The Diffiety Institute Preprint Series

Preprint DIPS–4/2002 June 18, 2004

Available via INTERNET: http://diffiety.ac.ru; http://diffiety.org

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Towards classification of conditionally integrable evolution

systems in (1 + 1) dimensions

by

A. Sergyeyev

TOWARDS CLASSIFICATION OF CONDITIONALLY INTEGRABLE

EVOLUTION SYSTEMS IN (1 + 1) DIMENSIONS

A. SERGYEYEV

Abstract. We generalize earlier results of Fokas and Liu and find all locally analytic (1+1)-dimensional evolution equations of order n that admit an N-shock type solution with N ≤ n + 1.

To this end, we develop a refinement of the technique from our earlier work (A. Sergyeyev, J. Phys. A: Math. Gen, 35 (2002), 7653–7660), where we completely characterized all (1+1)-dimen-sional evolution systems u_{t} = F (x, t, u, ∂u/∂x, …, ∂^{n}u/∂x^{n}) that are conditionally invariant under a given generalized (Lie– Bäcklund) vector field Q(x, t, u, ∂u/∂x, …, ∂^{k}u/∂x^{k})∂/∂u under the assumption that the system of ODEs Q = 0 is totally nondegenerate. Every such conditionally invariant evolution system admits a reduction to a system of ODEs in t, thus being a nonlinear counterpart to quasi-exactly solvable models in quantum mechanics.

Introduction. In order to be relevant, mathematical models of natural phenomena are often required to have solutions of prescribed form (e.g. travelling wave solutions, kinks, solitons, etc.). As was first shown by Fokas and Liu [4, 5] (see also Zhdanov [15]), a natural way to achieve this for evolution systems in (1+1) dimensions is to require that the system in question admit a generalized conditional symmetry (GCS) such that the solutions invariant under this symmetry have the desired form. It turns out that the set of systems admitting GCS (such systems, or even more broadly, the systems possessing integrable reductions, are often referred to as conditionally integrable, see e.g. [9]) is considerably larger than that of systems integrable via the inverse scattering transform or direct linearization, see e.g. [4, 5]. Notice [16]

2000 Mathematics Subject Classification. 35A30, 35G25, 81U15, 35N10, 37K35, 58J70, 58J72, 34A34.

Key words and phrases. Exact solutions, nonlinear evolution equations, conditional integrability, generalized symmetries, reduction, generalized conditional symmetries.

The research was partially supported by the Czech Grant Agency grant No. 201/04/0538 and the Ministry of Education, Youth and Sports of Czech Republic grant MSM:J10/98:192400002.

This paper is submitted for publication in the Issue on Geometry of PDE in: Fundamental’naya i Prikladnaya Matematika / Jour. of Math. Sci. (2004).

that if an (1+1)-dimensional evolution equation admits a reduction to a system of ODEs in the evolution parameter (time t), then it admits a GCS, and vice versa.

This naturally leads to the following problem: how to describe all evolution systems that admit a given GCS? For the case of linear GCS with time-independent coefficients this problem, restated in the terms of the so-called invariant modules, was completely solved in the seminal paper [6] (see also an important earlier work [13]). For the nonlinear GCS, some partial results were obtained in [2], and the complete solution of the problem was obtained in [12] under certain nondegeneracy assumptions, see below for details; see also [6] and [12] for the survey of earlier results in the field and the discussion of the role played by the GCS in the search of exact solutions of nonlinear PDEs.

However, the formulas found in [12] are not very convenient for applications. In the present paper we obtain alternative, easier to use, formulas for the locally analytic (1+1)-dimensional evolution systems that are conditionally invariant under a given analytic GCS. This is done in Section 3. In Section 4 we illustrate the application of our results by a number of examples. In particular, we completely characterize all locally analytic (1+1)-dimensional evolution equations of order n that admit an N-shock type solution with N ≤ n + 1.

1. Preliminaries

Consider an evolution system

∂u/∂t = F (x, t, u, u_{1}, …, u_{n}), n ≥ 0,  (1)

for an s-component vector function u = (u^{1}, …, u^{s})^{T}, where u_{l} = ∂^{l}u/∂x^{l}, l = 0, 1, 2, …, u_{0} ≡ u, and the superscript ‘T ’ denotes the matrix transposition.

