Generalized Solvable Structures and First Integrals for ODEs Admitting an 𝔰𝔩(2, ℝ) Symmetry Algebra

Paola Morando; Concepción Muriel; Adrián Ruiz

doi:10.1080/14029251.2019.1591712

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Volume 26, Issue 2, March 2019, Pages 188 - 201

Generalized Solvable Structures and First Integrals for ODEs Admitting an 𝔰𝔩(2, ℝ) Symmetry Algebra

Authors

Paola Morando

DISAA, Università degli Studi di Milano, Via Celoria, 2 Milano, 20133, Italy,paola.morando@unimi.it

Concepción Muriel

Departamento de Matemáticas, Universidad de Cádiz, Puerto Real (Cádiz), 11510, Spain,concepcion.muriel@uca.es

Adrián Ruiz

Departamento de Matemáticas, Universidad de Cádiz, Puerto Real (Cádiz), 11510, Spain,adrian.ruiz@uca.es

Received 11 May 2018, Accepted 24 October 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2019.1591712 How to use a DOI?
Keywords: Generalized solvable structure; first integral; nonsolvable symmetry algebra
Abstract: The notion of solvable structure is generalized in order to exploit the presence of an 𝔰𝔩(2, ℝ) algebra of symmetries for a kth-order ordinary differential equation ℰ with k > 3. In this setting, the knowledge of a generalized solvable structure for ℰ allows us to reduce ℰ to a family of second-order linear ordinary differential equations depending on k −3 parameters. Examples of explicit integration of fourth and fifth order equations are provided in order to illustrate the procedure.
Copyright: © 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction

It is well known that the existence of a k-dimensional solvable Lie symmetry algebra for a kth-order ordinary differential equation (ODE) ensures the (local) integrability of the equation by quadratures. However, it is not difficult to provide examples of equations lacking of Lie point symmetries but such that their solutions can be computed by quadratures [4, 15, 22, 23, 27]. This fact triggered, in recent decades, a number of generalizations of the classical Lie reduction method such as hidden symmetries [1, 2], nonlocal symmetries [3, 18], λ-symmetries [20, 21], μ-symmetries [12, 13], σ-symmetries [14, 19], and solvable structures [6–8, 11, 16, 26].

In this paper we focus on the concept of solvable structure, which provides a useful tool for integrating by quadratures ODEs lacking of local symmetries or admitting a nonsolvable symmetry algebra [23]. Although, under some regularity assumptions, the local existence of a solvable structure for any given ODE is guaranteed, its explicit determination is, in general, a quite difficult task. A noteworthy simplification may be obtained by looking for solvable structures which are adapted to some admitted symmetry algebra. For example, in [9, 10] solvable structures adapted to local and nonlocal symmetry algebras are considered, while in [24] a solvable structure for any third-order ODE admitting a nonsolvable symmetry algebra isomorphic to 𝔰𝔩(2, ℝ) is constructed exploiting the symmetry generators. In this paper we address a generalization of the results of [24], considering ODEs of arbitrary order k > 3 admitting a nonsolvable symmetry algebra isomorphic to 𝔰𝔩(2, ℝ). For this class of equations we introduce the notion of generalized solvable structure and we prove that the knowledge of a generalized solvable structure allows the construction of a complete set of first integrals for the equation, given in terms of two independent solutions to a corresponding second-order linear ODE. As a consequence, the general solution to the equation can be expressed in parametric form in terms of a fundamental set of solutions to a (k − 3)-parameter family of second-order linear equations. It is important to remark that, despite the symmetry algebra is nonsolvable, the first integrals can be computed by quadratures. Moreover, the generators of the symmetry algebra 𝔰𝔩(2, ℝ) and the vector fields belonging to the generalized solvable structure do not form a (standard) solvable structure.

The paper is organized as follows. We start Section 2 briefly recalling the main definitions and facts about distributions of vector fields and their symmetries and the concept of solvable structure, as well as its role in the integration by quadratures of ODEs. Afterwards we define generalized solvable structures for kth-order ODEs admitting an 𝔰𝔩(2, ℝ) symmetry algebra and we prove a result which allows us to exploit the vector fields belonging to the generalized solvable structure for computing by quadratures k − 3 functionally independent first integrals shared by the equation and the symmetry generators of 𝔰𝔩(2, ℝ). In particular, it is proved that the restriction of the ODE to the generic leaf defined by these first integrals provides a third-order equation which inherits 𝔰𝔩(2, ℝ) as Lie symmetry algebra. Since the classical Lie reduction of order for this kind of equations ends with a Riccati-type equation that does not permit to recover a closed form expression for the general solution to the third-order ODE [5, 17]. We exploit the recent results on the first integrals and general solutions to this class of equations [24, 25], in order to complete the set of first integrals. In particular we obtain the remaining three functionally independent first integrals in terms of the solutions to a related second-order ODE. The complete set of k independent first integrals can be used to provide the general solution to the original kth-order equation, expressed, in explicit or parametric form, in terms of a fundamental set of solutions to a (k −3)-parameter family of second-order linear equations.

