Symmetry Classification of Third-Order Nonlinear Evolution Equations. Part I: Semi-simple Algebras

Abstract

We give a complete point-symmetry classification of all third-order evolution equations of the form u _t =F(t,x,u,u _x ,u _xx )u _xxx +G(t,x,u,u _x ,u _xx ) which admit semi-simple symmetry algebras and extensions of these semi-simple Lie algebras by solvable Lie algebras. The methods we employ are extensions and refinements of previous techniques which have been used in such classifications.

Appendices

Appendix A: Representations of so(3,ℝ) and sl(2,ℝ)

$\operatorname {sl}(2, \mathbb{K})$ with $\mathbb {K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$: The irreducible finite-dimensional representations of $\operatorname {sl}(2, \mathbb{K})$ are well-known (see for instance [22]). An irreducible representation space is defined by a half-integer $J=0,\frac{1}{2}, 1, \frac{3}{2}, 2,\ldots$ and a vector space k _J =〈Q ₁,…,Q _2J+1〉 and dimk _J =2J+1. Denoting by e _k ⋅Q _l the representation of e _k on Q _l , we have the relations

$$Q_k=\frac{e_3^{k-1}}{(k-1)!}\cdot Q_1,\quad e_3 \cdot Q_{2J+1}=0, $$

as well as

$$Q_{2J+1-k}=\frac{e_2^k}{k!}\cdot Q_{2J+1},\quad e_2\cdot Q_1=0, $$

and

$$e_1\cdot Q_k=2(J-k+1)Q_k. $$

These relations hold for both real and complex vector spaces k _J .

$\operatorname {so}(3, \mathbb{R})$: $\operatorname {so}(3, \mathbb {R})=\langle e_{1}, e_{2}, e_{3}\rangle$ is defined by the commutation relations $[e_{1}, e_{2}]=e_{3},\; [e_{2}, e_{3}]=e_{1},\; [e_{3}, e_{1}]=e_{2}$. In this case, we use the trick of complexifying. If V is the real irreducible representation space of $\operatorname {so}(3, \mathbb{R})$, then define the complex vector space W=V+iV as the space spanned by vectors u+iv with u,v∈V. Next define $h=2ie_{1},\; x=e_{2}+ie_{3},\; y=-e_{2}+ie_{3}$. Then 〈h,x,y〉 is just $\operatorname {sl}(2, \mathbb {C})$ and it acts irreducibly on W. So we have W=〈Q ₁,…,Q _2J+1〉 and [h,Q _2J−k+1]=−2(J−k)Q _2J−k+1 for k=0,1,…,2J. Further, [y,Q _2J+1]=0 and we also have [h,Q _2J+1]=−2JQ _2J+1. With Q _2J+1=X+iY for some X,Y∈V, we then find from [h,Q _2J+1]=−2JQ _2J+1 that $[e_{1}, X]=-JY,\; [e_{1}, Y]=JX$. Note also, that if J is an integer, then there exists a vector Z∈V with [e ₁,Z]=0 (this follows from [h,Q _2J−k+1]=−2(J−k)Q _2J−k+1). Thus, in any irreducible representation of $\operatorname {so}(3, \mathbb{R})$ on a real Lie algebra A (whether abelian or not) we have dimA=2J+1 for some half-integer J, non-zero vectors X,Y with $[e_{1}, X]=\alpha Y, \; [e_{1}, Y]=-\alpha X$ for α>0 for any J>0 and non-zero vectors Z with [e ₁,Z]=0 if J is an integer.

Finally, we note that $\operatorname {so}(3, \mathbb{R})$ has no two-dimensional irreducible representations: in fact if this were the case then there would be a representation by three real 2×2 trace-free matrices of 〈e ₁,e ₂,e ₃〉. These would then form a basis for $\operatorname {sl}(2, \mathbb{R})$, contradicting the commutation relations for $\operatorname {so}(3, \mathbb{R})$.

Appendix B: Proof of Theorem 2.1

In this appendix we give a proof of Theorem 2.1. To this end we begin with some informal preliminaries on contact structures.