A smooth function of x, t, u, u_{1}, u_{2}, … is called local (see [7]; cf. also [14] and [8]) if it depends on a finite number of u_{j}. The greatest integer m such that ∂f/∂u_{m} ≠ 0 is called the order of a differential function f, and we shall write that as m = ord f. If f depends only on x and t, then we shall assume that ord f = 0. Unless otherwise explicitly stated, all functions considered below will be assumed to be local.

A generalized vector field Q = Q ∂/∂u, where Q is an s-component local vector function is called [4, 5, 15] a generalized conditional symmetry (GCS) for (1) if the system Q = 0 is compatible with (1).

Eq.(1) is compatible with Q = 0 if and only if (cf. e.g. [4, 15])

D_{t}(Q)|_{M} = 0,  (2)

where D_{t} = ∂/∂t+∑_{i=0}^{∞} D^{i}(F)∂/∂u_{i} and D = ∂/∂x+∑_{i=0}^{∞} u_{i+1}∂/∂u_{i} are total t- and x-derivatives, and M is the solution manifold of the system Q = 0.

In what follows we shall treat Q = 0 as a system of ODEs involving an extra parameter t, and assume that this system is totally nondegenerate, i.e., the systems D^{j}(Q) = 0 are locally solvable and have maximal rank for all j = 0, 1, 2, …, see Ch. 2 of [8] for more details on this. Then the condition (2) is equivalent [8] to the following: there exist s-component local vector functions η_{α,j} and an integer p such that

D_{t}(Q) = ∑_{α=1}^{s} ∑_{j=0}^{p} η_{α,j}D^{j}(Q^{α}).  (3)

Finding all GCS admitted by a given system (1) is an extremely difficult task, whose complexity is comparable to that of the problem of finding all solutions for the system in question. Surprisingly enough, we succeeded [12] in completely solving the inverse problem of describing all systems (1) that admit a given GCS Q = Q∂/∂u, provided the system Q = 0, considered as a system of ODEs, is totally nondegenerate.

Let us briefly review the results of [12]. Consider a system of ODEs

Q(x, t, u, u_{1}, …, u_{k}) = 0,  (4)

involving t as a parameter. Here Q = (Q^{1}, …, Q^{s})^{T} is an s-component local vector function.

Assume that the general solution of (4) in implicit form can be written as

G(x, t, u, c_{1}(t), …, c_{N}(t)) = 0.  (5)

The number N is sometimes called the total order of (4). It is tacitly assumed here that G depends on N arbitrary functions c_{i}(t), i = 1, …, N, in an essential way, and that det ∂G/∂u ≠ 0.

Then using the implicit function theorem we can, at least locally, obtain the general solution in the explicit form as u = P (x, t, c_{1}(t), …, c_{N}(t)), whence

u_{j}^{α} = ∂^{j}P^{α}(x, t, c_{1}(t), …, c_{N}(t))/∂x^{j}.

Using these equations we can express c_{i} as functions of x, t, u^{1}, …, u_{n_{1}−1}^{1}, …, u^{s}, …, u_{n_{s}−1}^{s} for some numbers n_{1}, …, , n_{s}:

c_{i} = h_{i}(x, t, u^{1}, …, u_{n_{1}−1}^{1}, …, u^{s}, …, u_{n_{s}−1}^{s}), i = 1, …, N.

Let B̃_{i} = − (∂G/∂u)^{−1} ∂G/∂c_{i}, R̃ = − (∂G/∂u)^{−1} ∂G/∂t, B_{i} = B̃_{i}(x, t, h_{1}, …, h_{N}), and R(x, t, u, u_{1}, …, u_{r}) ≡ R̃(x, t, h_{1}, …, h_{N}), i.e., B_{i} and R are obtained from B̃_{i} and R̃ by substituting h_{i} for c_{i}. Here it is understood that c_{i} are not differentiated with respect to t while evaluating ∂G/∂t.

Let V be an open domain in the space V of variables x, t, u, u_{1}, …, and let W be the set of all points in V satisfying the equations D^{j}(Q) = 0, j = 0, 1, 2, …, considered as algebraic equations.

Theorem 1 ([12]). Suppose that Q(x, t, u, …, u_{k}) = 0, considered as a system of ODEs, is analytic on V, totally nondegenerate on W, and has the same total order N on the whole of W.