In Section 3 our results are applied to fourth and fifth order ODEs, whose symmetry algebra is three-dimensional and isomorphic to 𝔰𝔩(2, ℝ). Since the symmetry algebra is nonsolvable and its dimension is strictly lower than the order of the ODE, the integrability by quadratures cannot be guaranteed by the classical Lie theory. For each one of the two examples of fourth-order equations, the construction of a generalized solvable structure requires the determination of a single vector field, and allows us to provide the general solutions to the ODEs (in explicit and parametric form, respectively) in terms of the solutions to a one-parameter family of second-order linear equations. For the fifth-order ODE presented in Section 3.3, the generalized solvable structure is formed by two vector fields and the general solution to the equation is expressed in parametric form in terms of the solutions to a two-parameter family of Schrödinger-type equations.

2. Solvable Structures and Their Generalizations

2.1. Distributions of vector fields and their symmetries

Given a set of vector fields {A₁,...,A_n−k} defined on an n-dimensional manifold N, we denote by 𝒜 := 〈A₁,...,A_n−k〉 the distribution generated by {A₁,...,A_n−k}. Moreover, given a set of one-forms {β₁,...,β_k}, we denote by 〈β₁,...,β_k〉 the corresponding Pfaffian system (i.e. the sub-module over C^∞(N)) generated by {β₁,...,β_k}.

The distribution 𝒜 is integrable (in Frobenius sense) if [A, B] ∈ 𝒜, for any A, B ∈ 𝒜. If U is an open domain of N where the vector fields {A₁,...,A_n−k} are pointwise linearly independent, we say that 𝒜 is a distribution of maximal rank n − k (or of codimension k) on U. It is well known that any integrable distribution 𝒜 of maximal rank on U ⊆ N determines a (n − k)-dimensional foliation of U. If this foliation is described through the vanishing of k functions of the form I_h − c_h, where I_h ∈ 𝒞^∞(U) and c_h ∈ ℝ, we can chose 〈dI₁,...,dI_k〉 as generators for the Pfaffian system annihilating the distribution 𝒜. A submanifold S ⊂ N is an integral manifold for 𝒜 if 𝒜|_S ⊆ TS. If in particular 𝒜|_S = TS we say that S is a maximal integral manifold for 𝒜.

Given a distribution 𝒜, a vector field X is a symmetry of 𝒜 if [X, A] ∈ 𝒜, for any A ∈ 𝒜. Let 𝒜 and ℬ be two distributions on N. We say that 𝒜 and ℬ are transversal at p ∈ N if they do not vanish at p and 𝒜 (p) ∩ ℬ(p) = {0}. Analogously, 𝒜 and ℬ are transversal in U if they are transversal at any point of U. An algebra 𝒢 of symmetries for a distribution 𝒜 is nontrivial if 𝒢 generates a distribution which is transversal to 𝒜.

2.2. Solvable structures for ODEs

It is well known that, given a (n − k)-dimensional integrable distribution 𝒜 on an n-dimensional manifold N, the knowledge of a solvable k-dimensional algebra 𝒢 of nontrivial symmetries for 𝒜 guarantees that a complete set of first integrals for 𝒜 can be found by quadratures. Solvable structures provide an extension of this classical result, significantly enlarging the class of vector fields which can be used to integrate by quadratures a distribution of vector fields.

In this section we recall some basic definitions and facts on solvable structures. The interested reader is referred to [6, 8, 16, 26] for further details.

Definition 2.1.

Let 𝒜 be a (n − k)-dimensional distribution on an n-dimensional manifold N. A set of vector fields {Y₁,...,Y_k} is a solvable structure for 𝒜 in an open domain U ⊆ N if, denoting by 𝒜₀ = 𝒜 and 𝒜_h = 𝒜 ⊕ 〈Y₁,...,Y_h〉 (h ≤ k), the following conditions hold:

(1)
The distribution 〈Y₁, Y₂,...,Y_h〉 has maximal rank h and is transversal to 𝒜 in U, for any h ≤ k;
(2)
𝒜_h has maximal rank (n − k + h) in U;
(3)
ℒ_{Y_h}𝒜_h−1 ⊆ 𝒜_h−1, for 1 ≤ h ≤ k.

