Integrable discretizations for the short wave model of the Camassa-Holm equation

In the present paper, we consider integrable discretizations of the nonlinear partial differential equation w T XX -2κ 2 w X + 2w X w XX + ww XXX = 0, (1) which belongs to the Harry-Dym hierarchy [1,2,3]. Here κ is a real parameter and, as shown subsequently, can be normalized by the scaling transformation when κ = 0. A connection between Eq.( 1) and the sinh-Gordon equation was established in [4]. When κ = 0, Eq.( 1) is called the Hunter-Saxton equation and is derived as a model for weakly nonlinear orientation waves in massive nematic liquid crystals [5]. The Lax pair and bi-Hamiltonian structure were discussed by Hunter and Zheng [6]. The dissipative and dispersive weak solutions were discussed in details by the same authors [7,8]. Equation ( 1) can be viewed as a short-wave model of the Camassa-Holm equation [9] w T + 2κ 2 w Xw T XX + 3ww X = 2w X w XX + ww XXX .

(

Following the procedure in [10,11,12], we introduce the time and space variables T and X

where ε is a small parameter. Then w is expanded as w = ε 2 (w 0 + εw 1 + • • •) with w i (i = 0, 1, • • •) being functions of T and X. At the lowest order in ε, we obtain w 0, T X X -2κ 2 w 0, X + 2w 0, X w 0, X X + w 0 w 0, X X X = 0 ,

which is exactly Eq.( 1) after writing back into the original variables. Based on this fact, Matsuno obtained the N-cuspon solution of Eq.( 1) by taking the short-wave limit on the Nsoliton solution of the Camassa-Holm equation [13,14]. Note that the parameter κ of Eq.( 1) can be normalized to 1 under the transformation x = κX , t = κT , which leads to

We call Eq.( 4) the short wave model of the Camassa-Holm equation (SCHE). Without loss of generality, we will focus on Eq. ( 4) and its integrable discretizations, since the solution of Eq.( 1) with arbitrary nonzero κ, its integrable discretizations and the corresponding solutions can be recovered through the above transformation.

The reminder of the present paper is organized as follows. In section 2, we reveal a connection between the SCHE and the bilinear form two-dimensional Toda-lattice (2DTL) equations. The parametric form of N-cuspon solution expressed by the Casorti determinant is given, which is consistent with the solution given in [13]. Based on this fact, we propose an integrable semi-discrete analogue of the SCHE in section 3, and further its integrable fulldiscrete analogue in section 4. The concluding remark is given in section 5.

In this section, we will show that the SCHE can be derived from the bilinear form of twodimensional Toda lattice (2DTL) equations

where D x is the Hirota D-derivative defined as

and D -1 and D 1 represent the Hirota D derivatives with respect to variables x -1 and x 1 , respectively. It is shown that the N-soliton solution of the 2DTL equations ( 5) can be expressed as the Casorati determinant [16,17]

(n) i satisfying the following dispersion relations:

automatically satisfies the above dispersion relations.

Applying the two-reduction τ n-1 = (∏ N i=1 p 2 i ) -1 τ n+1 , i.e., enforcing

where the gauge transformation τ n → (∏ N i=1 p i ) n τ n is used. Letting τ 0 = f , τ 1 = g and x -1 = s, x 1 = y, the above bilinear equation ( 8) takes the following form:

Introducing u = g/ f , Eqs.( 9) and ( 10) can be converted into

Subtracting Eq.( 12) from Eq.( 11), one obtains ρ

by letting ρ = u 2 .

Introducing the dependent variable transformation

or

by differentiating Eq.( 12) with respect to s.

In view of Eq.( 14), Eq.( 13) becomes

Introducing the hodograph transformation

and referring to Eq.( 12), we have

Thus, Eqs.( 14) and ( 15) can be cast into

By eliminating ρ, we arrive at

which is actually the SCHE (4).

Based on the link of the SCHE with the two-reduction of 2DTL equations, the N-cuspon solution of the SCHE ( 4) is given as follows:

Moreover, the N-cuspon solution of the SCHE (1) with non-zero κ is given as follows:

where

We remark here that to assure the regularity of the solution, the τ-function is required to be positive definite. In what follows, we list the one-cuspon and two-cuspon solutions. For

The profiles of one-cuspon with κ = 1.0 and κ = 0.1 are plotted in Fig. 1. The τ-function corresponding to the two-cuspon solution is

Here a 1,1 /a 1,2 = -1 and a 2,1 /a 2,2 = 1 are chosen to assure the regularity of the solution.

Based on the link of the SCHE with the two-reduction of 2DTL equations clarified in the previous section, we attempt to construct the integrable semi-discrete analogue of the SCHE.

Consider a Casorati determinant

where ∆ k is defined as

. In particular, we can choose ψ

which automatically satisfies the dispersion relations (20) and (21). The above Casorati determinant satisfies the bilinear form of the semi-discrete 2DTL equation (the Bäcklund transformation of the bilinear equation of the 2DTL equation) [17,18]

Applying a two-reduction condition

by letting τ 0 23) and ( 24) are equivalent to

Subtracting Eq.( 26) from Eq.( 25), one obtains

Introducing the discrete ana

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