If Q = 0 is compatible with (1), i.e., Q∂/∂u is a generalized conditional symmetry for (1), then F on V can be represented in the form

\begin{matrix}F = R + ∑_{i=1}^{N} ζ_{i}(t, h_{1}, …, h_{N})B_{i} \\ + ∑_{p=0}^{m} ∑_{α=1}^{s} χ_{p,α}(t, x, u, …, u_{j_{p,α}})D^{p}(Q^{α}), \end{matrix}  (6)

where m and j_{p,α} are nonnegative integers, and ζ_{i} and χ_{p,α} are smooth functions of their arguments.

By construction [12], the system u_{t} = F with F given by (6) admits a solution of the same form as the general solution (5) of Q = 0, that is,

G(x, t, u, c_{1}(t), …, c_{N}(t)) = 0,

but now the c_{i}(t) must satisfy

dc_{i}/dt = ζ_{i}(t, c_{1}, …, c_{N}), i = 1, …, N,  (7)

rather than be arbitrary functions of t.

In other words, under the assumptions of Theorem 1 if Q∂/∂u is a GCS for u_{t} = F, then the substitution of the general solution (5) of Q = 0 into u_{t} = F reduces the latter to the system of ODEs (7).

2. Solving the inverse problem: which systems admit a given GCS?

How to pick among the F ’s (6) those of order ≤ n, where n is a given natural number? This can be done using the following result.

Theorem 2. Under the assumptions of Theorem 1 on Q, suppose that

Q^{α} = u_{n_{α}}^{α} − g^{α}(x, t, ũ),  (8)

where g^{α} are analytic functions of their arguments and

ũ = (u^{1}, …, u_{n_{1}−1}^{1}, …, u^{s}, …, u_{n_{s}−1}^{s}).

Then the most general locally analytic F of order n ≥ max(ord R, max_{i=1,...,N} ord B_{i}, max_{j=1,...,N} ord h_{j}) such that (1) is compatible with Q = 0

can be locally written as

F = R + ∑_{i=1}^{N} ζ_{i}(t, h_{1}, …, h_{N})B_{i} + ∑_{α=1}^{s} ∑_{m=0}^{n−n_{α}} D^{m}(Q^{α})K_{α,m}(x, t, ũ, Q̃),

(9)

where

Q̃ = (Q^{1}, D(Q^{1}), …, D^{n−n_{1}}(Q^{1}), …, Q^{s}, D(Q^{s}), …, D^{n−n_{s}}(Q^{s})), ζ_{i} and K_{α,m} are arbitrary locally analytic functions of their arguments, N = ∑_{α=1}^{s} n_{α}, and for n < n_{α} K_{α,m} ≡ 0.

Proof. We have u_{n_{α}}^{α} = Q^{α}+g^{α}(x, t, ũ). Using this equality and proceeding inductively with usage of the formulas found at the previous steps, we obtain u_{j}^{α} = ψ_{j}^{α}(x, t, ũ, Q^{1}, …, D^{j−n_{1}}(Q^{1}), …, Q^{s}, …, D^{j−n_{s}}(Q^{s})) for j ≥ max_{α} n_{α} and similar formulas for max_{α} n_{α} > j > n_{α}.

Plug the above formulas for u_{j}^{α} into (6) and expand the resulting expression in Taylor series with respect to D^{j}(Q^{α}). As the order of F is n and F is locally analytic, it must be independent of D^{j}(Q^{α}) with j > n − n_{α}. Taking this into account, we can rewrite (6) in the form (9), and the result follows. □

It is easily seen that the most general locally analytic F of order

n < ñ ≡ max(ord R, max_{i=1,...,N} ord B_{i}, max_{j=1,...,N} ord h_{j}) such that u_{t} = F

admits a given GCS Q∂/∂u meeting the requirements of Theorem 2 also has the form (9) with n replaced by ñ and K_{α,m} and ζ_{i} subjected to the extra conditions ∂F /∂u_{j} = 0 for j > n.

Note that for Q = u_{k} − g(x, t, u, …, u_{k−1}) we have N = s · k, and (9) boils down to

\begin{matrix}F = R + ∑_{i=1}^{N} ζ_{i}(t, h_{1}, …, h_{N})B_{i} \\ + ∑_{α=1}^{s} ∑_{m=0}^{n−k} D^{m}(Q^{α})K_{α,m}(x, t, u, u_{1}, …, u_{k−1}, Q, D(Q), …, D^{n−k}(Q)). \end{matrix}

(10)

3. Examples

Example 1. Let

Q^{α} = u_{n_{α}}^{α} − ∑_{β=1}^{s} ∑_{j=0}^{n_{β}−1} g_{β,j}^{α}(x, t)u_{j}^{β}, α = 1, …, s,  (11)

and hence N = ∑_{α=1}^{s} n_{α}. Then a general solution of Q = 0 reads u = ∑_{i=1}^{N} c_{i}(t) · f_{i}(x, t), where c_{i}(t) are arbitrary functions of t and f_{i} ≡ (f_{i}^{1}, …, f_{i}^{s})^{T} are linearly independent solutions of Q = 0.

