A Few New 2+1-Dimensional Nonlinear Dynamics and the Representation of Riemann Curvature Tensors

1 Introduction

It has been an important task to generate integrable systems in soliton theory. A great number of (1+1)-dimensional integrable systems had been found in the past decades by applying zero-curvature equations, Lax pairs, and other techniques (see, e.g. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12]). However, it is more difficult to search for (2+1)-dimensional integrable systems than the (1+1)-dimensional case [13], [14], [15], [16], [17], [18]. Athorne and Dorfman [15] and Dorfman and Fokas [16] constructed Hamiltonian operators to generate (2+1)-dimensional integrable systems over noncommutation rings. Moreover, Fokas and Tu [17] and Tu et al. [18] introduced a residue operator over an associative algebra to generate the Kadomtsev-Petviashvili (KP) equation and the Davey-Stewartson (DS) equation. This method was proposed by Tu et al. [18], which was called the Tu-Andrushkiw-Huang scheme, briefly called the TAH scheme. By applying the TAH scheme, some (2+1)-dimensional hierarchies and their corresponding Hamiltonian structures were obtained by Zhang et al. [19], [20], [21]. However, there exists an open problem that the integrability of the (2+1)-dimensional hierarchies obtained by the TAH scheme cannot be determined. Another approach is that Ablowitz et al. [22] applied some reduced equations of the self-dual Yang-Mills equations to generate some (1+1)- and (2+1)-dimensional integrable equations, such as the Kortweg-de Vries equation and the KP equation. On the basis of this procedure, Zhang et al. [23], [24] generated some (2+1)-dimensional integrable systems, including a (2+1)-dimensional integrable coupling, which was the first result on (2+1)-dimensional integrable coupling, to our best knowledge. In addition, Athorne and Fordy [25] applied the symmetric and homogeneous spaces to generate the N-wave, the KP equation, and the DS equation. Actually, we once adopted such symmetric space to generate nonlinear integrable couplings and some (2+1)-dimensional integrable equations [26], [27]. In this paper, we first recalled some basic notions on the symmetric and homogeneous spaces, then we introduced a stationary linear equation with a quadratic operator in ∂_x and ∂_y. An evolution equation is also introduced whose compatibility with the stationary linear equation can generate higher dimensional integrable equations. In particular, a second-order flow, which is a (2+1)-dimensional matrix heat equation, is obtained, which is associative with the symmetric space. A third flow in (2+1) dimensions is obtained, which is a generalized KP equation associated with the symmetric space. Under the framework of the homogeneous space, a second-order flow, which is a (2+1)-dimensional hyperbolic equation, is expressed by an element P of the Lie subalgebra m. For the third flow associated with the homogeneous space, we generated the m^± components of the Lie subalgebra m, which is a generalized KP equation. It is an extended form of the result presented by Athorne and Fordy [25]. Finally, we extended a pair of spectral problems by Fordy and Kulish [28] with the quadratic operators with respect to the operators ∂_x and ∂_y to further derive a new (2+1)-dimensional nonlinear Schrödinger equation associative with the symmetric space, which generalizes a main result by Fordy and Kulish [28]. It is remarkable that the method for generating (2+1)-dimensional nonlinear equations presented here is different from the Adler-Gelfand-Dikii (AGD) scheme, the TAH scheme, and the binomial residue representation scheme by Zhang et al. [29], [30].

2 The Symmetric and Homogeneous Spaces

We first recall some basic notions on the Hermitian symmetric and reductive homogeneous spaces [25], [31]. A homogeneous space of a Lie group G is a differentiable manifold M on which G acts transitively. The subgroup of G that leaves a given point p₀∈M fixed is called the isotropy group at p₀ and is defined by

K ≡ Kp0 = {g ∈ G:g • p0 = p0}.

Such manifold M can be identified with a coset space G/K. In this paper, we only consider the decompositions of the corresponding Lie algebras of the Lie group G and the isotropy group K.