We work on k-th order jet spaces J ^k(ℝⁿ,ℝ^m), with local coordinates $(x,u,\underset{(1)}{u}, \underset {(2)}{u},\ldots, \underset{(k)}{u})$ where x=(x ¹,…,x ⁿ), u=(u ¹,…,u ^m) and where $\underset{(j)}{u}$ stands for the collection of all the j-th order partial derivatives of (u ¹,…,u ^m) with respect to the (x ¹,…,x ⁿ). On these spaces we introduce the contact one-forms

for l=1,…,m and μ _i =1,…,n, and we sum over repeated indices. The symbols $u^{l}_{\mu_{i}},\ldots, u^{l}_{\mu_{1}\cdots\mu_{k}}$ are then the partial derivatives of the functions u ^l. These one-forms vanish on solutions of differential equations. They are known as the Cartan distribution (see [23]). Note that a k-th order contact form

$$\omega^l_{\mu_1\cdots\mu_k}=du^l_{\mu_1\cdots\mu_k}-u^l_{\mu_1\cdots\mu _k\mu_{k+1}}dx^{\mu_{k+1}} $$

is not a properly defined form on J ^k(ℝⁿ,ℝ^m), because the $u^{l}_{\mu_{1}\cdots\mu_{k}\mu_{k+1}}$ are coordinates on J ^k+1(ℝⁿ,ℝ^m). To remedy this, we may work on the infinite jet bundle J ^∞(ℝⁿ,ℝ^m) [23–25], which may be thought of as a space of infinite sequences $(x, u, \underset {(1)}{u},\ldots,\underset{(k)}{u},\ldots)$, and we have projections $\pi^{\infty}_{k}: J^{\infty}(\mathbb{R}^{n}, \mathbb{R}^{m})\to J^{k}(\mathbb {R}^{n}, \mathbb{R}^{m})$ as well as projections $\pi^{k}_{l}: J^{k}(\mathbb {R}^{n}, \mathbb{R}^{m})\to J^{l}(\mathbb{R}^{n}, \mathbb{R}^{m})$ for l≤k. In terms of the sequences $(x, u, \underset{(1)}{u},\ldots,\underset {(k)}{u},\ldots)$, we have $\pi^{\infty}_{k}(x, u, \underset{(1)}{u},\ldots ,\underset{(k)}{u},\ldots)=(x, u, \underset{(1)}{u},\ldots,\underset {(k)}{u})$ and $\pi^{k}_{l}(x, u, \underset{(1)}{u},\ldots,\underset {(k)}{u})=(x, u, \underset{(1)}{u}, \ldots,\underset{(l)}{u})$. The infinite jet space J ^∞(ℝⁿ,ℝ^m) is well-defined, as the inverse limit of the k-th order jet bundles J ^k(ℝⁿ,ℝ^m) (see [23–25]). A function F:J ^∞(ℝⁿ,ℝ^m)→ℝ is then defined to be smooth if F is a smooth function of only $(x, u, \underset {(1)}{u},\ldots,\underset{(k)}{u})$ for some natural number k∈ℕ. We work within this infinite jet bundle unless otherwise stated.

A tangent transformation is then a transformation

$$\bigl(x, u, \underset{(1)}{u},\ldots, \underset{(k)}{u} \bigr)\to \bigl(x', u', \underset {(1)}{u'}, \ldots, \underset{(k)}{u'} \bigr) $$

for any k≥1 such that the transformed contact forms

vanish when $\omega^{l},\, \omega^{l}_{\mu_{i}},\ldots, \omega^{l}_{\mu_{1}\cdots\mu _{k}}$ vanish (see [13]). Then the following result holds [13]:

Theorem B.1

A tangent transformation is given by a prolongation of a point transformation

$$(x,u)\to \bigl(x', u' \bigr) $$

if u=(u ¹,…,u ^m) for m≥2. It is given by the prolongation of a contact transformation

$$\bigl(x,u, \underset{(1)}{u} \bigr)\to \bigl(x',u', \underset{(1)}{u'} \bigr) $$

if m=1, and then there is a (locally smooth) function $W(t,x,u, \underset{(1)}{u})$ such that

$$x'^l=-W_{q^l},\qquad u'=W-pW_p-q^lW_{q^l}, \qquad u'_{x'^l}=W_{x^l} + q^lW_u $$

with $q^{l}=u_{x^{l}},\; l= 1,\ldots, k$.