We have [10, 11] h_{i} = Z_{i}/Z, where

Z = | \begin{matrix}f_{1}^{1} & … & f_{i}^{1} & … & f_{N}^{1} \\ ∂f_{1}^{1}/∂x & … & ∂f_{i}^{1}/∂x & … & ∂f_{N}^{1}/∂x \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ ∂^{n_{1}−1}f_{1}^{1}/∂x^{n_{1}−1} & … & ∂^{n_{1}−1}f_{i}^{1}/∂x^{n_{1}−1} & … & ∂^{n_{1}−1}f_{N}^{1}/∂x^{n_{1}−1} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ f_{1}^{s} & … & f_{i}^{s} & … & f_{N}^{s} \\ ∂f_{1}^{s}/∂x & … & ∂f_{i}^{s}/∂x & … & ∂f_{N}^{s}/∂x \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ ∂^{n_{s}−1}f_{1}^{s}/∂x^{n_{s}−1} & … & ∂^{n_{s}−1}f_{i}^{s}/∂x^{n_{s}−1} & … & ∂^{n_{s}−1}f_{N}^{s}/∂x^{n_{s}−1} \end{matrix} |

and Z_{i} are obtained from Z by replacing ∂^{j}f_{i}^{α}/∂x^{j} by u_{j}^{α}. It is easily seen that B_{i} = f_{i}(x, t), and R = ∑_{i=1}^{N}(Z_{i}/Z)∂f_{i}(x, t)/∂t.

If g_{β,j}^{α} are analytic in x and t, then by Theorem 2 the most general locally analytic F of order n ≥ max_{i} ord Z_{i}/Z that admits a GCS Q∂/∂u with Q (11) locally has the form

F = ∑_{i=1}^{N} \frac{Z_{i}}{Z} \frac{∂f_{i}(x, t)}{∂t} + ∑_{i=1}^{N} ζ_{i}(t, Z_{1}/Z, …, Z_{N}/Z)f_{i}(x, t)

+ ∑_{α=1}^{s} ∑_{m=0}^{n−n_{α}} D^{m}(Q^{α})K_{α,m}(x, t, ũ, Q̃),  (12)

where ζ_{i} and K_{α,m} are arbitrary locally analytic functions of their arguments,

Q̃ = (Q^{1}, D(Q^{1}), …, D^{n−n_{1}}(Q^{1}), …, Q^{s}, D(Q^{s}), …, D^{n−n_{s}}(Q^{s})),
ũ = (u^{1}, …, u_{n_{1}−1}^{1}, …, u^{s}, …, u_{n_{s}−1}^{s}),

and K_{α,m} ≡ 0 for n < n_{α}.

In the rest of this section we assume that s = 1, so for simplicity we shall write u ≡ u, Q ≡ Q, F ≡ F.

Example 2. Let Q = L(u), where L = ∏_{j=1}^{N}(D − k_{j}), and k_{i}, i = 1, …, N, are distinct (k_{i} ≠ k_{j} if i ≠ j) nonzero constants. Then the general solution of Q = 0 is ∑_{i=1}^{N} c_{i}(t) exp(k_{i}x), where c_{i}(t) are arbitrary functions of t, and using the results of Example 1 we obtain

h_{i} = \frac{exp(−k_{i}x)}{∏_{j=1,j≠i}^{N}(k_{i} − k_{j})}L_{i}(u),

where L_{i} = ∏_{j=1,j≠i}^{N}(D − k_{j}). Thus, the most general locally analytic F of order n ≥ N − 1 such that u_{t} = F admits a GCS Q∂/∂u with Q = L(u) can be locally written as

\begin{matrix}F = ∑_{i=1}^{N} ζ_{i}(t, h_{1}, …, h_{N}) exp(k_{i}x) \\ + ∑_{m=0}^{n−N} D^{m}(Q)K_{m}(x, t, u, u_{1}, …, u_{N−1}, Q, D(Q), …, D^{n−N}(Q)), \end{matrix}

(13)

where ζ_{i} and K_{m} are arbitrary locally analytic functions of their arguments, and K_{m} ≡ 0 if n < N.

