ISRN.MATHEMATICAL.PHYSICS ISRN Mathematical Physics 2090-4681 Hindawi Publishing Corporation 109170 10.1155/2013/109170 109170 Research Article Approximate Symmetries of the Harry Dym Equation Nadjafikhah Mehdi http://orcid.org/0000-0002-1847-3629 Kabi-Nejad Parastoo Bagchi B. Qiao Z. School of Mathematics Iran University of Science and Technology Narmak Tehran 1684613114 Iran iust.ac.ir 2013 23 12 2013 2013 27 10 2013 17 11 2013 2013 Copyright © 2013 Mehdi Nadjafikhah and Parastoo Kabi-Nejad. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We derive the first-order approximate symmetries for the Harry Dym equation by the method of approximate transformation groups proposed by Baikov et al. (1989, 1996). Moreover, we investigate the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. We compute the one-dimensional optimal system of subalgebras as well as point out some approximately differential invariants with respect to the generators of Lie algebra and optimal system.

1. Introduction

The following nonlinear partial differential equation (1)ut=-12u3uxxx is known as the Harry Dym equation . This equation was obtained by Harry Dym and Martin Kruskal as an evolution equation solvable by a spectral problem based on the string equation instead of Schrödinger equation. This result was reported in  and rediscovered independently in [3, 4]. The Harry Dym equation shares many of the properties typical of the soliton equations. It is a completely integrable equation [5, 6], which can be solved by inverse scattering transformation . It has a bi-Hamiltonian structure and an infinite number of conservation laws and infinitely many symmetries [10, 11].

In this paper, we analyze the perturbed Harry Dym equation (2)ut+12u3uxxx+ɛux=0, where ɛ is a small parameter, with a method which was first introduced by Baikov et al. [12, 13]. This method which is known as “approximate symmetry” is a combination of Lie group theory and perturbations. There is a second method which is also known as “approximate symmetry” due to Fushchich and Shtelen  and later followed by Euler et al. [15, 16]. For a comparison of these two methods, we refer the interested reader to [17, 18]. Our paper is organized as follows. In Section 2, we present some definitions and theorems in the theory of approximate symmetry. In Section 3, we obtain the approximate symmetry of the perturbed Harry Dym equation. In Section 4, we discuss the structure of its Lie algebra. In Section 5, we construct the one-dimensional optimal system of subalgebras. In Section 6, we compute some approximately differential invariants with respect to the generators of Lie algebra and optimal system. In Section 7, we summarize our results.

2. Notations and Definitions

In this section, we will provide the background definitions and results in approximate symmetry that will be used along this paper. Much of it is stated as in . If a function f(x,ɛ) satisfies the condition (3)limf(x,ɛ)ɛp=0, it is written f(x,ɛ)=o(ɛp) and f is said to be of order less than ɛp. If (4)f(x,ɛ)-g(x,ɛ)=o(ɛp), the functions f and g are said to be approximately equal (with an error o(ɛp)) and written as (5)f(x,ɛ)=g(x,ɛ)+o(ɛp) or, briefly, fg when there is no ambiguity. The approximate equality defines an equivalence relation, and we join functions into equivalence classes by letting f(x,ɛ) and g(x,ɛ) be members of the same class if and only if fg. Given a function f(x,ɛ), let (6)fo(x)+ɛfl(x)++ɛpfp(x) be the approximating polynomial of degree p in ɛ obtained via the Taylor series expansion of f(x,ɛ) in powers of ɛ about ɛ=0. Then any function gf (in particular, the function f itself) has the form (7)g(x,ɛ)=fo(x)+ɛfl(x)++ɛpfp(x)+o(ɛp). Consequently the expression (6) is called a canonical representative of the equivalence class of functions containing f. Thus, the equivalence class of functions g(x,ɛ)f(x,ɛ) is determined by the ordered set of p+1 functions f0(x),fl(x),,fp(x). In the theory of approximate transformation groups, one considers ordered sets of smooth vector-functions depending on x’s and a group parameter a: (8)f0(x,a),fl(x,a),,fp(x,a), with coordinates (9)f0i(x,a),f1i(x,a),,fpi(x,a),i=1,,n. Let us define the one-parameter family G of approximate transformations (10)x¯if0i(x,a)+ɛf1i(x,a)++ɛpfpi(x,a),i=1,,n, of points x=(x1,,xn)Rn into points x¯=(x¯1,,x¯n)Rn as the class of invertible transformations (11)x¯=f(x,a,ϵ), with vector-functions f=(f1,,fn) such that (12)fi(x,a,ϵ)f0i(x,a)+ϵf1i(x,a)++ɛpfpi(x,a),i=1,,n. Here a is a real parameter, and the following condition is imposed: (13)f(x,0,ϵ)x.