Theorem 2.1.

Let 𝒜 = 〈A₁,...,A_n−k〉 be an integrable (n − k)-dimensional distribution defined on an orientable n-dimensional manifold N and let {Y₁,...,Y_k} be a solvable structure for 𝒜. Denoting by Ω a volume form on N and by α the k-form A₁⌋ ...⌋A_n−k⌋Ω, the distribution 𝒜 can be described as the annihilator of the Pfaffian system generated by

ωi=1Δ(Y1⌋…⌋Y^i⌋…⌋Yk⌋α), (i=1,…,k)(2.1)

where the hat denotes omission of the corresponding vector field and Δ is the function on N defined by

Δ=Y1⌋Y2⌋…⌋Yk⌋α.

Moreover, the forms ω_i satisfy

dωk=0,dωi=0 mod{ωi+1,…,ωk}

for i ∈ {1,...,k − 1}. Therefore the integral manifolds of the distribution 𝒜 can be described in implicit form as the level manifolds I₁ = c₁, I₂ = c₂, ..., I_k = c_k, c_i ∈ ℝ, where

ωk=dIk, ωk−1|{Ik=ck}=dIk−1,…,ω1|{Ik=ck,Ik−1=ck−1,…,I2=c2}=dI1.

Proof.

The interested reader is referred to the original papers [6–8, 16, 26] for a proof of this theorem.

We remark that, if the distribution 𝒜 admits an Abelian Lie algebra of symmetries generated by the vector fields Y₁,...,Y_k, then all the 1-forms ω_i are closed, i.e. the function 1/Δ provides an integrating factor for all the 1-forms (Y₁⌋... ⌋Ŷ_i⌋...⌋Y_k⌋α), for 1 ≤ i ≤ k.

The main difference between a solvable structure and a solvable symmetry algebra for a completely integrable distribution 𝒜 is that the fields belonging to a solvable structure do not need to be symmetries of 𝒜. This, of course, gives more freedom in the choice of the vector fields which can be exploited to find integral manifolds of 𝒜 by quadratures.

Let (x, u^(k−1)) = (x, u, u₁,...,u_k−1) denote the coordinates of the jet space J^k−1(ℝ, ℝ), for k ≥ 2. In order to apply Theorem 2.1 to the integration of ODEs, we recall that, with any kth-order ODE in normal form

uk=F(x,u(k−1)),(2.2)

we can associate the vector field A defined on a suitable domain U ⊂ J^k−1(ℝ, ℝ) and given by

A=∂x+u1+∂u+⋯+F(x,u(k−1))∂uk−1.(2.3)

Definition 2.2.

If ℰ is a kth-order ODE of the form (2.2) and A is the corresponding vector field given by (2.3), the vector fields {Y₁, Y₂,...,Y_k} defined on a domain U ⊂ J^k−1(ℝ, ℝ) are a solvable structure for ℰ if they are a solvable structure for the one-dimensional distribution 𝒜 = 〈A〉.

Corollary 2.1.

The knowledge of a k-dimensional solvable structure for a kth-order ODE, allows us to obtain the solution to the ODE by quadratures.

Proof.

The distribution 𝒜 is obviously integrable, being one-dimensional. Moreover the functions I_h of Theorem 2.1 provide a complete set of first integrals for A.

2.3. Generalized solvable structures for ODEs

In this section we generalize the notion of solvable structure for ODEs, in order to exploit the knowledge of a symmetry algebra isomorphic to 𝔰𝔩(2, ℝ) for kth-order ODEs with k > 3. In particular we give the following

Definition 2.3.

Let ℰ be a kth-order ODE of the form (2.2) with k > 3 and let A be the corresponding vector field defined by (2.3). If ℰ admits a symmetry algebra isomorphic to 𝔰𝔩(2, ℝ) with generators {X₁, X₂, X₃}, we call generalized solvable structure for ℰ a set of vector fields {Y₁,...,Y_k−3} which form a standard solvable structure for the integrable distribution 𝒜 = 〈A, X₁, X₂, X₃〉.

We remark that the k vector fields {X₁, X₂, X₃, Y₁,...,Y_k−3} do not form a solvable structure for the distribution generated by A, due to the commutation relations between X_i. Despite this fact we have the following result.