Example 3. Let Q = u_{2} − f(u, t), where f is an arbitrary analytic function of u and t. Set a(z, t) = ∫ f(z, t)dz and ψ(y, z, t) = ∫ (2a(y, t) + z)^{−1/2}dy. Then the general solution of Q = 0 in implicit form reads ψ(u, c_{1}(t), t) = x + c_{2}(t), where c_{i}(t) are arbitrary functions of t, and we have h_{1} = u_{1}^{2} − 2a(u, t), h_{2} = ψ(u, z, t)|_{z=h_{1}} − x.

Hence by Theorem 2 the most general locally analytic F or order n ≥ 1 such that u_{t} = F admits a GCS Q∂/∂u with Q = u_{2} − f(u, t) locally has the form

F = −(2a(u, t) + h_{1})^{1/2}((\frac{∂ψ(u, z, t)}{∂t})|_{z=h_{1}}

−ζ_{1}(t, h_{1}, h_{2})∫ ^{u} \frac{dy}{2(2a(y, t) + h_{1})^{3/2}} − ζ_{2}(t, h_{1}, h_{2}))

+ ∑_{m=0}^{n−2} D^{m}(Q)K_{m}(x, t, u, u_{1}, Q, D(Q), …, D^{n−2}(Q)),

where ζ_{i} and K_{m} are arbitrary locally analytic functions of their arguments; if n < 2, then K_{m} ≡ 0 for all m.

Example 4. As a somewhat more elaborated example, consider Q = u_{2} − φ(x, t)f(u_{1}, t), where φ and f are arbitrary analytic functions of their arguments, and f ≢ 0. Let a(z, t) = ∫ dz/f(z, t), φ̃(x, t) = ∫ φ(x, t)dx, and let b(y, t) denote a solution of the equation a(z, t) = y with respect to z, so that a(b(z, t), t) ≡ z. Then the general solution of Q = 0 reads u = c_{1}(t) + χ(x, c_{2}(t), t), where χ(x, z, t) = ∫ b(φ̃(x, t) + z, t)dx, c_{i}(t) are arbitrary functions of t, and we have h_{1} = u − χ(x, z, t)|_{z=h_{2}}, h_{2} = a(u_{1}, t) − φ̃(x, t).

Hence, the most general locally analytic F of order n ≥ 1 such that u_{t} = F admits a GCS Q∂/∂u with Q = u_{2} − φ(x, t)f(u_{1}, t) locally has the form

F = (\frac{∂χ(x, z, t)}{∂t})|_{z=h_{2}}+ζ_{1}(t, h_{1}, h_{2}) + ζ_{2}(t, h_{1}, h_{2})(\frac{∂χ(x, z, t)}{∂z})|_{z=h_{2}}

+ ∑_{m=0}^{n−2} D^{m}(Q)K_{m}(x, t, u, u_{1}, Q, D(Q), …, D^{n−2}(Q)),

where ζ_{i} and K_{m} are arbitrary locally analytic functions of their arguments, and if n < 2, then K_{m} ≡ 0 for all m.

Example 5. Assume now that Q = M(1), where M = ∏_{j=1}^{N+1}(D − k_{j} − u). The general solution of Q = 0 has the form −v_{x}/v, where v ≡ ∑_{i=1}^{N+1} b_{i}(t) exp(k_{i}x), and b_{i}(t) are arbitrary functions of t, cf. [4].

Notice that −v_{x}/v actually involves only N independent arbitrary functions of t. Indeed, renumbering b_{i} if necessary, we can assume without loss of generality that b_{N+1}(t) ≠ 0. Then we can rewrite u = −v_{x}/v in the form

u = −D(ln( exp(k_{N+1}x) + ∑_{i=1}^{N} c_{i}(t) exp(k_{i}x))),  (14)

where c_{i}(t) = b_{i}(t)/b_{N+1}(t), whence

h_{i} = \frac{exp((k_{N+1} − k_{i})x)M_{i}(1) ∏_{j=1}^{N}(k_{N+1} − k_{j})}{M_{N+1}(1) ∏_{j=1,j≠i}^{N+1}(k_{i} − k_{j})}, i = 1, …, N. (15)

Here M_{i} = ∏_{j=1,j≠i}^{N+1}(D − k_{j} − u).

