Symmetries and solutions of a third order equation

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.


1.
Introduction. This work corresponds to a talk given by the second author in the session SS 129: Qualitative and Quantitative Techniques for Differential Equations arising in Economics, Finance and Natural Sciences, during The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications, which took place in Madrid, Spain, from July 07 -July 11, 2014. We would like to thank the organizers of the session and conference for the opportunity to discuss our results in that nice event.
Among other results we discussed about the Lie symmetries and some solutions obtained for the equation which was introduced in [11]. Taking = 1 and the dilation (x, t) → ( 3 √ ax, t), equation (1) is reduced to In this paper we shall consider equation (1) with the restriction a = 0. According to [11], (2) was discovered by using a genetic programming. Such program was employed for deducing model equations once a solution had been given. Then, (2) was obtained when it was given the function

JÚLIO CESAR SANTOS SAMPAIO AND IGOR LEITE FREIRE
where c > 0 and x 0 ∈ R. Function (3) is an well known solution of the celebrated KdV equation. Additionally, in [11] the authors showed that equation (1) can be transformed in a linear one when = −2/3. In fact, for this value of , equation (1) can be rewritten as the following conservation law Therefore, under the change (x, t) → ( 3 √ ax, t) and introducing ρ := u 2 , the last equation is equivalent to the Airy's equation which is an approximation of the KdV equation without nonlinear effects. Furthermore, letting w = w(x, t) be a function such that u = e w , equation (1) becomes Differentiating (6) with respect to x, defining v := w x , making (x, t) → ( 3 √ ax, 3t /2) and letting = 2/3, one obtains the modified KdV equation In regard to solutions, in [11] the authors found some similarity solutions of (1). An interesting case arise at the dispersionless limit → 0. Using the similarity variable z = x 3 /54t, two solutions of (1) were obtained, given by where I a (z) is the modified Bessel function of the first kind. We would like to emphasise that these solutions where obtained when = 0 in (1). Further solutions of (1), with = 0, were also discovered in [11].
In this work we present, in section 2, the Lie point symmetries of (1) using the classical approach [1,7,9,10,12]. Then, our next step is to use the generators of point symmetries for constructing solutions of (1), which is done in section 3. Particularly, we obtain solutions of (1), which were not found in [11]. In section 4 we use differential invariants to reduce the order of certain ODEs obtained in section 3. Additionally, we employ the direct method to construct conservation laws for (1) in section 5. Consequently, this enables us to find more conserved currents for (1) than those obtained in [11]. Finally, conclusions are presented at the end.
2. Lie symmetries. Here we present a very short recall on Lie symmetries. For further details, see [1,7,9,10,12]. Let x = (x 1 , · · · , x n ) ∈ R n and u = u(x) ∈ R be, respectively, n independent and a dependent variable. The set of all jth derivatives of u is denoted by u (j) . We also assume the summation over the repeated indices and all functions in this paper are assumed to be smooth.
The operator is called total derivative and, in particular, given a function u, we have u i1···ij = D i1 · · · D ij (u). We denote by A the set of all locally analytic functions of a finite number of the variables x, u and u (j) and, given a F ∈ A, we consider the equation An operator is called Lie point symmetry generator of the equation (8) if for a certain function λ ∈ A. The operator X is the generator of a one-parameter transfor- The set of these transformations is an additive group when endowed with the composition of functions, since T ε •T δ = T ε+δ , whose identity is given by the identity map I(x, u) = (x, u).
The operator X (k) in (10) is given by In this case, we say that (x, u) → (x,ū) is a Lie point symmetry of (8) and (10) is called invariance condition. Given a differential equation, from the invariance condition we obtain an overdetermined, linear system of partial differential equations for the coefficients of the Lie point symmetry generator. The solution of such system gives the set of all Lie point symmetry generators of the equation.  (1) is a linear combination of the operators Proof. Let be a Lie point symmetry generator of (1). Using the results of Chapter 4 of [1], we can affirm that the coefficients ξ and τ in (13) do not have any dependence with respect to u. Now, define Then we have

JÚLIO CESAR SANTOS SAMPAIO AND IGOR LEITE FREIRE
Condition (10) can therefore be written as the following The coefficients of the terms u t , u x , u xxx , · · · , give System (15) − (21) can easily be solved and its solution is given by Substituting (22) into (13) we obtain (12).
are the one-parameter symmetry groups of equation (1). 3. Invariant solutions. In this section we construct invariant solutions of (1). We assume that < 2. We say that a function ϕ = ϕ(x, t) is an invariant solution of (1) with respect to the symmetry generated by X if and u = ϕ(x, t) is a solution of (1). Condition (23) can be rewritten in the characteristic form where it is assumed that the generator in (23) is given by (13). Let us firstly consider a linear combination of the generators X 1 , X 2 and X 4 , that is where c and γ are constants. From (24) we have Consequently, we can look for a solution of the form u = e γt φ(x − ct). Substituting this function into (1), we obtain We do the ansatz φ = Ae m(x−ct) , with A = 0. Substituting φ = Ae m(x−ct) into (26), we obtain the following algebraic equation Since < 2, we can write (27) as In particular, when γ = 0, the roots of (28) are 0 and ± c a(2 − ε) .
The first root corresponds to a constant solution. The last two cases lead us to the solutions which are traveling wave solutions.
In the general case, we have three different solutions u i (x, t) = e mix+(γ−mic)t , i = 1, 2, 3, each one corresponding to the different roots of the cubic equation (28). Now we consider the generator Again, from (24) we obtain dx which enables us to seek for a solution of the type u = φ(z), where z = xt − 1 3 . Then, substituting this u into (1), we have Multiplying (32) by φ and taking the expression φφ = (φφ ) − 3φ φ into account, we can write If = − 2 3 we obtain the Airy's equation where y = φφ . The solution of (34) is given by where a 0 and a 1 are constants. Therefore, for = − 2 3 we have a solution of (1) given by where k = const., H = Y and Y is given by (35).