Definition 1.

The set of transformations (10) is called a one-parameter approximate transformation group if (14)f(f(x,a,ɛ),b,ϵ)f(x,a+b,ɛ) for all transformations (11).

Definition 2.

Let G be a one-parameter approximate transformation group: (15)z¯if(z,a,ɛ)f0i(z,a)+ɛf1i(z,a),i=1,,N. An approximate equation (16)F(z,ɛ)F0(z)+ɛF1(z)0 is said to be approximately invariant with respect to G or admits G if (17)F(z¯,ɛ)F(f(z,a,ɛ),ɛ)=o(ɛ) whenever z=(zl,,zN) satisfies (16). If z=(x,u,u(1),,u(k)) then (16) becomes an approximate differential equation of order k, and G is an approximate symmetry group of the differential equation.

Theorem 3.

Equation (16) is approximately invariant under the approximate transformation group (15) with the generator (18)X=X0+ɛX1ξ0i(z)zi+ɛξ1izi, if and only if (19)[X(k)F(z,ɛ)]F0=o(ɛ), or (20)[X0(k)F0(z)+ɛ(X1(k)F0(z)+X0(k)F1(z))](2.5)=o(ɛ), where X(k) is the prolongation of X of order k. The operator (18) satisfying (20) is called an infinitesimal approximate symmetry of or an approximate operator admitted by (16). Accordingly, (20) is termed the determining equation for approximate symmetries.

Theorem 4.

If (16) admits an approximate transformation group with the generator X=X0+ɛX1, where X00, then the operator (21)X0=ξ0i(z)zi is an exact symmetry of the equation (22)F0(z)=0.

Definition 5.

Equations (22) and (16) are termed an unperturbed equation and a perturbed equation, respectively. Under the conditions of Theorem 4, the operator X0 is called a stable symmetry of the unperturbed equation (22). The corresponding approximate symmetry generator X=X0+ɛX1 for the perturbed equation (16) is called a deformation of the infinitesimal symmetry X0 of (22) caused by the perturbation ɛF1(z). In particular, if the most general symmetry Lie algebra of (22) is stable, we say that the perturbed equation (16) inherits the symmetries of the unperturbed equation.

3. Approximate Symmetries of the Perturbed Harry Dym Equation

Consider the perturbed Harry Dym equation (23)ut+12u3uxxx+ɛux=0. By applying the method of approximate transformation groups, we provide the infinitesimal approximate symmetries (18) for the perturbed Harry Dym equation (2).

3.1. Exact Symmetries

Let us consider the approximate group generators in the form (24)X=X0+ɛX1=(ξ0+ɛξ1)x+(τ0+ɛτ1)t+(ϕ0+ɛϕ1)u, where ξi, τi, and ϕi for i=0,1 are unknown functions of x, t, and u. Solving the determining equation (25)X0(3)(ut-12u3uxxx)|ut-(1/2)u3uxxx=0=0, for the exact symmetries X0 of the unperturbed equation, we obtain (26)ξ0=(A1+A2x+A32x2),τ0=(A4+A5t),ϕ0=(A2-13A5+xA3)u, where A1,,A5 are arbitrary constants. Hence, (27)X0=(A1+A2x+A32x2)x+(A4+A5t)t+((A2-13A5+xA3)u)u. Therefore, the unperturbed Harry Dym equation admits the five-dimensional Lie algebra with the basis (28)X01=x,X02=t,X03=xx+uu,X04=3tt-uu,X05=x2x+2xuu.

3.2. Approximate Symmetries

At first, we need to determine the auxiliary function H by virtue of (19), (20), and (16), that is, by the equation (29)H=1ɛ[X0(k)(F0(z)+ɛF1(z))|F0(z)+ɛF1(z)=0]. Substituting the expression (27) of the generator X0 into (29) we obtain the auxiliary function (30)H=ux(A5-A2)+A3(u-xux). Now, calculate the operators X1 by solving the inhomogeneous determining equation for deformations: (31)X1(k)F0(z)|F0(z)=0+H=0. So, the above determining equation for this equation is written as (32)X1(3)(ut+12u3uxxx)|ut+(1/2)u3uxxx=0+ux(A5-A2)+A3(u-xux)=0. Solving the determining equation yields (33)ξ1=(A5-A2)t-A3xt+C4x-C5+C32x2,τ1=(C1t+C2),ϕ1=(-A3t+C4+C3x+C13)u, where C1,,C5 are arbitrary constants.