The most general locally analytic F of order n ≥ N − 1 such that u_{t} = F admits a GCS Q∂/∂u with Q = M(1) (and hence an N-shock type solution (14)) locally has the form

F = − ∑_{i=1}^{N} ζ_{i}(t, h_{1}, …, h_{N}) exp(k_{i}x)\frac{( ∑_{j=1}^{N+1}(k_{i} − k_{j})h_{j} exp(k_{j}x))}{( ∑_{q=1}^{N+1} h_{q} exp(k_{q}x))^{2}}

+ ∑_{m=0}^{n−N} D^{m}(Q)K_{m}(x, t, u, u_{1}, …, u_{N−1}, Q, D(Q), …, D^{n−N}(Q)),

(16)

where ζ_{i} and K_{m} are arbitrary locally analytic functions of their arguments, and for n < N we have K_{m} ≡ 0 for all m. This time the h_{i} are of the form (15), and for the sake of brevity we set h_{N+1} = 1. The corresponding N-shock type solution of the evolution equation u_{t} = F is of the form (14) with c_{i}(t) that satisfy (7).

Notice that one can readily pick among the F ’s (16) those independent of x and t. They are of the form

\begin{matrix}F = − ∑_{i=1}^{N} η_{i}(\frac{h̃_{1}^{k_{N+1}−k_{N}}}{h̃_{N}^{k_{N+1}−k_{1}}} , …, \frac{h̃_{N−1}^{k_{N+1}−k_{N}}}{h̃_{N}^{k_{N+1}−k_{N−1}}} )\frac{h̃_{i} ∑_{j=1}^{N+1}(k_{i} − k_{j})h̃_{j}}{( ∑_{q=1}^{N+1} h̃_{q})^{2}} \\ + ∑_{m=0}^{n−N} D^{m}(Q)K_{m}(u, u_{1}, …, u_{N−1}, Q, D(Q), …, D^{n−N}(Q)), \end{matrix} (17)

where h̃_{i} = h_{i} exp((k_{i} − k_{N+1})x), and η_{i} and K_{m} are arbitrary locally analytic functions of their arguments.

The substitution of (14) into the equation u_{t} = F with F of the form (17) reduces this equation to the following system of ODEs:

\frac{dc_{i}}{dt} = c_{i}η_{i}(\frac{c_{1}^{k_{N+1}−k_{N}}}{c_{N}^{k_{N+1}−k_{1}}} , …, \frac{c_{N−1}^{k_{N+1}−k_{N}}}{c_{N}^{k_{N+1}−k_{N−1}}} ), i = 1, …, N.

If ζ_{i} = −h_{i} ∑_{j=0}^{m} α_{j}(k_{i}^{j} − k_{N+1}^{j}), where α_{j} are arbitrary constants, then (16) represents the most general F of order n such that the equation u_{t} = F admits an N-shock solution of the form [4]

u = −D(ln( ∑_{i=1}^{N+1} A_{i} exp(k_{i}x − t ∑_{j=0}^{m} α_{j}k_{i}^{j}))),  (18)

where A_{i} are arbitrary constants.

In other words, the class of evolution equations admitting the Nshock solutions of the form (18) contains not only the equation [4]

u_{t} = ∑_{j=0}^{m} α_{j}D(D − u)^{j}(u)

but infinitely many other equations u_{t} = F as well. In particular, if the order n of F is greater than N − 2, then Theorem 2 implies that the corresponding F ’s have the form

\begin{matrix}F = ∑_{i=1}^{N} ∑_{r=0}^{m} α_{r}(k_{i}^{r} − k_{N+1}^{r})\frac{h̃_{i} ∑_{j=1}^{N+1}(k_{i} − k_{j})h̃_{j}}{( ∑_{q=1}^{N+1} h̃_{q})^{2}} \\ + ∑_{m=0}^{n−N} D^{m}(Q)K_{m}(x, t, u, u_{1}, …, u_{N−1}, Q, D(Q), …, D^{n−N}(Q)). \end{matrix}

4. Conclusions and discussion

In Theorem 2 of the present paper we completely characterized locally analytic evolution systems (1) of a given order n ≥ k − 1 that admit a generalized conditional symmetry Q(x, t, u, …, u_{k})∂/∂u with a given analytic Q of the form (8). In particular, for the case of linear GCS, i.e., when Q is linear in u_{j} for all j, the right-hand sides F of such systems (1) are given by (12).

