Realizations of the Witt and Virasoro Algebras and Integrable Equations

Qing Huang; Renat Zhdanov

doi:10.1080/14029251.2020.1683964

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Volume 27, Issue 1, October 2019, Pages 36 - 56

Realizations of the Witt and Virasoro Algebras and Integrable Equations

Authors

Qing Huang

School of Mathematics, Center for Nonlinear Studies, Northwest University, Xi’an 710127, P.R. China,hqing@nwu.edu.cn

Renat Zhdanov^*

CyberOptics Corporation, 5900 Golden Hills Drive, Minneapolis, MN 55416, USA,rzhdanov@cyberoptics.com

^*Corresponding author.

Corresponding Author

Renat Zhdanov

Received 6 September 2018, Accepted 2 July 2019, Available Online 25 October 2019.

DOI: 10.1080/14029251.2020.1683964 How to use a DOI?
Keywords: Witt algebra; Virasoro algebra; Lie vector field; equivalence transformation; integrable equation
Abstract: In this paper we study realizations of infinite-dimensional Witt and Virasoro algebras. We obtain a complete description of realizations of the Witt algebra by Lie vector fields of first-order differential operators over the space ℝ³. We prove that none of them admits non-trivial central extension, which means that there are no realizations of the Virasoro algebra in ℝ³. We describe all inequivalent realizations of the direct sum of the Witt algebras by Lie vector fields over ℝ³. This result enables complete description of all possible (1+1)-dimensional partial differential equations that admit infinite dimensional symmetry algebras isomorphic to the direct sum of Witt algebras. In this way we have constructed a number of new classes of nonlinear partial differential equations admitting infinite-dimensional Witt algebras. So new integrable models which admit infinite symmetry algebra are obtained.
Copyright: © 2020 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

Methods and ideas of the group approach to analyzing differential equations go back to the pioneering works of Sophus Lie [1, 10, 39]. It is nothing short of amazing that two centuries later these methods have not significantly changed in their content. What has changed dramatically is the range of applications of these methods. The Lie group approach is used to analyze linear and nonlinear algebraic, differential and integro-differential equations modeling natural phenomena in many application areas such as physics, chemistry, biology and finance, to name just a few.

A necessary prerequisite for applicability of Lie group methods to a specific model is that the model should possess nontrivial symmetry. The ultimate success of the Lie theory relies heavily upon the fact that symmetry is one of the fundamental properties of nature, which is reflected in the symmetries of the models describing it.

It comes as no surprise that the richer the symmetry properties of a model under study are, the more efficient the application of group methods for its analysis. The best possible scenario is when the model admits infinite symmetries, which in many cases may lead to a linearizing transformation or even general solution of the model in question. Bluman and Kumei analyzed the connection between infinite-dimensional symmetry and linearization of partial differential equations (PDEs) in [7, 36]. They have established necessary and sufficient conditions of the linearizability of nonlinear PDEs admitting infinite dimensional symmetry algebra and described the method for constructing invertible mappings of nonlinear differential equations into linear ones.

A classical example is the hyperbolic type Liouville equation

utx=exp(u).(1.1)

It admits the infinite-parameter Lie group

t˜=t+f(t), x˜=x+g(x), u˜=u−f˙(t)−g˙(x),(1.2)

where f and g are arbitrary smooth functions. Note that an alternative term for these kinds of groups is Lie pseudo-group [14, 43, 44, 46, 55].

The general solution of Eq. (1.1) can be obtained by the action of the transformation group (1.2) on its particular traveling wave solution of the form u(t, x) = φ(x + t) (see, e.g., [16]). An alternative way to solve Eq. (1.1) is reducing it to the linear form U_tx = 0 by the nonlocal transformation u = ln(2U_tU_x/U²) [16].

In most cases application of Lie methods boils down to solving one of the two basic problems [6, 16, 17, 27, 45, 48]. The first one is computing the maximal symmetry group admitted by the model under study. The second (classification) problem is describing all possible models of some prescribed form invariant with respect to a given Lie group or Lie algebra. Typically classification problems split into two sub-problems, (i) describing all inequivalent realizations of the Lie algebra in question, and, (ii) constructing corresponding classes of invariant models (see, [4, 18, 19, 28, 37, 54, 56] and the references therein).

Lie group classification of PDEs is a very popular topic and there are literally hundreds of publications devoted to applying it to various classes of linear and nonlinear PDEs. However, the overwhelming majority of these tackle the classification with respect to finite-dimensional Lie algebras.

The situation is drastically different for the case of generalized (higher) Lie symmetries, which have played a critical role in the success of the theory of nonlinear integrable systems in (1+1)- and (1 + 2)-dimensions (see, e.g. [27,45]). Progress in this area has been complemented by advances in development of the theory of infinite-dimensional Lie algebras such as loop [30, 50], Kac-Moody [8, 32] and Virasoro [3, 20, 29, 34] algebras.

Virasoro algebra continues to play an increasingly important role in mathematical physics in general [5, 24] and in the theory of integrable systems in particular. A number of nonlinear evolution equations in (1 + 2)-dimensions modeling various phenomena in modern physics are invariant with respect to Virasoro algebras. An incomplete list of these models includes Kadomtsev-Petviashvili (KP) [15, 26], modified KP, cylindrical KP [38], Davey-Stewartson [13, 25], Nizhnik-Novikov-Veselov, stimulated Raman scattering, (1 + 2)-dimensional Sine-Gordon [52] and the KP hierarchy [47] equations.

It is a common belief that nonlinear PDEs admitting symmetry algebras of a Virasoro type are prime candidates for the role of an integrable system. This is why systematic classification of Virasoro algebra realizations could come in handy in constructing new integrable systems by the symmetry approach (see, e.g., [40, 41]).

It should be pointed out that there are integrable equations which do not admit Virasoro algebras, for example, breaking soliton and Zakharov-Strachan equations [52].

Classification of realizations of Lie algebras by vector fields of differential operators within the action of a local diffeomorphism group was pioneered by Sophus Lie himself. It is still a very popular and efficient tool for group analysis of nonlinear PDEs. Relatively recent applications of this approach include geometric control theory [31], theory of systems of nonlinear ordinary differential equations possessing superposition principle [53], and an algebraic approach to molecular dynamics [2, 51]. Analysis of realizations of Lie algebras by first-order differential operators is at the core of almost every approach to group classification of PDEs (see, e.g., [1, 10, 17, 22, 23, 33, 35]). Let us emphasize that systematic and exhaustive description of realizations of infinite-dimensional Lie algebras within the context of group classification of nonlinear PDEs is an important problem (see, e.g., [14, 43, 44, 55] and the references therein).