Theorem 2.2.

Let ℰ be a kth-order ODE with k > 3 admitting a symmetry algebra isomorphic to 𝔰𝔩(2, ℝ) and let A be the corresponding vector field defined by (2.3). The knowledge of a generalized solvable structure {Y₁,...,Y_k−3} for ℰ allows us to construct k first integrals for the ODE (2.2) in terms of two independent solutions to a linear second order ODE.

Proof.

Since {Y₁,...,Y_k−3} provide a solvable structure for the integrable distribution 𝒜 = 〈A, X₁, X₂, X₃〉, we can use Theorem 2.1 in order to find suitable functions {I₁,...,I_k−3} such that Σ := {I_i = c_i}_i_=1,...,k−3 is an integrable manifold for 𝒜. Since in this setting the manifold N = J^k−1(ℝ, ℝ) has dimension (k + 1), we have that the dimension of Σ is k + 1 − (k − 3) = 4. Moreover, since the vector fields A, X₁, X₂, X₃ are tangent to Σ, equation (2.2) restricted to Σ turns out to be a third-order ODE admitting 𝔰𝔩(2, ℝ) as symmetry algebra. Therefore, we can exploit the results of [24,25] in order to obtain the remaining three first integrals. In particular, starting from the restriction of equation (2.2) to the 4-dimensional submanifold Σ, we get I_k−2, I_k−1 and I_k in terms of two dependent solutions to a second-order linear ODE and of the real constants c_i, i = 1,...,k − 3.

3. Examples

3.1. Example 1

Let us consider the fourth-order equation

u4u12−2(3u2u3u1−3u22−uu16)=0(3.1)

which corresponds, when u₁ ≠ 0, to the vector field

A=∂x+u1∂u+u2∂u1+u3∂u2+2(3u2u3u1−3u22−uu16)u12∂u3.

The symmetry algebra of equation (3.1) is spanned by

v1=∂x, v2=x2∂x, v3=x∂x,(3.2)

and hence it is three-dimensional and isomorphic to 𝔰𝔩(2, ℝ). Since equation (3.1) lacks of further symmetries, we can look for a generalized solvable structure. In particular we have to find a vector field Y on a suitable domain U ⊂ J³(ℝ,ℝ) such that Y is a symmetry of the distribution 𝒜 = 〈A, X₁, X₁, X₃〉, where X_i stands for the third-order prolongation of v _i, for i = 1, 2, 3, i.e.

X1=v1(3)=∂x,X2=v2(3)=x2∂x−2xu1∂u1−(2u1+4u2x)∂u2−(6u2+6u3x)∂u3,X3=v3(3)=x∂x−u1∂u1−2u2∂u2−3u3∂u3.(3.3)

It is worthwhile to remark that finding a symmetry for the distribution 𝒜 = 〈A, X₁, X₁, X₃〉 is definitely simpler than finding a symmetry for the vector field A. For instance, it is easy to check that the vector field

Y=u13∂u3,(3.4)

provides a generalized solvable structure for ODE (3.1), due to the following commutation relations

[X1,Y]=[X2,Y]=[X3,Y]=0,[A,Y]=(12u12x2)X1+(12u12)X2−u12xX3.(3.5)

Once we have got Y we can apply Theorem 2.2. by considering the volume form Ω = dx ∧ du ∧ du₁ ∧ du₂ ∧ du₃ and the differential 1-form α = A⌋X₁⌋X₂⌋X₃⌋Ω given by

α=4uu16du−(6u3u12−12u22u1)du1−6u12u2du2+2u13du3.

Since Y is a symmetry of α, the function 1/Δ, with

Δ=Y⌋α=2u16,

is an integrating factor of α. Therefore, the one-form

ω=1Δα=2udu−3(u3u1−2u22)u15du1−3u2u14du2+1u13du3

is closed and so (locally) exact, i.e. ω = dI, where the function I = I(x, u⁽³⁾) is given (up to an additive constant) by:

I=u2+u3u13−3u222u14.(3.6)

Function (3.6) provides a first integral for the integrable distribution 〈A, X₁, X₂, X₃〉 and the restriction of (3.1) to the submanifold I = c₁, where c₁ ∈ ℝ, leads to the reduced equation

2u2u14+2u1u3−3u22=2u14c1.(3.7)