Thus, we derive the following approximate symmetries of the perturbed Harry Dym equation: (34)v1=x,v2=t,v3=xx+uu,v4=3tt-uu,v5=x2x+2xuu,v6=ɛx,v7=ɛt,v8=ɛ(xx+uu),v9=ɛ(3tt-uu),v10=ɛ(x2x+2xuu). Table 1 of commutators, evaluated in the first order of precision, shows that the operators (34) span a ten-dimensional approximate Lie algebra and hence generate a ten-parameter approximate transformation group.

Approximate commutators of approximate symmetry of perturbed Harry Dym equation.

v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10
v 1 0 0 v 1 0 2 v 3 0 0 v 6 0 2 v 8
v 2 0 0 0 12 v 2 0 0 0 0 3 v 7 0
v 3 - v 1 0 0 0 v 5 - v 6 0 0 0 v 10
v 4 0 - 12 v 2 0 0 0 0 - 3 v 7 0 0 0
v 5 - 2 v 3 0 - v 5 0 0 - 2 v 8 0 - v 10 0 0
v 6 0 0 v 6 0 2 v 8 0 0 0 0 0
v 7 0 0 0 3 v 7 0 0 0 0 0 0
v 8 - v 6 0 0 0 v 10 0 0 0 0 0
v 9 0 - 3 v 7 0 0 0 0 0 0 0 0
v 10 - 2 v 8 0 - v 10 0 0 0 0 0 0 0
Remark 6.

Equations (34) show that all symmetries (28) of (1) are stable. Hence, the perturbed equation (2) inherits the symmetries of the unperturbed equation (1).

4. The Structure of the Lie Algebra of Symmetries

In this section, we determine the structure of the Lie algebra of symmetries of the perturbed Harry Dym equation. The Lie algebra g is nonsolvable, since (35)g(1)=[g,g]=SpanR{v1,v2,v3,v5,v6,v7,v8,v10},g(2)=[g(1),g(1)]=SpanR{v1,v3,v5,v6,v8,v10},g(3)=[g(2),g(2)]=g(2). The Lie algebra g admits a Levi decomposition as the following semidirect product g=rs, where (36)r=SpanR{v2,v4,v6,v7,v8,v9,v10} is the radical of g (the largest solvable ideal contained in g) and (37)s=SpanR{v1,v3,v5} is a semisimple subalgebra of g.

The radical r is solvable with the following chain of ideals: (38)r(1)r(2)r(3)={0}, where (39)r(1)=SpanR{v2,v4,v6,v7,v8,v9,v10},r(2)=SpanR{v2,v7}. The semisimple subalgebra s of g is isomorphic to the Lie algebra A3,8 of the classification of three-dimensional Lie algebras in , by the following isomorphism: (40)𝒯:{v1,v3,v5}{v1,-v2,-v3}.

5. Optimal System for Perturbed Harry Dym Equation Definition 7.

Let G be a Lie group. An optimal system of s-parameter subgroups is a list of conjugacy inequivalent s-parameter subgroups with the property that any other subgroup is conjugate to precisely one subgroup in the list. Similarly, a list of s-parameter subalgebras forms an optimal system if every s-parameter subalgebra of g is equivalent to a unique member of the list under some element of the adjoint representation: h~=Ad(g(h)), gG.

Proposition 8.

Let H and H~ be connected, s-dimensional Lie subgroups of the Lie group G with corresponding Lie subalgebras h and h~ of the Lie algebra g of G. Then H~=gHg-1 are conjugate subgroups if and only if  h~=Ad(g(h)) are conjugate subalgebras (Proposition 3.7 of ).

Actually, the proposition says that the problem of finding an optimal system of subgroups is equivalent to that of finding an optimal system of subalgebras. For one-dimensional subalgebras, this classification problem is essentially the same as the problem of classifying the orbits of the adjoint representation, since each one-dimensional subalgebra is determined by a nonzero vector in g. To compute the adjoint representation one uses the Lie series: (41)Ad(exp(μvi))vj=vj-μ[vi,vj]+μ22[vi,[vi,vj]]-, where [vi,vj], i,j=1,,10 is the commutator for the Lie algebra and μ is a parameter. In this manner, one constructs Table 2 with the (i,j)th entry indicating Ad(exp(μvi))vj.

Adjoint representation of approximate symmetry of the perturbed Harry Dym equation.