Let g and k be the Lie algebras of G and K, respectively, and let m be the vector space complement of k in g. Then we have

g = k ⊕ m, [k, k] ⊂ k,

where m is identified with the tangent space Tp0M of M=G/K at point p₀. When g satisfies the following conditions:

g = k ⊕ m, [k, k] ⊂ k, [k, m] ⊂ m,

then G/K is called a reductive homogeneous space [25]. Such space possesses the defined connections with curvature and torsion. At fixed point p₀, the curvature and torsion tensors are given by the Lie bracket operation on m:

(R(X, Y)Z)p0 = −[[X, Y]k, Z], T(X, Y)p0  = −[X, Y]m, ∀X, Y, Z ∈ m.

When g satisfies the following conditions:

g = k ⊕ m, [k, k] ⊂ k, [k, m] ⊂ m, [m, m] ⊂ k,

then g is called a symmetric algebra and G/K is a symmetric space. At fixed point p₀, the curvature is given by

(R(X, Y)Z)p0 = −[[X, Y], Z], ∀X, Y, Z ∈ m,

and the torsion is free. Assuming h is a Cartan subalgebra of g, there exists an element A∈h such that

k = Cg(A) = {B ∈ g:[B, A] = 0}.

If A is regular, C_g(A)=h. Otherwise, C_g(A)⊃h. The operator representation α(A) has three distinct eigenvalues: 0, ±a. In particular, we have

m = m+ + m−, [A, k] = 0, [A, X±] = ±aX±.

For any X∈g, X=X⁰+X⁺+X⁻, and X+ = ∑αXαeα, X− = ∑αX−αe−α, where α is summed over a special subset θ⁺ of the positive root system Φ⁺. In Hermitian symmetric spaces, we have the convenient property that [X⁺, Y⁺]=0, ∀X⁺, Y⁺∈m⁺, and similarly for m⁻. Besides, X⁺Y⁺=0 for all pairs of elements of m⁺, similarly for m⁻. However, for the homogeneous space, we still have m=m⁺⊕m⁻, but each of m^± is further split into blocks, each of which is an eigenspace of α(A). θ⁺ splits into several θj+ subsets, each of which α(A) = aj, α ∈ θj+. Assuming Q∈m, we can write

Q± = ∑jQj±, [A, Qj±] = ±ajQj±.

3 Applications of the Symmetric Space

Athorne and Fordy [25] once introduced the following operator:

L = Dx + A∂y + Q,

where Q is a matrix function, and D_x is a derivative with respect to x. Now we consider a stationary linear equation with a quadratic operator

where A and B are diagonal matrices that have the same sizes with the matrix Q; here Q∈m.

A time evolution linear equation is given by

Equating coefficients of ∂xii∂xjj(xi, xj = x, y; i + j = 0, 1, …, N + 1) in the commutator

can cause some evolution equations concerning S⁽ⁱ⁾(i=0, 1, …, N+1), A, B, Q, where A, B, and Q are the same-order matrices with the matrices S⁽ⁱ⁾. In what follows, we consider the cases where N=2 and N=3 associated with the symmetric spaces.

By taking N=2 and by comparing the coefficients of ∂xi∂yj, i + j = 0, 1, 2, 3, 4, (3) leads to the following equations:

Assuming A, B∈k, B=I, we can take

where α and β are constants independent of x, y, and t. Equation (6) admits

and (8) gives

Equations (9–11) can be written as

which indicates that there exist traveling-wave solutions for the function Q. Equation (12) can be written together with (16) as follows:

Qt = αQxx + (2β − α)Qxy + (αA + βI)Qy,

which is a (2+1)-dimensional generalized heat equation with parameters α and β. To our best knowledge, it is a new (2+1)-dimensional linear matrix equation.