It is straightforward to verify that the Lie algebra of the group (1.2) is a direct sum of two infinite-dimensional Witt algebras. This observation is the starting point of our search for nonlinear generalizations of the wave equation possessing the same (infinite) symmetries. Namely, we are looking for all possible PDEs in two dimensions admitting a direct sum of two Witt algebras. In order to solve this classification problem we first need to construct all inequivalent realizations of the Witt algebra. The next step is to describe all possible inequivalent realizations of the direct sum of two Witt algebras. The remarkable fact is that this problem has a complete and elegant solution and there are only four inequivalent classes of PDEs that admit a direct sum of the Witt algebras. The wave and Liouville equations are particular cases of the so obtained PDEs.

The paper is organized as follows. In Section 2, we give a brief account of necessary facts and definitions. We also describe the algorithmic procedure for classification of inequivalent realizations of the Virasoro algebra in full detail. We construct all inequivalent realizations of the Witt algebra (a.k.a. centerless Virasoro algebra) in Section 3. Section 4 is devoted to analysis of realizations of the Virasoro algebra. We prove that there are no central extensions of the Witt algebra over the space ℝ³. In Section 5 we construct broad classes of nonlinear PDEs admitting the Witt algebras. All inequivalent realizations of the direct sum of two Witt algebras are obtained in Section 6. This enables us to completely solve the classification problem of second-order PDEs whose invariance algebras contain a direct sum of the Witt algebras. We prove that any such PDE is equivalent to one of the four canonical equations (6.2)–(6.5). The last section, Section 7, contains a brief summary of the main results of the paper and concluding remarks regarding future work.

2. Notations and Definitions

Definition 2.1 ([12,21,29,42]).

The Virasoro algebra 𝔙 is the infinite-dimensional Lie algebra with basis elements {L_n, C | n ∈ 𝕑} satisfying the following commutation relations:

[Lm,Ln]=(m−n)Lm+n+112m(m2−1)δm,−nC, [Lm,C]=0

for m, n ∈ 𝕑. Here [Q, P] = QP − PQ is the standard commutator of Lie vector fields P and Q, and the symbol δ_a,b stands for the Kronecker delta

δa,b={1,a=b,0,otherwise.

The operator C commutes with all other basis elements. It is called the central element or the central charge of the Virasoro algebra 𝔙. When the central element C is zero, the algebra 𝔙 reduces to centerless Virasoro algebra, which is usually called the Witt algebra 𝔚 [11, 49]. Consequently, the full Virasoro algebra is a nontrivial one-dimensional central extension of the Witt algebra.

Consider the Virasoro algebra as a linear subspace of the infinite-dimensional Lie algebra 𝔏_∞ spanned by the basis elements of the form

Q=τ(t,x,u)∂t+ξ(t,x,u)∂x+η(t,x,u)∂u(2.1)

over ℝ³. Applying the transformation

t˜=T(t,x,u), x˜=X(t,x,u), u˜=U(t,x,u)(2.2)

with D(T, X, U)/D(t, x, u) ≠ 0 to (2.1), we get

Q˜=(τTt+ξTx+ηTu)∂t˜+(τXt+ξXx+ηXu)∂x˜+(τUt+ξUx+ηUu)∂u˜.

Evidently, Q˜∈𝔏∞. Consequently the set of operators (2.1) is invariant with respect to the transformation (2.2).

The correspondence, Q∼Q˜, is an equivalence relation on (2.1). It splits the set of differential operators (2.1) into several equivalence classes. Any two elements within the same equivalence class are related by a transformation (2.2), while two elements belonging to different classes cannot be transformed into each other by a transformation of the form (2.2). Thus to describe all possible realizations of the Virasoro algebra, it is sufficient to construct a representative of each equivalence class. The remaining realizations can be obtained by applying transformations (2.2) to the representatives.

Our method for the construction of all inequivalent realizations of the Virasoro algebra is implemented as a three-step process.

The first step is to describe all inequivalent forms of L₀, L₁ and L₋₁ such that the commutation relations

[L0,L1]=−L1, [L0,L−1]=L−1, [L1,L−1]=2L0,(2.3)

hold together with [L_i, C] = 0, (i = 0, 1, −1). Note that algebra 〈L₀, L₁, L₋₁〉; is isomorphic to sl(2, ℝ).

At the second step, we construct all inequivalent realizations of the operators L₂ and L₋₂, which commute with C and satisfy the relations

[L0,L2]=−2L2, [L−1,L2]=−3L1, [L1,L−2]=3L−1,[L0,L−2]=2L−2, [L2,L−2]=4L0+12C.(2.4)

The third step is to derive the forms of the remaining basis operators of the Virasoro algebra using the recurrence relations

Ln+1=(1−n)−1[L1,Ln], L−n−1=(n−1)−1[L−1,L−n]

with

[Ln+1,L−n−1]=2(n+1)L0+112n(n+1)(n+2)C, [Li,C]=0,

where i = n + 1, −n − 1 and n = 2, 3, 4,....

In Section 3 and 4, we use this procedure to obtain all inequivalent realizations of the Witt and Virasoro algebras by Lie vector fields of the general form (2.1).

3. Realizations of the Witt Algebra

We now turn to analysis of realizations of the Witt algebra 𝔚. We note that the algebra 𝔚 is obtained from the Virasoro algebra by putting C = 0.

Let vector field L₀ be of the general form (2.1), namely,

L0=τ(t,x,u)∂t+ξ(t,x,u)∂x+η(t,x,u)∂u,

where τ² + ξ² + η² ≠ 0 since otherwise L₀ is trivial. Transformation (2.2) maps L₀ into

L˜0=(τTt+ξTx+ηTu)∂t˜+(τXt+ξXx+ηXu)∂x˜+(τUt+ξUx+ηUu)∂u˜.

By choosing solutions of the equations

τTt+ξTx+ηTu=1, τXt+ξXx+ηXu=0, τUt+ξUx+ηUu=0

as T, X and U, we can reduce L₀ to the form ∂_t. Thus L₀ is equivalent to the canonical operator ∂_t, which generates group of displacements by t. From now on, we drop the tildes.

With L₀ in hand, we proceed to constructing L₁ and L₋₁ which obey the commutation relations (2.3). Taking L₁ in the general form (2.1) and inserting it into [L₀, L₁] = −L₁ yield

L1=e−tf(x,u)∂t+e−tg(x,u)∂x+e−th(x,u)∂u,

where f, g, h are arbitrary smooth functions of their arguments.

In what follows we use only those equivalence transformations (2.2) which preserve L₀. Applying (2.2) to L₀ gives

L0→L˜0=Tt∂t˜+Xt∂x˜+Ut∂u˜=∂t˜.