Now let us consider smooth, real-valued functions $F(x,u,\underset {(1)}{u},\ldots, \underset{(k)}{u})$. Then one can show [23, 25] that the exterior derivative dF may be written as

$$dF = DF + \sum_{j=1}^{k} \varLambda_l^{\mu_1\cdots\mu_j}\omega^l_{\mu_1\cdots \mu_j} $$

for some functions $\varLambda_{l}^{\mu_{1}\cdots\mu_{j}}$, when we work on the infinite jet bundle, where we sum over repeated indices. Here we have

$$DF=D_{x^1}Fdx^1 + \cdots+ D_{x^n}Fdx^n $$

and $D_{x^{i}}$ is the operator of total differentiation with respect to x ⁱ:

$$D_{x^{\mu_i}}F=F_{x^i} + u^l_{\mu_i}F_{u^l} + u^l_{\mu_i\mu}F_{u^l_{\mu }}+\cdots+ u^{l}_{\mu_i\mu_1\cdots\mu_k}F_{u^{l}_{\mu_1\cdots\mu_k}}. $$

From this, it follows easily that if $D_{x^{i}}F=0$ then $F_{u^{l}}=F_{u^{l}_{\mu_{1}\cdots\mu_{j}}}=0$ for l=1,…,m and j=1,…,k, and then DF=0 implies $F=\operatorname {constant}$, and so DF=0 if and only if dF=0. A further result in this direction is the following:

Lemma B.1

Suppose that

$$\bigl(x,u,\underset{(1)}{u},\ldots, \underset{(k)}{u} \bigr)\to \bigl(x',u',\underset {(1)}{u'},\ldots, \underset{(k)}{u'} \bigr) $$

defines an invertible tangent transformation, with x′¹=X ¹,…,x′ⁿ=X ⁿ, then we have

$$DX^1\wedge\cdots\wedge DX^n\neq0. $$

Proof

There are two cases: if u=(u ¹,…,u ^m) with m≥2, then the transformation is the prolongation of an invertible point transformation (x,u)→(X(x,u),U(x,u)). Thus we have dX ¹∧⋯∧dX ⁿ∧dU ¹∧⋯∧dU ^m≠ by invertibility.

Now suppose that DX ¹∧⋯∧DX ⁿ=0. We know from the previous remarks that DX ⁱ≠0 for i=1,…,n. Hence DX ¹∈I(DX ²,…,DX ⁿ) where I(DX ²,…,DX ⁿ) is the module generated by the forms DX ²,…,DX ⁿ, that is, I(DX ²,…,DX ⁿ) consists of linear combinations of the form λ ₂ DX ²+⋯+λ _n DX ⁿ for some functions λ ₂,…,λ _n . Now each X ⁱ is a function of (x ¹,…,x ⁿ,u ¹,…,u ^m) and then we have

$$dX^i=DX^i+\varLambda_l\omega^l. $$

From this it now follows that dX ¹∈I(dX ²,…,dX ⁿ,ω ¹,…,ω ^m). Further, since the transformation is a tangent transformation, we have $dU^{l}-U^{l}_{i}dX^{i}=\lambda^{l}_{j}\omega^{j}$ for l=1,…,m, so that dU ^l∈I(dX ¹,dX ²,…,dX ⁿ,ω ¹,…,ω ^m) and consequently dU ^l∈I(dX ²,…,dX ⁿ,ω ¹,…,ω ^m) since we know that dX ¹∈I(dX ²,…,dX ⁿ,ω ¹,…,ω ^m). Hence each of the m+n one-forms dX ¹,…,dX ⁿ,dU ¹,…,dU ^m is a sum of m+n−1 one-forms, so that dX ¹∧⋯∧dX ⁿ∧dU ¹∧⋯∧dU ^m=0, which contradicts the invertibility of the transformation.