Equation (3.7) inherits a symmetry algebra isomorphic to 𝔰𝔩(2, ℝ) and spanned by the restrictions of the vector fields (3.2) to I = c₁. Moreover (3.7) corresponds to the equation presented in [25], Case 1 in Table 2, for the function C(u) = u² − c₁. By Proposition 4.1 in [25], a complete set of first integrals for equation (3.7) can be expressed in terms of two independent solutions to the second-order linear ODE (see Case 1, Table 6 in [25]):

Ψc1″(u)+(u2−c1)Ψc1(u)=0.(3.8)

Let Ψ_c₁;1(u) and Ψ_c₁;2(u) denote two linearly independent solutions to (3.8), such that the corresponding Wronskian is equal to 1 (note that the Wronskian is constant by the Liouville’s formula).

According to Table 7 (Case 1) in [25], three independent first integrals for equation (3.7) are given by

I1(x,u,u1,u2;c1)=u2Ψc1;1(u)−2u12Ψc1;1′(u)u2Ψc1;2(u)−2u12Ψc1;2′(u),I2(x,u,u1,u2;c1)=(2u1+xu2)Ψc1;1(u)−2xu12Ψc1;1′(u)(2u1+xu2)Ψc1;2(u)−2xu12Ψc1;2′(u),I3(x,u,u1,u2;c1)=((2u1+xu2)Ψc1;2(u)−2xu12Ψc1;2′(u))24u13.(3.9)

If we put c1=u2+u3u13−3u222u14 in (3.9), the resulting functions I₁, I₂, I₃ provide, together with (3.6), a complete set of first integrals for equation (3.1). Therefore, the general solution to (3.1) is given by (see Section 5 in [25]):

x=c4(c2−c3)c2Ψc1;1(u)−Ψc1;2(u)c3Ψc1;1(u)−Ψc1;2(u),(3.10)

where c_i ∈ ℝ for i = 1, 2, 3, 4, c₄ ≠ 0, c₂ ≠ c₃.

We finally remark that the solutions to (3.8) can be expressed in terms of Whittaker functions: if M_μ,ν(z) and W_μ,ν(z) denote the Whittaker functions of parameters μ=ι4c1 and v=14 , then two linearly independent solutions to (3.8) are given by

Ψc1;1(u)=1uMμ,ν(iu2) and Ψc1;2(u)=1μWμ,ν(iu2).

Consequently, the general solution (3.10) can be expressed as follows:

x=c4(c2−c3)c2Wμ,ν(iu2)−Mμ,ν(iu2)c3Wμ,ν(iu2)−Mμ,ν(iu2).

3.2. Example 2

Let us consider the fourth-order ODE

u1(x−u)u4−2xu2u3−6u22u1+4u12u3+2u2u3u−12u22+8u3u1=0(3.11)

and let A denote the corresponding vector field of the form (2.3) on a suitable domain U ⊂ J⁴(ℝ, ℝ) such that u₁(x − u) ≠ 0.

The Lie symmetry algebra of equation (3.11) is three-dimensional and isomorphic to 𝔰𝔩(2, ℝ). The third-order prolongations of the corresponding Lie symmetry generators are given by

X1=∂x+∂u,X2=x2∂u+u2∂u−2u1(x−u)∂u1−(2u1+4xu2−2u12−2uu2)∂u2 −(6u2+6xu3−6u2u1−2u3u)∂u3,X3=x∂x+u∂u−u2∂u2−2u3∂u3.

A symmetry of the integrable distribution 〈A, X₁, X₂, X₃〉 can be easily found searching, for instance, a symmetry of the form η(x, u⁽³⁾)∂_u. In particular, if we consider the vector field

Y=(u12u3(x−u)+3u2(u1+1))∂u,(3.12)

we have the following commutation relations

[X1,Y]=[X3,Y]=0,[X2,Y]=μ0A+μ1X1+μ2X2+μ3X3,[A,Y]=ρ0A+ρ1X1+ρ2X2+ρ3X3,(3.13)

for suitable functions μ_i, ρ_i (i = 0, 1, 2, 3) whose expressions are omitted, not being involved in the following discussion. Since the vector field (3.12) provides a generalized solvable structure for equation (3.11), an integrating factor for the 1-form α = A⌋ X₁⌋X₂⌋X₃⌋Ω is given by 1/Δ, where

Δ=A⌋X1⌋X2⌋X3⌋Y⌋Ω=2u1(3u22−2u3u1)

and we assume that Δ ≠ 0.