Hence, the change of variables

t˜=t+T(x,u), x˜=X(x,u), u˜=U(x,u)

is the most general transformation that does not alter the form of L₀. It transforms Lie vector field L₁ into

L˜1=e−t(f+gTx+hTu)∂t˜+e−t(gXx+hXu)∂x˜+e−t(gUx+hUu)∂u˜.

To further simplify L₁, we need to consider two inequivalent cases g² + h² = 0 and g² + h² ≠ 0 separately.

Case 1. If g² + h² = 0, we have L˜1=e−tf(x,u)∂t˜. Choosing t˜=t−ln|f(x,u)| yields L₁ = e^−t∂_t. Taking into account (2.1) and (2.3) yields L₋₁ = e^t∂_t.
Case 2. Given the condition g² + h² ≠ 0, we can select T(x, u), X(x, u) and U(x, u) to satisfy the relation
e−T=f+gTx+hTu, gXx+hXu=e−T, gUx+hUu=0.

As a consequence, L₁ takes the form e^−t(∂_t + ∂_x). Taking into account (2.1) and (2.3) we arrive at the formula

L−1=et[1−e−2xf1(u)]∂t+et[−1−e−2xf1(u)+e−xg1(u)]∂x+et−xh1(u)∂u,

where f₁, g₁, h₁ are arbitrary smooth functions of the variable u.

Applying the transformation

t˜=t, x˜=x+X(u), u˜=U(u),(3.1)

which keeps L₀ and L₁ invariant, to L₋₁ gives

L˜−1=et[1−e−2xf1(u)]∂t˜+et[−1−e−2xf1(u)+e−xg1(u)+e−xh1X˙]∂x˜+et−xh1(u)U˙∂u˜.

Consider the cases f₁(u) ≠ 0 and f₁(u) = 0 separately.

Assuming f₁(u) ≠ 0 we set

X(u)=−ln|f1(u)|, φ(u)=[g1(u)+h1(u)X˙(u)]|f1(u)|

and

U(u)={∫1h1(u)du,h1≠0,arbitary non−constant function,h1=0, (3.2)

whence

L−1=et[1+αe−2x]∂t+et[−1+αe−2x+e−xφ(u)]∂x+βet−x∂u

with α = ±1 and β = 0, 1.

The case f₁(u) = 0 gives rise to the realization

L˜−1=et∂t˜+et[−1+e−xg1(u)+e−xh1(u)X˙]∂x˜+et−xh1(u)U˙∂u˜.

Selecting X = 0 and U satisfying (3.2) we get

L−1=et∂t+et[−1+e−xg1(u)]∂x+βet−x∂u,

where β = 0, 1.

Hence we conclude that the following assertion holds.

Lemma 3.1.

Any realization of the Lie algebra 〈L₀, L₁, L₋₁〉; over ℝ³ is equivalent to one of the following canonical realizations:

〈∂t,e−t∂t,e−t∂t〉;,(3.3)

〈∂t,e−t(∂t+∂x),e−t(1+αe−2x)∂t+et[−1+φ(u)e−x+αe−2x]∂x+βet−x∂u〉;.(3.4)

Here α = 0, ±1, β = 0, 1 and ϕ(u) is an arbitrary smooth function.

To obtain the complete description of all inequivalent Witt algebras, we need to implement the last two steps of the classification procedure outlined in Section 2 and extend (3.3) and (3.4) up to full realizations of the Witt algebra.

The following assertion holds.

Theorem 3.1.

There are at most eleven inequivalent realizations of the Witt algebra 𝔚 over the space ℝ³. The representatives, 𝔚_i, (i = 1, 2,...,11), of each equivalence class are listed below.

𝔚₁ : 〈e^−nt∂_t〉,
𝔚₂ : 〈e−nt∂t+e−nt[n+12n(n−1)αe−x]∂x〉,
𝔚₃ : 〈e−nt+(n−1)x[e2x−(n+1)γex+12n(n+1)γ2](ex−γ)−n−1∂t +e−nt+(n−1)x[nex−12n(n+1)γ](ex−γ)−n∂x〉,
𝔚₄ : L0=∂t,L1=e−t∂t+e−t∂x,L−1=et(1+γe−2x)∂t+et(−1+γe−2x+e−xφ˜)∂x,L2=e−2tf(x,u)∂t+e−2tg(x,u)∂x,L−2=e2t[1+3γe−2x−12e−3x(6γφ˜+φ˜3±(4γ+φ˜2)3/2)]∂t +e2t[−2+3e−xφ˜+6γe−2x−12e−3x(6γφ˜+φ˜3±(4γ+φ˜2)3/2)]∂x,Ln+1=(1−n)−1[L1,Ln], L−n−1=(n−1)−1[L−1,L−n], n≥2,
𝔚₅ : 〈e^{−nt+(n−1)x} (e^x ± n)(e^x ± 1)⁻ⁿ∂_t + ne^{−nt+(n−1)x} (e^x ± 1)¹⁻ⁿ∂_x〉,
𝔚₆ : 〈e^−nt∂_t + γe^−nt [e^nx − (e^x − γ)ⁿ](e^x − γ)¹⁻ⁿ∂_x〉,
𝔚₇ : L0=∂t,L1=e−t∂t+e−t∂x,L−1=et(1+γe−2x)∂t+et(−1+γe−2x+e−xφ˜)∂x,L2=e−2t+xex−φ˜e2x−exφ˜−γ∂t+e−2t+x2ex−φ˜e2x−exφ˜−γ∂x,L−2=e2t−3x(e3x+3γex−γφ˜)∂t+e−2t−3x(2ex−φ˜) (−e2x+exφ˜+γ)∂x,Ln+1=(1−n)−1[L1,Ln], L−n−1=(n−1)−1[L−1,L−n], n≥2,
𝔚₈ : 〈e−nt∂t+e−nt[n−sgn(n)γ2∑j=1|n|−1j(j+1)e−2x]∂x〉 ,
𝔚₉ : 〈e−nt+(n−1)x(ex−1)n+1[(−1+∑j=1|n|−1(2j+1))n+(2n+1)ex−(n+2)e2x+e3x+sgn(n)φ˜2∑j=1|n|−1j(j+1)]∂t+e−nt+(n−1)x(ex−1)n+1[(−1+∑j=1|n|−1(2j+1))n−2nex+ne2x−sgn(n)φ˜2∑j=1|n|−1j(j+1)]∂x〉,
𝔚₁₀ : 〈e−nt∂t+ne−nt∂x+sgn(n)2∑j=1|n|j(j−1)e−nt−2x∂u〉,
𝔚₁₁ : 〈e−nt∂t+e−nt[n+αn(n−1)2e−x]∂x+n(n−1)2e−nt−x∂u〉.