The case of just one function is treated similarly: in this case the tangent transformation is a prolongation of an invertible contact transformation

$$\bigl(x^1,\ldots, x^n, u, u_1,\ldots u_n \bigr)\to \bigl(x'^1,\ldots, x'^n, u', u'_1, \ldots u'_n \bigr), $$

where $u_{i}=u_{x^{i}}$, which is a particular case of a point transformation with n+1 functions u,u ₁,…,u _n . Hence we must have DX ¹∧⋯∧DX ⁿ≠0.

We now come to the proof of Theorem 2.1: □

Theorem B.2

Any invertible contact transformation

$$(t,x,u,p,q)\to \bigl(t', x', u', p', q' \bigr) $$

with $p=u_{t},\;\; q=u_{1}$ and $p'=u'_{t'},\;\, q'=u'_{x'}$, preserving the form of an evolution equation of order n

$$u_0=F(t,x,u,u_1,\ldots, u_n), $$

with n≥2, is such that t′=T(t) with $\dot{T}(t)\neq0$. Then the contact transformation has the form

for some smooth function W.

Proof

Our contact transformation is

and there is a function W(t,x,u,p,q) so that

(see [13]). An evolution equation of order n+1 becomes in our notation

$$p=F(t, x, u, q, q_1,\ldots, q_n) $$

where $q_{1}=D_{x}q, q_{2}=D^{2}_{x}q,\ldots, q_{n}=D^{n}_{x}q$. We shall show that D _t T≠0, D _x T=0, from which it follows that T=T(t).

We define the sequence of functions Q ^k, k≥0, by Q ⁰=Q and for k≥1 they are given by

$$DT\wedge DQ^k=Q^{k+1}DT\wedge DX. $$

The Q ^k are well-defined since we have DT∧DX≠0 by Lemma B.1. These functions are just the transformations of the functions q _k =∂ ^k q/∂x ^k defined by the contact conditions:

$$dQ^k-Q^{k+1}dX - Q^k_0dT=0\quad \text{mod}\ I $$

where $Q^{k}_{0}=D_{T}Q^{k}$, Q ^k+1=D _X Q ^k and where I is the module of contact one-forms. From the fact that dF=DF mod I we obtain $DQ^{k}-Q^{k+1}DX-Q^{k}_{0}DT=0$ on imposing the contact conditions. Then DT∧DQ ^k=Q ^k+1 DT∧DX follows easily.

Then we make the substitution $q'_{l}=Q^{l}$ in the evolution equation

$$p'=F' \bigl(t',x',u',q',q'_1 \ldots, q'_n \bigr). $$

The result is to be another evolution equation

$$p=F(t,x,u,q,q_1,\ldots, q_n) $$

so that the right-hand side contains the highest spatial derivative $q_{n}=D^{n}_{x}q$ and none of the derivatives $p_{n}=D^{n}_{x} p,\; p_{0}^{n}=D^{n}_{t}p,\; q_{0}^{n}=D^{n}_{t}q$ nor the mixed derivatives $q_{k,l}=D^{k}_{t}D^{l}_{x} q$ for k≥1, and $p_{k, l}=D^{k}_{t}D^{l}_{x} p$. Note that since $p=D_{t}u,\; q=D_{x} u$ we have p _k,l =q _k+1,l−1 and in particular q _1,0=q ₀=p _0,1=p ₁. Thus, we require that $Q^{n}_{p_{0}^{n}}=0$ and $Q^{n}_{p_{n}}=0$ as well as $Q^{n}_{q_{n}}\neq0$.

We now show that D _t T≠0. For if D _t T=0 then T=T(x). Then the relation Q ^k+1 DT∧DX=DT∧DQ ^k gives Q ^k+1 T′(x)D _t Xdt∧dx=D _t Q ^k T′(x)dt∧dx, so that $Q^{k+1}=\frac{D_{t}Q^{k}}{D_{t}X}$. We have for k=0 that $Q^{1}=\frac {D_{t}Q}{D_{t}X}$ which shows that Q ¹=Q ¹(t,x,u,p,q,p ₀,q ₀). Thus, by induction, we find that Q ⁿ is independent of the spatial derivatives of $p_{n},\; q_{n}$, and hence we contradict the requirement that $Q^{n}_{p_{n}}\neq0$. Consequently we must have D _t T≠0.

