Some Wadati-Konno-Ichikawa type integrable systems and their constructions

A standard-form Wadati-Konno-Ichikawa(WKI) type integrable hierarchy is derived from a corresponding matrix spectral problem associated with the Lie algebra sl(2, R). Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the trace identity. Then, the higher grading affine algebraic construction of some special cases is proposed. We also show that eneralized short pulse equation arises naturally from the negative WKI flow.


Introduction
Integrable systems (or integrable hierarchies) have attracted extensive attention in natural science because of successful description and explanation of nonlinear phenomena. Matrix spectral problems associated with Lie algebras are crucial keys to construct integrable hierarchies. There has been a lot of work on the generation of integrable hierarchies from matrix spectral problems and interesting examples contain the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy, the Kaup-Newell (KN) hierarchy, the Wadati-Konno-Ichikawa (WKI) hierarchy, the Korteweg-de Vries (KdV) hierarchy, the modified KdV hierarchy, the Benjamin-Ono hierarchy, the Boiti-Pempinelli-Tu (BPT) hierarchy, the Dirac hierarchy and the coupled Harry-Dym hierarchy [1][2][3][4][5][6][7][8]. These integrable hierarchies usually possess nice properties such as having hereditary recursion operators, being multi-Hamiltonian, and carrying infinitely many commuting symmetries and conservation laws. The so-called trace identity (or variational identity) provide a systematic construction approach for establishing Hamiltonian structures of integrable hierarchies [9][10][11][12].
WKI type integrable systems [3,13,14] not only represent the classical WKI equation [3], but also represent a large class of related equations such as the following short pulse (SP) equation, SP equation is proposed as a model to describe ultra-short optical pulses traversing within a nonlinear media [15][16][17].
Recently, how to use some specific algebra as a tool to systematically construct integrable systems began to develop, which aroused great interest of scholars. In Refs. [18][19][20], the algebraic construction based on Toda field theory is generalized by the addition of fields associated to higher grading operators, yielding the generalized affine Toda models. The higher grade fields are physically interpreted as matter fields with the usual Toda fields coupled to them. In Ref. [21], authors propose a general higher grading construction for the zero curvature equation, containing the WKI hierarchy as a particular case. In the construction, the zero grade Toda fields are completely removed, remaining the higher grade fields only. Based on this method, one can obtain a series of mixed integrable systems such as the mixed mKdV-sine-Gordon equation, the mixed AKNS-Lund-Regge equation and the mixed super symmetric mKdV-sinh-Gordon equation. Integrable systems obtained from negative flows have important physical and mathematical significance, such as the Camassa-Holm (CH) equation, the Degasperis-Procesi (DP) equation and the Vakhnenko equation. A mixed WKI-SP model has been found by combining the positive flows and extend the WKI hierarchy to incorporate negative flows [21,22].
The remainder of this paper is organized as follows. In section 2, we would like to construct a standard-form WKI type integrable hierarchy and bi-Hamiltonian structure by using the trace identity. In section 3, we introduce the higher grading affine algebraic construction method and some special cases in the obtained hierarchy are considered. The local and non-local conserved charges are obtained from the Riccati form. In section 4, higher order SP equation and mixed WKI-SP equation are derived by considering the negative flow and mixed flow. The last section is devoted to conclusions and discussions.

A standard-form WKI type integrable hierarchy
For the sake of readability, let's first introduce the three-dimensional real special linear Lie algebra sl(2, R). This algebra consisting of trace-free 2 × 2 matrices, has the basis We can also define the corresponding matrix loop algebra sl(2, R), Thus, a brief account of the procedure for building integrable hierarchies associated with sl(2, R) is described below.
Step 1: One needs to select an appropriate spectral matrix U to form a spatial spectral problem φ x = Uφ.
Step 2: Construct a particular Laurent series solution W = W (u, λ), to the stationary zero curvature equation W x = [U, W ], based on which one can also prove the localness property for W .
Step 3: By means of the solution W obtained in the above step, introduce temporal spectral Step 4: Finally, furnish Hamiltonian structures u tm = K m (u) = J δHm δu , m ≥ 0 by trace identity. In Ref. [13], a WKI type spatial spectral problem, which is associated with sl(2, R), is defined by When α = 0, it is exactly the classical WKI spatial spectral problem [3]. In order to construct recursion relations, we consider the following standard-form of matrix W , Remark 1. In Ref. [13], Matrix W is taken as follows Eq. (2.3) is easy to show the correspondence between the Tu scheme method and the higher grading structure construction method.
Firstly, we solve the stationary zero curvature equation Substituting the following Laurent series expansion and To determine the recursion relation between {b k+1 , c k+1 } and {b k , c k }, we need to represent a k+1 by {b k , c k }. In order to achieve this purpose, we rewrite a kx as Thus we change a k+1,x to Let's rewrite the above equation again as pq + 1a k+1 It means that we arrive at So we can compute {a k , b k , c k , k ≥ 1} recursively from the following initial values by using Eq.(2.9) and the last two equations of (2.6). Here {a 0 , b 0 , c 0 } are determined by the initial conditions (2.7). To guarantee the uniqueness of {a k , b k , c k }, we also need impose the integration conditions Here we use Maple software to deal with complicated symbolic computations. The first two sets are listed as follows: In fact, values of functions {a k , b k , c k , k ≥ 0} are all local. We can prove this fact from the recursion relations of the last two equations in Eq.(2.6) and which is derived from a 2 + bc = (a 2 + bc) u=0 = 1. Now, taking Due to (2.6), we can easily see that Thus we can introduce and then p tm , q tm can be expressed as follows Therefore, we have obtained a WKI type integrable hierarchy associated with the Lie algebra sl(2, R): (2.12) When α = 0, it just is the classical WKI integrable hierarchy [3]. Next, we construct Hamiltonian structures of the above WKI type integrable hierarchy (2.12), which are furnished by using the following trace identity [9][10][11][12], where W solves the stationary zero curvature equation W x = [U, W ]. Thus, the corresponding trace identity becomes By means of Eq.(2.12), we can compute , Thus,the above integrable hierarchy (2.12) can be represented as the following Hamiltonian forms with the Hamiltonian operator It is now a direct computation that all members in Eq.(2.12) are bi-Hamiltonian. We compute So we arrive at Therefore, it is easy to see that the WKI type integrable hierarchy (2.12) is Liouville integrable. 8