Here n ∈ 𝕑, α = 0, ±1, γ = ±1, sgn(·) is the standard sign function, the symbol φ˜(u) stands for either u or an arbitrary real constant c, and besides

f(x,u)=ex[4e4x−10e3xφ˜−36γe2x+2ex(31γφ˜+6φ˜3±6(4γ+φ˜2)3/2) −64γ2−54γφ˜2−9φ˜4∓9φ˜(4γ+φ˜2)3/2]r−1g(x,u)=ex[8e4x−16e3xφ˜−2e2x(44γ+5φ˜2)+2ex(44γφ˜+9φ˜3±9(4γ+φ˜2)3/2) −64γ2−54γφ˜2−9φ˜4∓9φ˜(4γ+φ˜2)3/2]r−1,r=4e5x−10e4xφ˜−40γe3x+10e2x(6γφ˜+φ˜3±(4γ+φ˜2)3/2)−10ex(6γ2+6γφ˜2+φ˜4±φ˜(4γ+φ˜2)3/2)+30γ2φ˜+20γφ˜3+3φ˜5±(2γ+3φ˜2)(4γ+φ˜2)3/2.

Proof.

To prove the theorem we need to construct all possible inequivalent extensions of the realizations (3.3) and (3.4).

Case 1. Inserting (3.3) into (2.4) we have
L2=e−2t∂t, L−2=e2t∂t.

The remaining basis elements of the corresponding Witt algebra are readily obtained through recursion, which yields L_n = e^−nt∂_t, n ∈ 𝕑. Thus the realization 𝔚₁ is obtained.
Case 2. Turn now to realization (3.4). Inserting L₀, L₁, L₋₁ into the commutation relations [L₀, L₋₂] = 2L₋₂ and [L₁, L₋₂] = 3L₋₁ and solving the obtained PDEs give rise to the following form of L₋₂
L−2=e2t[1+3αe−2x+ψ1(u)e−3x]∂t+e2t[−2+3φ(u)e−x+ψ2(u)e−2x +ψ1(u)e−3x]∂x+e2t[3βe−x+ψ3(u)e−2x]∂u,
where ψ₁, ψ₂, ψ₃ are arbitrary smooth functions of u.

Utilizing commutation relations [L₀, L₂] = −2L₂ and [L₋₁, L₂] = −3L₁ we arrive at
L2=e−2tf(x,u)∂t+e−2tg(x,u)∂x+e−2th(x,u)∂u
with f, g, h satisfying the following system of PDEs:
−3(αe−2x+1)f+2αe−2xg+(φe−x+αe−2x−1)fx+βe−xfu+3=0,(3.5a)

(1−φe−x−αe−2x)f+(φe−x−2)g−φue−xh+(φe−x+αe−2x)gx+βe−xgu+3=0,(3.5b)

βe−xf−βe−xg+2(1+αe−2x)h−(φe−x+αe−2x−1)hx−βe−xhu=0.(3.5c)

Inserting above L₂ and L₋₂ into the commutation relation [L₂, L₋₂] = 4L₀ gives three more PDEs
4(ψ1e−3x+3αe−2x+1)f−3e−2x(ψ1e−x+2α)g+e−3xψ˙1h −(ψ1e−3x+ψ2e−2x+3φe−x−2)fx−e−x(ψ3e−x+3β)fu−4=0,2(ψ1e−3x+ψ2e−2x+3φe−x−2)f−(ψ1e−3x−2(3α−ψ2)e−2x+3φe−3−2)g +e−x(ψ˙1e−2x+ψ˙2e−x+3φ˙)h(ψ1e−3x+ψ2e−2x+3φe−x−2)gx−e−x(ψ3e−x+3β)gu=0,2e−x(ψ3e−x+3β)f−e−x(2ψ3e−x+3β)g+(2ψ1e−3x+(6α+ψ˙3)e−2x+2)h −(ψ1e−3x+ψ2e−2x+3φe−x−2)hx−e−x(ψ3e−x+3β)hu=0.(3.6)

To determine the forms of L₂ and L₋₂, we have to solve Eqs. (3.5) and (3.6). It is straightforward to verify that the relation
Δ=e−t−4x[βe3x+ψ3e2x+(βψ2−φψ3−3αβ)ex+βψ1−αψ3]≠0
is the necessary and sufficient condition for (3.5) and (3.6) to have a unique solution in terms of f_x, f_u, g_x, g_u, h_x and h_u. For this reason, we need to differentiate between the cases Δ = 0 and Δ ≠ 0.
- Case 2.1. Let Δ = 0 or, equivalently, β = ψ₃ = 0. Eqs. (3.5) and (3.6) do not contain derivatives of the functions f, g, h with respect to u.
  
  Solving (3.5) and (3.6) with respect to the derivatives f_x, g_x, h_x we obtain two expressions for each of them. Equating the right-hand sides of equations containing h_x yields
  hexe4x−2φe3x−ψ2e2x−2ψ1ex+3α2+φψ1−αψ2(e2x−φex−α)(2e3x−3φe2x−ψ2ex−ψ1)=0.
  
  Hence h = 0.
  
  Analyzing compatibility conditions for equations containing f_x and g_x we derive two more equations, which are linear in f and g. These equations form a system of two linear equations with non-vanishing determinant. Consequently, the system in question has a unique solution for f and g. Computing the derivatives of the expressions for f and g with respect to x and comparing the results with the previously obtained formulas for f_x and g_x, we arrive at the equations
  (ψ2−6α)(φ3+φψ2+2ψ1)e11x+F10[x,u]=0,(3.7)
  and
  [10φ3ψ1−3αφ2(3ψ2−8α)+3φψ1(2α+3ψ2)+2(5ψ12−4α(2α2−3ψ2+ψ22))]e10x +F9[x,u]=0.(3.8)
  
  Hereafter F_n[x, u] with n ∈ ℕ denotes an nth degree polynomial in exp(x).
  
  To determine admissible forms of f and g we need to construct the most general ϕ and ψ_i (i = 1, 2, 3) satisfying Eqs. (3.7) and (3.8).
  
  If (3.7) holds, then at least one of the equations ψ₂ = 6α and ψ₁ = −(ϕ³ +ϕψ₂)/2 is identically satisfied.
  - Case 2.1.1. Given ψ₂ = 6α, Eqs. (3.7) and (3.8) hold if and only if
    16α3+3α2φ2−6αφψ1−φ3ψ1−ψ12=0,
    whence ψ1=[−6αφ−φ3±(4α+φ2)32]/2. Choice of plus or minus in this formula leads to different realizations. So we need to consider those cases separately.
    - Case 2.1.1.1. Let ψ1=[−6αφ−φ3−(4α+φ2)32]/2. If α = 0, then we have either ψ₁ = 0 or ψ₁ = −ϕ³.
      
