximant of this Taylor expansion, we have Let us assume now that there are subsetsŜ 0 ⊂ S 0 andŜ 1 ⊂ S 1 , such that the Padé approximant ofŜ =Ŝ 0 ×Ŝ 1 leads to an exact solution, i.e. v(z;Ŝ) = P L (z;Ŝ) This ansatz allows us to seek solutions in the rational form without a need to consider further properties of the differential equation. Substituting (3) in (2), we have where the degree Λ L,M depends of L, M and the particular form of the differential equation.
In order to determine all elements s j ∈Ŝ for which the ansatz (3) is true, we need to solve a system of algebraic equations,  (3) is determined by the choice of parametersŜ 0 and initial conditionŝ S 1 .
In the next section, we apply this approach to the traveling-wave and self-similarity reductions of the generalized Hunter-Saxton equation. Moreover, for these specific reductions, we are able to extend the rational solutions obtained by the Padé approximant method to a more general set of parameters.
The Taylor expansion of a solution to the above equation near the origin (z = 0) can be written as where u 0 and u 1 are arbitrary constants. In terms of the formalism from the previous section we have S 0 = (k, µ) and S 1 = (u 0 , u 1 ). We now apply the Padé approximant approach to the two simplest cases [1/1] and [2/2]. Let us begin with the ansatz u ≡ u(z; (k, µ, u 0 , u 1 )) = P 1 (z;Ŝ) where For this ansatz, the system (6) is given bŷ with Λ 1,1 = 1. The condition (7) reads here as We see that k = −2 and k = 0 simultaneously solve conditions (11) and (12). Thus, by substituting these particular values of parameters on (10), we obtain exact solutions: Let us now take as an ansatz the Padé approximant of type [2/2] : u ≡ u(z; (k, µ, u 0 , u 1 )) = P 2 (z;Ŝ) where It follows that for this ansatz the system (6) is given bŷ supplemented by condition (7), i. e.
Substituting the solution of system (16-18) into the ansatz (15) yields exact solutions: Notice that with the above choice of parameters the order of the Padé approximants is reduced accordingly.
Observe that solutions (13) and (14) obtained from the first ansatz are included in the above set of solutions. This is not a general case as it will be evident in the next section. These ansatzes can be worked out by an algebraic manipulation software such as Mathematica or Maple so we omit details for the system (5) and (6) in the remaining part of the text. Following similar procedure as above for the ansatz: u ≡ u(z; (k, µ, u 0 , u 1 )) = P 3 (z;Ŝ) we obtain the solutions We can see that all non-trivial solutions of the above system are expressed in terms ofŜ 1 = S 1 andŜ 0 being independent of the initial conditions, therefore, they are general solutions.
A careful study of these solutions suggests a general expression that compactly expresses the whole setŜ. Indeed direct verification confirms that all solutions are reproduced by . (20)