The results of the present paper are somewhat easier to use in applications than the formulas found by us earlier in [12]. For instance, the usage of Theorem 2 enabled us to find the explicit form of all locally analytic evolution equations of order n ≥ N − 1 that admit N-shock (and, more broadly, N-shock type) solutions for a given number N, and thus generalize the results of Fokas and Liu [4].

Moreover, Theorem 2 can be employed for the classification of exactly solvable initial value problems along the lines of [3, 17] and for finding evolution equations whose symmetries are compatible with prescribed boundary conditions, and one can find exact solutions for the corresponding boundary value problems, cf. [1] and discussion in [12]. We plan to address these topics elsewhere.

Acknowledgments. I acknowledge with gratitude the support from the Czech Grant Agency under grant No. 201/04/0538 and the Ministry of Education, Youth and Sports of Czech Republic under grant MSM:J10/98:192400002.

References

[1] V.E. Adler, B. Gürel, M. Gürses, I. Habibullin, “Boundary conditions for integrable equations,” J. Phys. A: Math. Gen., 30, 3505–3513 (1997).

[2] A. Andreytsev, “Classification of systems of nonlinear evolution equations admitting higher-order conditional symmetries, ” in: Proc. 4th Int. Conf. Symmetry in Nonlinear Mathematical Physics (Kyiv, 2001), Part 1, Institute of Mathematics, Kyiv (2002), pp. 72–79.

[3] P. Basarab-Horwath, R.Z. Zhdanov, “Initial-value problems for evolutionary partial differential equations and higher-order conditional symmetries,” J. Math. Phys., 42, 376–389 (2001).

[4] A.S. Fokas, Q.M. Liu, “Generalized conditional symmetries and exact solutions of non-integrable equations,” Theor. Math. Phys., 99, 571–582 (1994).

[5] A.S. Fokas, Q.M. Liu, “Nonlinear interaction of traveling waves of nonintegrable equations,” Phys. Rev. Lett. 72, 3293–3296 (1994).

[6] N. Kamran, R. Milson and P. Olver, “Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems, ” Adv. Math., 156, 286–319 (2000).

[7] A. V. Mikhailov, A. B. Shabat, A. B.; V. V. Sokolov, “The symmetry approach to classification of integrable equations, ” in: What is integrability?, Springer, Berlin (1991), pp.115–184.

[8] P. J. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1986.

[9] J. Rubin and P. Winternitz, “Point symmetries of conditionally integrable nonlinear evolution equations,” J. Math. Phys. 31, 2085–2090 (1990).

[10] A.V. Samokhin, “Full Symmetry Algebra for ODEs and Control Systems,” Acta Appl. Math.; 72, no. 1-2, 87–99 (2002).

[11] A.V. Samokhin, “Symmetries of linear and linearizable systems of differential equations,” Acta Appl. Math., 56, 253–300 (1999).

[12] A. Sergyeyev, “Constructing conditionally integrable evolution systems in (1+ 1) dimensions: a generalization of invariant modules approach,” J. Phys. A: Math. Gen., 35, 7653–7660 (2002).

[13] Svirshchevskii S.R., “Lie-Bäcklund symmetries of linear ODEs and invariant linear spaces,” in: Modern Group Analysis, MPhTI, Moscow (1993), pp.75�83.

[14] Symmetries and conservation laws for differential equations of mathematical physics, by A.V. Bocharov et al., I.S. Krasil’shchik and A.M. Vinogradov eds., AMS, Providence, RI (1998).

[15] R.Z. Zhdanov, “Conditional Lie-Bäcklund symmetry and reduction of evolution equations,” J. Phys. A: Math. Gen., 28, 3841–3850 (1995).

[16] R.Z. Zhdanov, “Higher conditional symmetries and reduction of initial value problems for nonlinear evolution equations,” in: Proc. Int. Conf. Symmetry in nonlinear mathematical physics (Kyiv, 1999), Part 1, Institute of Mathematics, Kyiv (2000), pp.255–263.

[17] R.Z. Zhdanov, “Higher conditional symmetry and reduction of initial value problems,” Nonlinear Dynamics, 28, 17–27 (2002).

Silesian University in Opava, Mathematical Institute, Na Rybníčku 1, 746 01 Opava, Czech Republic.

E-mail address: Artur.Sergyeyev@math.slu.cz