4.
Reductions of order by differential invariants. In this section we use differential invariants to reduce the order of equations (26) and (32).
To begin with, we recall that if G is a local group of point transformations acting on M ⊆ R 2 , a differential invariant of order n of G is an invariant function of the prolonged action of the group.
The first order prolongation of (37) is They provide a set of first order differential invariants under the scaling transformation (z, φ) → (z, λφ), λ > 0. We observe that this last transformation is a point symmetry of equation (26).
From (38), we have From (39) we can obtain φ , φ and φ in terms of v and its derivatives. Next, substituting φ , φ and φ into (26) and after reckoning we obtain the following second order ODE: We now observe that equation (40) is invariant under the translation (u, v) → (u + ε, v). Then, let w = w(v) be a function such that Let w := dw dv .
Then equation (40) is reduced to the first order ODE which is an Abel differential equation of the second kind.
Remark 1. Solution (30) can now be obtained in a different way. Assume that w = 0. On one hand, (41) implies that v is a constant. On the other hand, from (42) and the fact that w = 0, we conclude that v is a solution of the algebraic equation (28). Then, using the transformations (38) and the fact that u(x, t) = e γt φ(x − ct), we arrived again at the solution (30). (42) can be found using different techniques such as those suggested in [13], even though it is not a trivial task. As example, consider the case = 2/3, γ = 0 and 3c = 2a. For this case we obtain

Remark 2. Solutions of equation
which has the solution 1 v = −cossec(u + 2B), where B is an arbitrary constant. Then, from where A is another arbitrary constant. Therefore, one has the solution of (1), provided that 3c = 2a and = 2/3.
Let us now reduce the order of equation (32). We firstly observe that it admits the scaling (z, φ) → (z, λφ), λ > 0, as a symmetry. This means that (38) are differential invariants of this equation and then, substituting (39) into (32) we arrive at 5. Conservation laws. We recall that a conservation law for an equation of the type (1) is a vanishing expression D t C 0 + D x C 1 = 0 on the solutions u = u(x, t) of (1).
The vector field C = (C 0 , C 1 ) is called conserved vector, while component C 0 is the conserved density and the component C 1 is the conserved flux. Defining from the divergence theorem we conclude that 2 Assuming that C 1 → 0 when x → ±∞, we conclude that Q (t) ≡ 0, which provides a time conserved quantity.
Let µ = µ(u) be a function such that Such function is called multiplier, see [3,4,5,6,8]. Equation (44) implies that on the solutions of (1), the vector field C = (C 0 , C 1 ) is a vanishing divergence, that is, it is a conservation law for (1).
On one hand, since δ δu is the Euler-Lagrange operator, from equation (44) we have On the other hand, after reckoning, we have Thus, substituting (46) into (45) and equating the coefficients of u 3 x and u x u xx to 0, one concludes that and Equation (47) is a consequence of (48) and it is then enough to solve this last one, which gives µ = c 1 u + c 2 u − 2 , if = −2, and µ = c 1 u + c 2 u ln u, in the case = −2.
This proves the following result.
Our next results are the conservation laws of (1) obtained via multipliers.
Theorem 5.2. Conservation laws of equation (1) are given by Proof. The proof follows from the fact that and u ln u u t + 2a 6. Conclusions. In this communication we found the Lie point symmetry generators of equation (1), given in the Theorem 2.1. From these generators, we obtained some explicit, exact solutions for the considered equation.
We would like to call attention to the fact that in [11] the authors found some solutions. In particular, they sought by solutions of the type u(x, t) = f (z), where z = x 3 /54t. A solution of this type is an invariant solution under the one-parameter group of symmetries (x, t, u) → (e ε x, e 3ε t, u), which is a solution obtained using the generator (31). However, in [11] the authors had obtained two different solutions in terms of the confluent hypergeometric function, but only to the dispersionless limit = 0. In our case, the invariance under the scaling (x, t, u) → (e ε x, e 3ε t, u) led us, when = −2/3, to the solution (36), where the function Y (z) is given by the serie (35).
On the other hand, although in [11] the authors had looked for travelling wave solutions, they only obtained implicit ones. We, however, assuming the ansatz u = e γt φ(x − ct), found two explicit solutions given by (29), as well as other three (not exactly travelling waves, of course), given by (30), where the constants m 1 , m 2 , m 3 are the roots of the equation (28). Additionally, from the results of section 4 we can obtain solutions of (1) in terms of the solutions of the Abel equation of the second kind (42) or the solutions of the second order ODE (43). Then our results complement those found in the interesting paper [11] with respect to the solutions of (1).
In regard to conservation laws, in [11] the authors established (52). Therefore, the conservation law (53) is new, even though it is restricted to the case = −2.