      The case α = ψ₁ = 0 gives L₋₁ = e^t∂_t + e^t(−1 + e^−xϕ)∂_x. Making equivalence transformation x˜=x+X(u), we can reduce ϕ to one of the three forms a = 0, ±1 and get
      f=1, g=2+ae−x.
      
      Utilizing the recurrence relations we derive the remaining basis elements and obtain the realization 𝔚₂.
      
      Provided α = 0 and ψ₁ = −ϕ³, we reduce the function ϕ to the form b = 0, ±1 by applying the equivalence transformation x˜=x+X(u). The case b ≠ 0 gives rise to the following f and g:
      f=ex(e2x−3bex+3b2)(ex−b)3, g=ex(2e2x−3b)(ex−b)2.
      
      Hence the realization 𝔚₃ is obtained. Note that the case b = 0 leads to the particular case of 𝔚₂.
      
      Assuming α = ±1, we get ψ1=[−6αφ−φ3−(4α+φ2)32]/2, which yields 𝔚₄.
    - Case 2.1.1.2. Let ψ1=[−6αφ−φ3+(4α+φ2)32]/2. Analysis similar to the one applied to the Case 2.1.1.1 gives the realizations 𝔚₂ and 𝔚₄.
      
      This completes analysis of the Case 2.1.1.
  - Case 2.1.2. If ψ₁ = −(ϕ³ + ϕψ₂)/2, then Eq. (3.7) takes the form
    (4α+φ2)[ψ2−(4α−5φ2)/4][ψ2−(2α−φ2)]e10x+F9[x,u]=0.
    
    Constructing the general solution of this equation boils down to analysis of the following three subcases.
    - Case 2.1.2.1. Given the relation ψ₂ = (4α − 5ϕ²)/4, Eqs. (3.7) and (3.8) hold if and only if
      4α+φ2=0.
      
      Consequently α ≤ 0.
      
      In the case when α < 0 we choose α = −b² so that b=±(−α)12 and ϕ = 2b. Solving (3.7) and (3.8), we immediately get the expression for f
      f=ex(ex−2b)(ex−b)2, g=2exex−b,
      which leads to 𝔚₅.
      
      If α = 0 and ϕ = ψ₁ = ψ₂ = 0, then the realization 𝔚₂ with α = 0 is obtained.
    - Case 2.1.2.2. Let ψ₂ = 2α − ϕ². Provided α = 0, we can choose ϕ to be b = ±1 without any loss of generality. We drop the case b = 0 since it has already been considered. Taking into account the above relations we get from Eqs. (3.7) and (3.8)
      f=1, g=2ex−bex−b.
      
      The realization 𝔚₆ is obtained.
      
      Provided α ≠ 0, we can apply an equivalence transformation to get α = ±1, whence
      f=ex(e2x−φ)e2x−exφ−b, g=ex(2ex−φ)e2x−exφ−b
      with b = ±1. Since ϕ can be reduced to the form ũ by the equivalence transformation ũ = ϕ with φ˙≠0, we thus get the realization 𝔚₇.
    - Case 2.1.2.3. If 4α + ϕ² = 0 then using Eqs. (3.7) and (3.8) we get α ≤ 0, whence α = 0, −1.
      
      Given the relation α = 0, we can reduce ϕ to the form a = 0, ±1. Inserting these expressions into (3.7), (3.8) yields f = 1 and g = 2 − ae^−x. The realization 𝔚₈ is obtained.
      
      In the case when α = −1, we have
      f=ex(e3x−4e2x+5ex+4+ψ2)(ex−1)4, g=ex(2e2x−4ex−4−ψ2)(ex−1)3.
      
      And furthermore, the function ψ₂ is reduced to the form ũ by equivalence transformation ũ = ψ₂, provided ψ₂ is a non-constant function. As a result, we get 𝔚₉.
      
      This completes analysis of Case 2.1. In summary we conclude that the case Δ = 0 leads to the realizations 𝔚_i, i = 2, 3,...,9.
- Case 2.2. Suppose now that Δ ≠ 0, or equivalently, β2+ψ32≠0. Given this condition we can solve Eqs. (3.5) and (3.6) with respect to f_x, f_u, g_x, g_u, h_x and h_u. The system obtained is overdetermined, so we need to analyze its compatibility.
  
  The compatibility conditions
  fxu−fux=0, gxu−gux=0, hxu−hux=0
  can be rearranged into the system of three linear equations for the functions f, g and h
  a1f+a2g+a3h+d1=0,b1f+b2g+b3h+d2=0,c1f+c2g+c3h+d3=0.
  
  Here a_i, b_i, c_i, d_i, (i = 1, 2, 3) are functions of t, x, ϕ, ψ₁, ψ₂ and ψ₃.
  
  It is straightforward to verify that the above system has a unique solution f, g, h when β2+ψ32≠0. We have solved the system in question using Mathematica and obtained very cumbersome formulas. For brevity, we do not exhibit them here. Inserting the expressions for f, g, h into Eq. (3.5a) yields
  αβ6e42x+F41[x,u]=0.
  
  Consequently, we have either β = 0 or α = 0.
  - Case 2.2.1. If β = 0, then Eq. (3.5a) takes the form
    αψ36e36x+F35[x,u]=0,
    which gives α = 0 and ψ₃ ≠ 0 (since otherwise Δ = 0). Now we can rewrite Eq. (3.6) as follows
    ψ1ψ36e36x+F35[x,u]=0,(15φ2+2ψ2)ψ36e37x+F36[x,u]=0,(57φ2−2ψ2)ψ37e35x+F34[x,u]=0.
    
    Hence we conclude that ϕ = ψ₁ = ψ₂ = 0. Inserting these formulas into the initial Eqs. (3.5) and (3.6) and solving the obtained system yield
    f=1, g=2, h=−e−2xψ3.
    
    The function ψ₃ can be reduced to −1 by the equivalence transformation ũ = − ∫ 1/ψ₃du. As a result, we get the realization 𝔚₁₀.
  - Case 2.2.2. Provided α = 0, Eq. (3.5c) takes the form
    β5(4βφψ3−6ψ32+β2ψ˙3)e41x+30β5φψ32e40x+F39[x,u]=0.
    
    Since the case α = β = 0 has already been analyzed in Case 2.2.1, we can restrict our considerations to the cases ψ₃ = 0, β = 1 and ϕ = 0, β = 1 without any loss of generality.
    
    If ψ₃ = 0 and β = 1, then it follows from (3.5c) and (3.6) that ψ₁ = ψ₂ = 0. Thus
    f=1, g=2+e−xφ, h=e−x.
    