Therefore, the differential one-form ω=1Δα is closed and, locally, exact. A corresponding first integral is given by

I1=12u12(6u2(u−x)−u3(x−u)2+6u2u1(u−x)−6u1−6u13).(3.14)

The restriction of equation (3.11) to a generic leaf I₁ = c₁, c₁ ∈ ℝ, provides the following third-order ODE:

u3=−2(c1u12+3u13+3u1−3uu1u2+3u1u2x−3u2x+3xu2)(x−u)2.(3.15)

Equation (3.15) inherits the symmetry algebra 𝔰𝔩(2, ℝ) and corresponds to the third-order ODE appearing in [25, Case 3 in Table 1], for the particular case of

C(s)=13s2+4c1−24, where s=2u1+2u12+u2(x−u)u13/2.

According to [25], a complete set of first integrals to (3.15) can be expressed in terms of a system of solutions to the second-order linear equation

Ψc1″(s)+7s3s2+4c1−24Ψc1′(s)+4(3s2+4c1−24)Ψc1(s)=0.(3.16)

In particular, if Ψ_c₁;1 = Ψ_c₁;1(s), Ψ_c₁;2 = Ψ_c₁;2(s) denote two linearly independent solutions to (3.16) and W = W(Ψ_c₁;1, Ψ_c₁;2)(s) denotes the corresponding Wronskian, a complete system of first integrals for equation (3.15) is given by (see [25, Case 3 in Table 8]):

I2(x,u,u1,u2;c1)=2u1Ψc1;1(s)+(3s2+4c1−24)Ψc1;1′(s)2u1Ψc1;2(s)+(3s2+4c1−24)Ψc1;2′(s),I3(x,u,u1,u2;c1)=2xu1Ψc1;1(s)+u(3s2+4c1−24)Ψc1;1′(s)2xu1Ψc1;2(s)+u(3s2+4c1−24)Ψc1;2′(s),I4(x,u,u1,u2;c1)=(2xu1Ψc1;2(s)+3u(3s2+4c1−24)Ψc1;2′(s))26u1(u−x)(3s2+4c1−24)W.(3.17)

By setting J_i = J_i(x, u⁽³)) = I_i(x, u, u₁, u₂; I₁), for i = 2, 3, 4, we finally obtain that {I₁, J₂, J₃, J₄} is a complete set of first integrals for the fourth-order equation (3.11). Hence, the general solution to (3.15), and therefore to the original equation (3.11), is implicitly defined by

I2(x,u,u1,u2;c1)=c2,I3(x,u,u1,u2;c1)=c3,I4(x,u,u1,u2;c1)=c4,(3.18)

where c_i ∈ ℝ for i = 1, 2, 3, 4.

In order to obtain a parametric expression for the implicit solution (3.18), we consider a new parameter t such that s = s(t) is determined as follows (see [25], Section 5 for further details):

s′(t)=1C(s(t))=3s(t)2+4c1−24.(3.19)

Three possible cases arise:

If c₁ > 6 then (3.19) yields
s(t)=2c−12tan(2t2c1−12),
and the second-order equation (3.16) is transformed by means of the change ϕ_c₁(t) = Ψ_c₁(s(t)) into
φc1″(t)+2c−12tan(2t2c1−12)φc1′(t)+4φc1(t)=0.(3.20)
If c₁ < 6 then (3.19) yields
s(t)=−12−2c1tanh(2t12−2c1),
and through the transformation ϕ_c₁(t) = Ψ_c₁(s(t)) the second-order equation (3.16) becomes
φc1″(t)−12−2ctanh(2t12−2c)φc1′(t)+4φc1(t)=0.(3.21)
If c₁ = 6 then a solution to (3.19) is locally given by
s(t)=−12t
and by means of the transformation ϕ_c₁(t) = Ψ_c₁(s(t)), equation (3.16) is mapped into
φc1″(t)−12tφc1′(t)+4φc1(t)=0.(3.22)

For each one of the three cases considered above, let ϕ_1;c₁(t) = Ψ_c₁;1(s(t)) and ϕ_2;c₁(t) = Ψ_c₁;2 (s(t)) denote two linearly independent solutions to the second-order linear equations (3.20), (3.21), and (3.22), respectively. In terms of these functions, the general solution to equation (3.11) can be expressed in parametric as follows [25]:

{x(t)=c4(c2−c3)c3φ2;c1′(t)−φ1;c1′(t)c2φ2;c1′(t)−φ1;c1′(t),u(t)=c4(c2−c3)c3φ2;c1(t)−φ1;c1(t)c2φ2;c1(t)−φ1;c1(t).(3.23)

3.3. Example 3

In this section we consider the fifth-order ODE

uu3u5+u1u3u4+2u2u32−uu42+4u3u34=0(3.24)

on a suitable domain U ⊂ J⁵(ℝ, ℝ) such that uu₃ ≠ 0 and its associated vector field

A=∂x+u1∂u+u2∂u1+u3∂u2+u4∂u3−(u4u1u3+2u2u32−uu42+4u3u34uu3)∂u4.