It follows from the fact that D _t T≠0 that $Q^{1}_{q_{1}}\neq0$: in fact, we note that if $Q^{k}_{q_{k}}\neq0$ then DQ ^k will depend linearly on q _k+1 and this appears only in the term D _x Q ^k as the coefficient of $Q^{k}_{q_{k}}$. So, differentiating Q ^k+1 DT∧DX=DT∧DQ ^k with respect to q _k+1 we find that $Q^{k+1}_{q_{k+1}}DT\wedge DX=Q^{k}_{q_{k}}D_{t}Tdt\wedge dx$. Since we have $Q^{n}_{q_{n}}\neq0$, D _t T≠0, DT∧DX≠0, it follows, by induction, that $Q^{1}_{q_{1}}\neq0$.

Our next step is to show that D _x T=0 and to this end we assume that D _x T≠0. Then $Q^{1}_{p_{0}}=0$. In fact, if $Q^{k}_{p^{k}_{0}}\neq0$ then DQ ^k will be linear in $p^{k+1}_{0}$, and this will appear in the coefficient of $Q^{k}_{p^{k}_{0}}$ in D _t Q ^k. So, differentiating $Q^{k+1}_{q_{k+1}}DT\wedge DX=Q^{k}_{q_{k}}D_{t}Tdt\wedge dx$ with respect to $p^{k+1}_{0}$ we find that $Q^{k+1}_{p^{k+1}_{0}}DT\wedge DX=-Q^{k}_{p^{k}_{0}}D_{x}Tdt\wedge dx$. Thus, if D _x T≠0 then $Q^{1}_{p_{0}}\neq0$ implies, by induction, that $Q^{n}_{p^{n}_{0}}\neq0$, and this contradicts our requirement $Q^{n}_{p^{n}_{0}}=0$. Hence $Q^{1}_{p_{0}}=0$.

Another consequence of D _x T≠0 is that $Q_{q_{0}}=Q_{p_{1}}=0$. For suppose $Q^{1}_{q_{0}}\neq0$. We know that $Q^{k}_{p_{0}^{k}}=0$ so that D _x Q ^k is independent of $q^{k+1}_{0}=D_{x}p^{k}_{0}$, and therefore, from Q ^k+1 DT∧DX=DT∧DQ ^k it follows that $Q^{k+1}_{q^{k+1}_{0}}DT \wedge DX=-Q^{k}_{q^{k}_{0}}D_{x}Tdt\wedge dx$ for k≥1. So, if $Q^{1}_{q_{0}}\neq0$, then $Q^{n}_{q^{n}_{0}}\neq0$, as follows from induction. Since we require $Q^{n}_{q^{n}_{0}}= 0$, we must have $Q^{1}_{q_{0}}=0$, because we also assume D _x T≠0. Note that q ₀=D _t q=D _t D _x u=D _x D _t u=D _x p=p ₁, and so we also have $Q^{1}_{p_{1}}=0$.

From all this we find that Q ¹=Q ¹(t,x,u,p,q,q ₁) and it then follows that Q ⁿ will contain the n-th order derivative $q_{n-1, 1}=D^{n-1}_{t}D_{x}q$ if D _x T≠0. In fact, putting $q_{k-1, 1}=D^{k-1}_{t}D_{x}q$ for k≥1, we have from Q ^k+1 DT∧DX=DT∧DQ ^k that

$$Q^{k+1}_{q_{k, 1}}DT\wedge DX=-Q^{k}_{q_{k-1, 1}}D_xTdt \wedge dx $$

for k≥1, since q _k,1 is a derivative of order k+1 which can only occur in D _t Q ^k. Now q _0,1=q ₁ and we know that $Q^{1}_{q_{1}}\neq0$, so it follows by induction that $Q^{n}_{q_{n-1, 1}}\neq0$ if D _x T≠0. However, this is a contradiction since we must have $Q^{n}_{q_{n-1, 1}}=0$ for our transformations to transform any given evolution equation into an evolution equation. Hence we conclude that D _x T=0 and it then follows easily that T=T(t).

Finally, noting that

for some function W(t,x,u,p,q), we find, on integrating this system, that the contact transformation has the stated form. This proves the result.