    Furthermore, the function ϕ can be reduced to 0, 1 or −1 by the equivalence transformations x˜=x+X(u) and ũ = U(u). Hence we get the realization 𝔚₁₁.
    
    In the case when ϕ = 0 and β = 1, Eqs. (3.5) and (3.6) are incompatible. This completes analysis of the Case 2.2.
    
    We check by direct computation that the realizations 𝔚_i (i = 1, 2,...,11) cannot be mapped into one another by any equivalence transformation. Consequently, they are inequivalent.
    
    While proving Theorem 3.1. we have also obtained the exhaustive description of the Witt algebras over the spaces ℝ¹ and ℝ².

Theorem 3.2.

𝔚₁ is the only inequivalent realization of the Witt algebra over the space ℝ.

Theorem 3.3.

The realizations 𝔚₁–𝔚₉ with φ˜=c∈ℝ exhaust the list of inequivalent realizations of the Witt algebra over the space ℝ².

4. Realizations of the Virasoro Algebra

After obtaining the full list of inequivalent realizations of the Witt algebra we can proceed to classification of realizations of the Virasoro algebra 𝔙. To this end we need to extend the inequivalent realizations of the Witt algebra listed in Theorem 3.1 by all possible nonzero central elements C. In this section we prove that realizations of the Virasoro algebra with nonzero central element over the space ℝ³ do not exist. We describe the major elements of the proof skipping cumbersome intermediate calculations.

Let us begin by constructing all possible central extensions of the subalgebra 〈L₀, L₁, L₋₁〉. According to Lemma 3.1 it suffices to consider the algebras (3.3) and (3.4) only.

Case 1. Given the realization (3.3) we have
L0=∂t, L1=e−t∂t, L−1=et∂t.

Choosing the basis element C in the general form (2.1) and inserting it into the commutation relations [L_i, C] = 0, (i = 0, 1, −1) yield
C=ξ(x,u)∂x+η(x,u)∂u, ξ2+η2≠0.

Applying the transformation
t˜=t, x˜=X(x,u), u˜=U(x,u),
which preserves L₀, L₁ and L₋₁, to the central element C we get
C→C˜=(ξXx+ηXu)∂x˜+(ξUx+ηUu)∂u˜.

Choosing solutions of the equations
ξXu+ηXu=0, ξUx+ηUu=1
as X and U yields C = ∂_u.

We now turn to the basis element L₂. Using the commutation relations [L₀, L₂] = −2L₂, [L₋₁, L₂] = −3L₁ and [L₂, C] = 0 yields L₂ = e^−2t∂_t.

Inserting the basis element L₋₂ of the general form (2.1) into (2.4), which involve L₋₂, results in incompatible over-determined system of PDEs for the functions τ, ξ and η. Hence the realization (3.3) cannot be extended to a realization of the Virasoro algebra with nonzero central element.
Case 2. Consider now the algebra (3.4). Since C commutes with L₀ and L₁, we have
C=e−xf(u)∂t+[g(u)+e−xf(u)]∂x+h(u)∂u,
where f, g and h are arbitrary smooth functions.

Acting by transformation (3.1), which does not alter L₀ and L₁, on C gives
C˜=e−xf(u)∂t˜+[g(u)+e−xf(u)+h(u)X˙(u)]∂x˜+h(u)U˙(u)∂u˜.

To further simplify C˜, we need to analyze the cases f(u) ≠ 0 and f(u) = 0 separately.

If f(u) ≠ 0, then choosing X(u) = −ln|f(u)| we get C˜=e−x∂t˜+[e−x˜+β(g+hX˙)]∂x˜+βhU˙∂u˜, where β = ±1.

Provided h = 0 and g˙≠0, we can make the equivalence transformation ũ = g(u) and thus get C₁ = e^−x∂_t + (e^−x + u)∂_x. The case h=g˙=0 leads to C₂ = e^−x∂_t + (e^−x + λ)∂_x, where λ is an arbitrary constant.

Next, if the condition h ≠ 0 holds, we choose the functions X and U satisfying g+hX˙=0 and hU˙=1/β thus getting C₃ = e^−x∂_t + e^−x∂_x + ∂_u.

Given f(u) = 0 we have C˜=(g+hX˙)∂x˜+hU˙∂u˜.

If h ≠ 0, we can reduce C˜ to the form C₄ = ∂_u by a suitable choice of X and U. The case when h vanishes yields C˜=g∂x˜, which can be simplified to one of the following inequivalent forms, C₅ = u∂_x or C₆ = ∂_x.

Therefore there exist six inequivalent nonzero central element C commuting with basis elements L₀ = ∂_t and L₁ = e^−t∂_t + e^−t∂_x.

The next step is extending the realizations 〈L₀, L₁, C_i〉, (i = 1, 2,...,6) up to realizations of the full Virasoro algebra. Here we present the calculation details for the case i = 1 only. The remaining five cases are handled in a similar fashion.

We begin by constructing all possible realizations of L₋₁. Taking into account (2.3), we have

L−1=et−2x(u2e2x−1)u2∂t−et−2x(uex+1)2u2∂x.

With L₋₁ in hand we can proceed to constructing L₂. Using the commutation relations (2.4) yields

L2=uex(uex+2)e2t(uex+1)2∂t+2uexe2t(uex+1)∂x.

Inserting the obtained expressions into the commutation relations for L₋₂, we arrive at an incompatible system of PDEs. Hence, the algebra 〈L₀, L₁, C₁〉 cannot be extended to a realization of the full Virasoro algebra.

The same statement is true for the remaining C_i, (i = 2, 3,...,6).

Theorem 4.1.

There are no realizations of the Virasoro algebra with nonzero central element C over the space ℝⁿ, (n = 1, 2, 3).

5. PDEs Invariant under the Witt Algebras

In this section, we construct a number of new nonlinear (1+1)-dimensional second-order PDEs whose invariance algebra contains the Witt algebra and is consequently infinite-dimensional.

Constructing invariant equations is a straightforward application of Lie’s infinitesimal method (see, e.g., [45, 48]). Below we give a brief description of the method and present an example of derivation of PDE invariant under 𝔚₁.

A second-order PDE of the form

F(t,x,u,ut,ux,utt,utx,uxx)=0

is invariant with respect to the Witt algebra 〈L_n〉 if and only if the condition

pr(2)Ln(F)|F=0=0

holds for any n ∈ ℕ, where pr⁽²⁾L_n is the second-order prolongation of the vector field L_n, that is

pr(2)Ln=Ln+η4∂ut+ηx∂ux+ηtt∂utt+ηtx∂utx+ηxx∂uxx

with

ηt=Dt(η)−utDt(τ)−uxDt(ξ),ηx=Dx(η)−utDx(τ)−uxDx(ξ),ηtt=Dt(ηt)−uttDt(τ)−utxDt(ξ),ηtx=Dx(ηt)−uttDx(τ)−utxDx(ξ),ηxx=Dx(ηx)−uxtDx(τ)−uxxDx(ξ).