The Lie symmetry algebra of equation (3.24) is three dimensional and isomorphic to 𝔰𝔩(2, ℝ). The corresponding fourth-order prolongations of the symmetry generators are

X1=∂x,X2=x2∂x+2xu∂u+2u∂u1−(2xu2−2u1)∂u2−4xu3∂u3−(4u3+6xu4)∂u4,X3=x∂x+u∂u−u2∂u2−2u3∂u3−3u4∂u4.

A symmetry Y₁ of the integrable distribution 〈A, X₁, X₂, X₃〉 can be easily determined by searching, for instance, a vector field of the form η(x, u⁽⁴⁾)∂_u₂. For example, the vector field

Y1=1u∂u2,

satisfies the following commutation relations:

[X1,Y1]=[X2,Y1]=[X3,Y1]=0,[A,Y1]=−x22u2X1−12u2X2+xu2X3.(3.25)

The next step in order to obtain a generalized solvable structure for equation (3.24) is looking for a symmetry Y₂ of the integrable distribution 〈A, X₁, X₂, X₃, Y₁〉. It is easy to check that the vector field

Y2=−2u∂u−3u1∂u1+u3∂u3

satisfies

[X1,Y2]=[X3,Y2]=0,[X2,Y2]=−x2X1−X2+2xX3+8uu1Y1,[Y1,Y2]=−2Y1,[A,Y2]=A−(u+2x2u2)uX1−2u2uX2+4xu2uX3+(4u2u1−2uu3)Y1.

Therefore 〈Y₁, Y₂〉 provides a generalized solvable structure for equation (3.24).

In order to apply Theorem 2.2. we consider the volume form Ω = dx∧du∧du₁ ∧du₂ ∧du₃ ∧du₄ and the two one-forms

α2=A⌋X1⌋X2⌋X3⌋Y1⌋Ω and α1=A⌋X1⌋X2⌋X3⌋Y2⌋Ω.

An integrating factor for α₂ is given by 1/Δ, where

Δ=Y2⌋α2=6(4u4u34+u2u42+4u4u1u3u+4u32u12)u

and we assume that Δ ≠ 0. Therefore we can chose as first integral for the (locally) exact form ω2=1Δα2 the function ln(I₂)/6, where

I2=u324u4u34+u2u42+4u4u1u3u+4u32u12(3.26)

is a first integral for the distribution 〈A, X₁, X₂, X₃, Y₁〉.

Since ω1=1Δα1 is closed on each leaf I₂ = c₂, with c₂ ∈ ℝ, c₂ > 0, we have that ω₁ is locally exact module ω₂ and a corresponding first integral is given by

I1=12u12−u2u+12arctan(2u32u22u3u1+uu4).(3.27)

Hence, by restricting to a generic leaf {I₁ = c₁, I₂ = c₂}, we get the reduced equation

u3=−sin(2c1+2uu2−u12)2u2c2,(3.28)

which is a third-order ODE admitting 𝔰𝔩(2, ℝ) as Lie symmetry algebra. Equation (3.28) corresponds to the third-order ODE given in [25, Case 2 in Table 2] for

C(s)=c24sin(2c1−s), where s=u12−2uu2.

According to the results obtained in [25], if Ψ_c₁,c₂;1 = Ψ_c₁,c₂;1(s) and Ψ_c₁,c₂;2 = Ψ_c₁,c₂;2(s) are two linearly independent solutions to the linear second-order equation [25, Case 2 in Table 6],