Ad v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10
v 1 v 1 v 2 v 3 - μ v 1 v 4 v 5 - 2 μ v 3 + μ 2 v 1 v 6 v 7 v 8 - μ v 6 v 9 v 10 - 2 μ v 8 + μ 2 v 6
v 2 v 1 v 2 v 3 v 4 - 12 μ v 2 v 5 v 6 v 7 v 8 v 9 - 3 μ v 7 v 10
v 3 e μ v 1 v 2 v 3 v 4 e - μ v 5 e μ v 6 v 7 v 8 v 9 e - μ v 10
v 4 v 1 e 12 μ v 2 v 3 v 4 v 5 v 6 e 3 μ v 7 v 8 v 9 v 10
v 5 v 1 + 2 μ v 3 + μ 2 v 5 v 2 v 3 + μ v 5 v 4 v 5 v 6 + 2 μ v 8 + μ 2 v 10 v 7 v 8 + μ + v 10 v 9 v 10
v 6 v 1 v 2 v 3 - μ v 6 v 4 v 5 - 2 μ v 8 v 6 v 7 v 8 v 9 v 10
v 7 v 1 v 2 v 3 v 4 - 3 μ v 7 v 5 v 6 v 7 v 8 v 9 v 10
v 8 v 1 + μ v 6 v 2 v 3 v 4 v 5 - μ v 10 v 6 v 7 v 8 v 9 v 10
v 9 v 1 v 2 + 3 μ v 7 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10
v 1 0 v 1 + 2 μ v 8 v 2 v 3 + μ v 10 v 4 v 5 v 6 v 7 v 8 v 9 v 10
Theorem 9.

An optimal system of one-dimensional approximate Lie algebras of the perturbed Harry Dym equation is provided by (42)v1=v8,v2=v7+av8,v3=v6+v8,v4=v6-v7+v8,v5=v6+v7+v8,v6=v2+av8,v7=v2-v6+av8,v8=v2+v6+av8,v9=v1+av2+bv7,v10=av1+bv2+v5+cv6+dv7,v11=av1+bv2+v3+cv5+dv7+ev8,v12=av1+bv3+v4+cv5+dv6+ev8,v13=av1+bv3+cv4+dv5+ev6+fv8+v9,v14=av1-v2+bv3+cv4+dv5+ev6+fv8+v9,v15=av1+v2+bv3+cv4+dv5+ev6+fv8+v9,v16=av1+v2+bv3+cv4+dv5+ev6+fv8+v9.

Proof.

Consider the approximate symmetry algebra g of the unperturbed Harry Dym equation, whose adjoint representation was determined in Table 2. Given a nonzero vector (43)v=i=110aivi, our task is to simplify as many of the coefficients ai as possible through judicious applications of adjoint maps to v.

6. Approximately Differential Invariants for the Perturbed Harry Dym Equation

In this section, we compute some approximately differential invariants of the perturbed Harry Dym equation with respect to the optimal system. Consider the operator v2. To determine the independent invariants I, we need to solve the first-order partial differential equation: (53)(ɛt+aɛxx+aɛuu)(I(x,t,u))=0; that is, (54)ɛIt+aɛxIx+aɛuIu=0, which is a first-order homogeneous PDE. The solution can be found by integrating the corresponding characteristic system of ordinary differential equation, which is (55)dxaɛx=dtɛ=duaɛu. Hence, the independent approximately differential invariants are as follows: (56)y=ux,v=lnx-ata. In this manner, we investigate some independent approximately differential invariants with respect to the optimal system which are listed in Table 3.

Approximately differential invariants for the perturbed Harry Dym equation.

Operator Approximate differential invariants
v 1 t , u
v 2 x , u
v 3 t , u x
v 4 x , u t 1 / 3
v 5 t , u x 2
v 7 + a v 8 - ln x a + t , u x
v 6 + v 8 t , u x + 1
v 6 - v 7 + v 8 ln ( x + 1 ) + t , u x + 1
v 6 + v 7 + v 8 - ln ( x + 1 ) + t , u x + 1
v 2 + a v 8 - ln x a ε + t , u x
v 2 - v 6 + a v 8 - ln ( a x - 1 ) a ε + t , u a x - 1
v 2 + v 6 + a v 8 - ln ( a x + 1 ) a ε + t , u a x + 1
v 1 + a v 2 + b v 7 - b ε x - a ε + t , u
a v 1 + b v 2 + v 5 +cv6+dv7 - d ε - b c ε + a arctan ( x c ε + a ) + t , u x 2 + c ε + a
7. Conclusions

In this paper, we investigate the approximate symmetry of the perturbed Harry Dym equation and discuss the structure of its Lie algebra. Moreover, we compute optimal system of one-dimensional approximate Lie algebras of the perturbed Harry Dym equation and derive some approximately differential invariants with respect to the generators of Lie algebra and optimal system.

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