Here the symbols D_t and D_x stand for the total differentiation operators with respect to t and x respectively, namely,

Dt=∂t+ut∂u+utt∂ut+uxt∂ux+⋯,Dx=∂x+ux∂u+utx∂ut+uxx∂ux+⋯.

Let us apply the above method to derivation of PDEs admitting the realization 𝔚₁. The second-order prolongation of the basis elements of 𝔚₁ reads

pr(2)Ln=e−nt∂t+ne−ntut∂ut+(2ne−ntutt−n2e−ntut)∂utt+ne−ntutx∂utx.(5.1)

The next step is computing the complete set of functionally-independent second-order differential invariants I_m(t, x, u, u_t, u_x, u_tt, u_tx, u_xx) (m = 1, 2,...,7) of the infinite set of first-order differential operators L_n. Integrating the characteristic equations

dte−nt=dx0=du0=dutne−ntut=dux0=dutt2ne−ntutt−n2e−ntut=dutxne−ntutx=duxx0,

which correspond to the operator L_n with arbitrary n, gives the following invariants

I1=x, I2=u, I3=ux, I4=uxx, I5=utxut, I6=e−ntut, I7=e−2ntutt−ne−2ntut.

Thus the most general L_n-invariant equation is of the form

F(I1,I2,…,I7)=0.

Since this equation should be invariant under every basis element of the Witt algebra 𝔚₁, it should be independent of n. To meet this requirement the function F has to be independent of I₆ and I₇. Consequently, the most general 𝔚₁ invariant second-order PDE is

F(I1,I2,I3,I4,I5)=0,

or, equivalently,

F(x,u,ux,uxx,utxut)=0.

Furthermore, we have succeeded in constructing the general forms of PDEs invariant under 𝔚₂, 𝔚₆, 𝔚₈ and 𝔚₁₀. The corresponding invariant equations are listed in Table 1, where F is an arbitrary smooth real-valued function.

Symmetry algebra	Invariant equation
𝔚₁	F(x,u,ux,uxx,utxut)=0
𝔚₂	F(u, u_x, u_xx, e^−x(u_tu_xx − u_xu_tx)) = 0, α = 0
𝔚₂	F(u,uxx−uxux2,utux−utuxx+uxutx+ux2exux−2αux)=0, α = ±1
𝔚₆	F(u,ex(ux+uxx)−γ(uxx+utx)ux(exux−γ(ut+ux)))=0
𝔚₈	F(u,uxx−2uxux2)=0
𝔚₁₀	F(u_x + 2u, u_xx − 4u) = 0

Table 1.

Second-order PDEs admitting the Witt algebra

6. The Direct Sums of the Witt Algebras

This section is devoted to the classification of realizations of the direct sum of the Witt algebras in ℝ³. We obtain a complete description of inequivalent realizations of the direct sum of two Witt algebras.

In view of Theorem 3.1. it suffices to consider realizations of the form

𝔚i⊕〈L˜n,n∈𝕑〉, i=1,2,…,11,

where 𝔚_i are given in Theorem 3.1 and L˜n, n ∈ 𝕑 are basis elements of the Witt algebra commuting with the realization 𝔚_i.

Consider first realizations of direct sum of the Witt algebras 𝔚1⊕〈L˜n〉. Taking into account that L˜n commutes with basis elements of 𝔚₁, we conclude that

L˜n=fn(x,u)∂x+gn(x,u)∂u.(6.1)

Here f_n and g_n are arbitrary smooth functions.

Now we can utilize the results of classification of inequivalent realizations of the Witt algebra over ℝ² and get the final form of the basis elements L˜n from Theorem 3.3 by replacing t, x with x, u respectively.

The remaining realizations 𝔚_i (i = 2,...,11) are analyzed analogously. We skip intermediate computations and present the final result.

Theorem 6.1.

Any realization of the direct sum of two Witt algebras in ℝ³ is equivalent to one of the following ten inequivalent realizations:

𝔇₁ : 〈e^−mt∂_t〉 ⊕ 〈e^−nx∂_x〉,
𝔇₂ : 〈e^−mt∂_t〉 ⊕ 〈e^−nx∂_x + ne^−nx∂_u〉,
𝔇₃ : 〈e^−mt∂_t + me^−mt∂_x〉 ⊕ 〈ne^−nu∂_x + e^−nu∂_u〉,
𝔇₄ : 〈e^−mt∂_t〉 ⊕ 〈e^−nx∂_x + γe^−nx[e^nu − (e^u − γ)ⁿ](e^u − γ)¹⁻ⁿ∂_u〉,
𝔇₅ : 〈e−mt∂t〉⊕〈e−nx∂x+e−nx[n−sgn(n)γ2∑j=1|n|−1j(j+1)e−2u]∂u〉,
𝔇₆ : 〈e^−mt∂_t〉 ⊕ 〈e^{−nx+(n−1)u}(e^u ± n)(e^u ± 1)⁻ⁿ∂_x + ne^{−nx+(n−1)u} (e^u ± 1)¹⁻ⁿ∂_u〉,
𝔇₇ : 〈e−mt∂t〉⊕〈e−nx+(n−1)u[e2u−(n+1)γeu+12n(n+1)](eu−γ)−n−1∂x+e−nx+(n−1)u[neu−12n(n+1)γ](eu−γ)−n∂u〉,
𝔇₈ : 〈e^−mt∂_t〉 ⊕ 〈J₁∂_x + J₂∂_u〉,
𝔇₉ : 〈e−mt∂t〉⊕𝔚˜4,
𝔇₁₀ : 〈e−mt∂t〉⊕𝔚˜7.

Here

J1=e−nx+(n−1)u(eu−1)n+2[(−1+∑j=1|n|−1(2j+1)n+(2n+1)eu−(n+2)e2u+e3u+sgn(n)c2∑j=1|n|−1j(j+1))],J2=e−nx+(n−1)u(eu−1)n+1[(−1+∑j=1|n|−1(2j+1)n−2neu+ne2u−sgn(n)c2∑j=1|n|−1j(j+1))],

n, m ∈ 𝕑, c ∈ ℝ and the symbols 𝔚˜4 and 𝔚˜7 stand for the realizations obtained from 𝔚₄ and 𝔚₇ listed in Theorem 3.3 by replacing (t, x) with (x, u).

When the paper has been submitted for publication, one of the referees drew the authors’ attention to the paper by Medolaghi [44]. Medolaghi had constructed fourteen types of infinite-parameter groups of point transformations over ℝ³ and obtained corresponding invariant PDEs.