−4sin2(2c1−s)Ψ″(s)+2sin(4c1−2s)Ψ′(s)+c2sΨ(s)=0,(3.29)

three functionally independent first integrals for equation (3.28) are given by

I3(x,u(2);c1,c2)=c2u1Ψc1,c2;1(s)+2sin(2c1−s)Ψc1,c2;1′(s)c2u1Ψc1,c2;2(s)+2sin(2c1−s)Ψc1,c2;2′(s),I4(x,u(2);c1,c2)=c2(−2u+u1x)Ψc1,c2;1(s)+2xsin(2c1−s)Ψc1,c2;1′(s)c2(−2u+u1x)Ψc1,c2;2(s)+2xsin(2c1−s)Ψc1,c2;2′(s),I5(x,u(2);c1,c2)=(c2(−2u+u1x)Ψc1,c2;2(s)+2xsin(2c1−s)Ψc1,c2;2′(s))24c2usin(2c1−s)W,

where W = W(Ψ_c₁,c₂;1, Ψ_c₁,_c₂;2)(s) denotes the Wronskian corresponding to the solutions Ψ_c₁,c₂;1 and Ψ_c₁,c₂;2. Therefore, as in the previous examples, the general solution to the original fifth-order equation (3.24) is implicitly given by

I3(x,u(2);c1,c2)=c3,I4(x,u(2);c1,c2)=c4,I5(x,u(2);c1,c2)=c5.(3.30)

In order to obtain a parametric general solution from (3.30), we consider a new parameter t such that s = s(t) is determined as follows:

s′(t)=1C(s(t))=4sin(2c1−s(t))c2,

which yields

s(t)=−arctan(2e−4t/c2e−8t/c2−1)+2c1.(3.31)

By means of the transformation ϕ(t) = Ψ(s(t)) equation (3.29) is transformed into the Schrödinger-type equation

φ″(t)+4(arctan(2e−4t/c2e−8t/c2−1)−2c1)φ(t)=0.(3.32)

If ϕ₁ = ϕ₁(t; c₁, c₂) and ϕ₂ = ϕ₂(t; c₁, c₂) are two linearly independent solutions to (3.32) such that W(ϕ₁, ϕ₂)(t; c₁, c₂) = 1, then the parametric general solution to equation (3.24) can be expressed as follows

{x(t)=c5(c3−c4)φ1(t;c1,c2)−c4φ2(t;c1,c2)φ1(t;c1,c2)−c3φ2(t;c1,c2),u(t)=c5(c3−c4)24(φ1(t;c1,c2)−c3φ2(t;c1,c2))2,(3.33)

where c_i ∈ ℝ for i = 1, 2, 3, 4, 5, c₂ > 0, c₃ ≠ c₄, c₅ ≠ 0.

4. Conclusions

Generalized solvable structures for ODEs of order k > 3 admitting a symmetry algebra isomorphic to 𝔰𝔩(2, ℝ) have been introduced, providing a powerful tool for the explicit determination of the general solutions to these equations.

A generalized solvable structure is formed by k − 3 vector fields, which, in general, are not symmetries of the equation. Moreover, the symmetry generators of 𝔰𝔩(2, ℝ) and the vector fields of the generalized solvable structure do not form a solvable structure. Nevertheless, the knowledge of a generalized solvable structure allows the determination by quadratures of a complete set of first integrals, and hence of the general solution to the equation, in terms of two independent solutions to an associated family of second-order linear ODEs. This link between equations admitting 𝔰𝔩(2, ℝ) and second-order linear ODEs is, to the best of our knowledge, new in the literature and generalizes the recent results obtained for third-order equations.

The concept of generalized solvable structure gives more freedom in the choice of the vector fields which can be used to integrate the equation under study. The effectiveness of the proposed method is illustrated by several examples of equations admitting only an 𝔰𝔩(2, ℝ) algebra of Lie symmetries. In these cases, after finding a generalized solvable structure, we provide a complete set of first integrals as well as the general solutions in parametric or explicit forms.

Acknowledgments

P. Morando acknowledges the financial support from Gruppo Nazionale di Fisica Matematica (GNFM-INdAM).

C. Muriel and A. Ruiz acknowledge the financial support from the University of Cádiz by means of the project PR2017-090 and from Grupo de Investigación de la Junta de Andalucía FQM-377.

A. Ruiz acknowledges the financial support from the Ministry of Education, Culture and Sport of Spain (FPU grant FPU15/02872).

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 2
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TY  - JOUR
AU  - Paola Morando
AU  - Concepción Muriel
AU  - Adrián Ruiz
PY  - 2021
DA  - 2021/01/06
TI  - Generalized Solvable Structures and First Integrals for ODEs Admitting an 𝔰𝔩(2, ℝ) Symmetry Algebra
JO  - Journal of Nonlinear Mathematical Physics
SP  - 188
EP  - 201
VL  - 26
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1591712
DO  - 10.1080/14029251.2019.1591712
ID  - Morando2021
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