Taking N=3, similar to the case of N=2, (3) admits some differential equations (see Appendix) and

Taking B=I, S⁽⁰⁾=A, A∈k, the equations in the Appendix have the following special solutions:

which implies that the solutions for Q, S⁽³⁾ are traveling waves. Equation (17) becomes

Because S⁽³⁾∈g, it can be decomposed into

S(3) = S0(3) + S+(3) + S−(3), [A, S0(3)  ] = 0.

Equation (18) can be presented as

[A, S+(3)  ] + [A, S−(3)  ] = −2AQy+ − 2AQy− − Qy+ − Qy− + [A, Q+] + [A, Q−] + 3AQ+ + 3AQ−.

Therefore, we obtain the m^± components of S⁽³⁾ as follows:

Hence, with the help of (20) and (21), S⁽³⁾ can be written as

The k component of (19) reads

Substituting (20) and (21) into (23) yields

The m⁺ component of (19) presents

Qt+ + AS+,yy(3) + S+,xy(3) + [Q+, S0(3)  ] − S+,xx(3) − AQyy+ − AQyyy+ = 0;

that is,

Similarly, the m⁻ component of (19) reads

where S0(3) is determined by (24). Obviously, (25) and (26) are all nonlocal, which are generalized KP equations with the same eigenvalues.

4 Applications of the Homogeneous Space

As applications of (4–12), we first consider a second-order flow by taking S⁽⁰⁾=C∈k, and introducing an element P∈m so that Q=[A, P]. Set B=I, then (4–12) can be solvable. In terms of (6) and (8), we get the solutions

S(1) = C, S(2) = [P, C].

Equation (11) gives

Py = 2Px.

Therefore, (12) can be written as

Equation (27) is a (2+1)-dimensional hyperbolic equation, which is different from the N-wave equation presented by Athorne and Fordy [25].

In what follows, we shall discuss the third-order flow by taking B=I, S⁽⁰⁾=S⁽¹⁾=A, A∈k, and S⁽²⁾=Q and introducing P∈m such that Q=[A, P]. According to the equations in Appendix, we have

In the following, we want to discover the k component and the m^± component of the matrix equation (28).

It is easy to see that

Because

Q± = ∑jQj±, [A, Qj±] = ±ajQj±.

Thus, the k component of (31) is that

It is easy to see that

[A, Qj+Qi,y−] = (aj − ai)Qj+Qi,y−.

Similar to the previously mentioned analysis, we can obtain the m^± components of (28) as follows:

where S0(3) satisfies

(A∂y2 + ∂x∂y)S0(3) = (I + 3A)∑j[Pj−, Qj+].

Equations (31) and (32) constitute a pair of new generalized KP equations, which they have various eigenvalues.

5 Applications of the Hermitian Symmetric Space

In this section, we shall introduce an isospectral Lax pair based on linear equations (1) and (2), whose integrability condition leads to a generalized nonlinear Schrödinger equation under the framework of the symmetric spaces, which can be expressed by the Reimann curvature tensors.

Consider the following isospectral problems:

where B, D∈k, and P∈g. The integrability condition of (33) and (34) reads

where T = 14B2∂y2 − B∂x∂y + ∂x2.

According to the definition of the symmetric space, (35) decouples to

where we have used [D, P_k]=0 because k=C_g(D), P=P_k+P_m, P_k∈k, and P_m∈m. Because

the first equation in (36) can be written as

Assuming P = ∑i = 0NP(i)λi, the recursion operator appearing on the right of (38) is that

where j=1, 2, …, N–1.

Substituting the P into the first equation in (36), we have

Because Q∈m=span{e_±a}, we suppose

Hence, we have derived from (41) and (42) that

where we have used the property [e_α, e_β]=0, ∀α, β∈θ⁺.

We decompose (40) into the following form:

According to the definition of the Riemann curvature tensor, (43) and (44) can be expressed by the Riemann curvature tensors as follows:

which is the generalized nonlinear Schrödinger equation. If we take T=∂_x, (45) reduces to the result of Fordy and Kulish [28].