The majority of the realizations listed in Theorems 3.1–3.3 can be obtained from those given in in Sec. 4 of [44]. For example, the algebra 〈ξ(t)∂_t〉 reduces to 𝔚₁ under ξ(t) = e^−nt. Additionally, the realization 𝔚₂ with α = 0 can be derived from 〈ξ(t)∂t+ξ˙(t)∂x〉 given in [44] by choosing ξ(t) = e^−nt and x → −x. The realizations 𝔚₄, 𝔚₇ and 𝔚₉ are, to the best of our knowledge, new.

Medolaghi had not considered the problem of classifying PDEs invariant under sums of infinite-dimensional algebras. However, some of the realizations listed in Theorem 6.1 can be represented as the direct sums of infinite-dimensional algebras given in Sec. 4 of the paper [44]. For example, the realizations 𝔇₁ and 𝔇₂ can be obtained from 〈ξ₁(t)∂_t〉 ⊕ 〈ξ₂(x)∂_x〉 and 〈ξ1(t)∂t〉⊕〈ξ2(x)∂x+ξ˙2(x)∂u〉 by choosing ξ₁(t) = e^−mt and ξ₂(x) = e^−nx, respectively. The realization 𝔇₃ is a direct sum of 〈ξ1(t)∂t+ξ˙1(t)∂x〉 and 〈ξ2(u)∂u+ξ˙2(u)∂x〉 with ξ₁(t) = e^−mt, ξ₂(u) = e^−nu, x → −x. The realizations 𝔇₇–𝔇₁₀ are new.

Analysis of the second-order PDEs invariant under the direct sum of the Witt algebras shows that the determining equations are incompatible for the realizations 𝔇₄, 𝔇₅, 𝔇₇–𝔇₁₀. The remaining four realizations give rise to four classes of equations admitting symmetry algebras which are direct sums of two Witt algebras.

𝔇1:F(u,utxutux)=0,(6.2)

𝔇2:F(utxute−u)=0,(6.3)

𝔇3:F(utuxx−uxutxux3e−x)=0,(6.4)

𝔇6:F(utx(1−ux±eu)+ut(uxx−ux2+ux)ut[e2u+(ux−1)(ux−1∓2eu)])=0.(6.5)

Here F is an arbitrary smooth real-valued function.

Let us reiterate, any second-order PDE, in two independent variables, which is invariant under the direct sum of two Witt algebras, is equivalent to one of the equations (6.2)–(6.5).

We refer to these equations as integrable, since they admit infinite-dimensional symmetry algebras involving two arbitrary functions. This is similar to the concept of integrable solitonic equations, since an overwhelming majority of these equations admit infinite generalized symmetry [27]. One more analogy comes from the theory of Hamitonian systems, which are completely integrable provided a sufficient number of integrals are known [45].

Applying the results by Bluman and Kumei [7, 36] to an integrable PDE one can, in theory, linearize these PDEs. This fact justifies usage of the term integrable. Note that integrability in this sense is closely related to the concept of C-integrability by Calogero [9]. Nonlinear PDE is called C-integrable if it can be linearized by a local or nonlocal change of variables.

Eq. (6.2) can be rewritten in the equivalent form

utx=f(u)utux.

By making the change of variable u = R(U), where R(U) is an arbitrary solution of the ordinary differential equation

f(R)R′2−R″=0,

we reduce the above nonlinear PDE to the linear wave equation U_tx = 0.

Without any loss of generality we can rewrite (6.3) as follows

utx=λeuut, λ∈ℝ.

Integrating it with respect to t gives

ux=λeu+g″(x)g′(x),

where g is an arbitrary smooth non-constant function of x. We rewrite the above equation in the formx

(u−lng′(x))x=λe(u−lng′(x))elng′(x).

Integrating this equation yields the general solution

u(t,x)=lng′(x)h(t)−λg(x)

of the initial nonlinear PDE (6.3).

Eq. (6.4) is equivalent to the equation

utuxx−uxutx=λexux3, λ∈ℝ.

The hodograph transformation x → u, u → x followed by re-scaling t → λt turns it into the Liouville equation (1.1), which is known to be integrable.

To the best of our knowledge, Eq. (6.5) is a new integrable nonlinear equation. The simplest PDE from the class of equations (6.5) is of the form

utx(1−ux±eu)+ut(uxx−ux2+ux)=0.

7. Concluding Remarks

In this paper, we have classified all possible inequivalent realizations of the Witt and Virasoro algebras by Lie vector fields over the space ℝⁿ with n = 1, 2, 3. The complete lists of these realizations are given in Theorems 3.1–3.3, 4.1 and 6.1.

The main results can be briefly summarized as follows:

There exists only one inequivalent realization of the Witt algebra in ℝ.
There are nine inequivalent realizations of the Witt algebra in ℝ².
There exist eleven inequivalent realizations of the Witt algebra in ℝ³ space.
There are no realizations of the Virasoro algebra with nonzero central element over the space ℝⁿ with n ≤ 3.
There exist ten inequivalent realizations of the direct sum of two Witt algebras in ℝ³.

As an application, we construct a number of nonlinear PDEs which are invariant under various realizations of the Witt algebra. This enables constructing broad classes of new nonlinear equations whose symmetry algebra involves, at least, one arbitrary function of one variable.

We have constructed all second-order PDEs in two independent variables whose invariance algebras contain a direct sum of the Witt algebras. As a result we get four canonical invariant equations (6.2)–(6.5). Each of these admits infinite-dimensional algebra involving two arbitrary functions. The massless wave and Liouville equations in (1+1)-dimensions are typical examples of such PDEs. They are particular cases of Eqs. (6.2)–(6.4), which are well-known, while model (6.5) is new.

Since the Virasoro algebra is a subalgebra of the Kac-Moody-Virasoro algebra, the results obtained here can be directly applied to classify the integrable KP type equations in (1 + 2) dimensions. The starting point would be describing inequivalent realizations of the Kac-Moody-Virasoro algebras by differential operators over ℝ⁴. This problem is in progress now and will be reported elsewhere.

Acknowledgments

Qing Huang is supported by National Natural Science Foundation of China (Grant Nos. 11871396 and 11771352) and Natural Science Foundation of Shaanxi Province of China (Grant No. 2018JM1005).

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 27 - 1
Pages: 36 - 56
Publication Date: 2019/10/25
ISSN (Online): 1776-0852
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TY  - JOUR
AU  - Qing Huang
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PY  - 2019
DA  - 2019/10/25
TI  - Realizations of the Witt and Virasoro Algebras and Integrable Equations
JO  - Journal of Nonlinear Mathematical Physics
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EP  - 56
VL  - 27
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