The higher grading construction
Firstly, we give a brief description of the procedure for the higher grading construction method. A more detailed description of the method process is given in Ref. [21]. Let G be an affine Kac-Moody algebra and Q an operator decomposing the algebra into the graded subspaces As a consequence of the Jacobi identity, [ Integrable systems can be constructed from the zero curvature equation where U and V lied on G and have the following forms With this algebraic structure, this zero curvature equation can be solved non trivially. The projection into each graded subspace yields the following set of equations where T j ∈ {H j , E j + , E j − } and j is an integer. For construction of integrable systems we just need to use the loop algebra, achieved by settingĉ = 0. The homogeneous gradation Q =d yields the grading subspaces Fixing the semi-simple element as E = αpH 0 + H 1 , we have K (j) = {H j } and M (j) = {E j + , E j − }. Thus the operator containing functions p ≡ p(x, t), q ≡ q(x, t) have the form By setting the Lax operator V which is a sum of elements in the form we have the following zero curvature equation of positive flow Here coefficients a j , b j , c j will be determined in terms of the field functions p and q. When n = 2, the grad-by-grad decomposing of the above equation (3.3) leads to Therefore, we can get the following WKI type integrable system, whose Lax pair is given by This equation (3.4) just is the first equation in the WKI type integrable hierarchy (2.12) with m = 0 by replacing −U, −V with U, V respectively. In fact, we can find that there is a correspondence between the method in section 2 and this method. In other words, this method gives a Kac-Moody algebraic interpretation of the Tu scheme method.
Similarly, when n = 3, we can construct the following WKI type integrable system, with Lax pair This equation (3.5) just is the second equation in the WKI type integrable hierarchy (2.12) with m = 1 by replacing −U, −V with U, V respectively..

Remark 2.
If setting E = α √ pq + 1H 0 + H 1 proposed in Ref. [14], we can get similar results according to the above method.
Next we shall derive local and nonlocal charges from the Riccati form of the spectral problem By using the matrix representation, we have where λ is the spectral parameter. Introducing the variables we can write the Riccati form of the spectral problem (3.6), whose compatibility yields the conservation laws Therefore, we can construct an infinite number of conserved charges by using pΓ and qΓ −1 , assuming a power series in λ. Let F = pΓ and G = qΓ −1 , we can obtain which are the generating equations for the conserved densities.
To get the local density, we expand F in the power series of 1/λ, Substituting (3.11) into (3.10) and equating the terms of the same powers of 1/λ, we obtain conserved densities, The charges associated to these densities are These charges are the Hamiltonians generating the WKI type integrable systems within the positive flows.
To consider furthermore, we set the most general expansion, (3.12) which can generate nonlocal densities. Substituting Eq.(3.12) into Eq.(3.10), we can get the following system Using the similar method in Ref. [21], we can get Here P and Q are defined as P = pe 2α∂ −1 x p , Q = qe −2α∂ −1 x p . The respective conserved charges are give by (3.13) Thus we believe that the charges are conserved, as can be explicitly checked, either from the positive or negative flows.

Remark 3.
For the spectral problem (3.6) with U = α √ pq + 1H 0 + H 1 + pE 1 + + qE 1 − , the local and nonlocal charges can be worked out in the same way.

SP type integrable systems
In this section, we study SP type integrable systems by using the Lax operator . Here we introduce the operator ∂ −1 x f (x) = x −∞ f (y)dy and assume that the fields and its derivatives of any order decay sufficiently fast when | x |→ ∞. Under this condition ∂ According to our construction, the negative flows can be constructed from the zero curvature equation When n = 1, we can obtain the following two-component SP equation [21] u xt = 4u + 2∂ x (uvu x ), with corresponding Lax pair Contrary to the positive flows of the WKI type equations, the SP equation does not seem do describe large amplitude solutions. A multi-component generalization of the above equation with the same structure, has also been proposed in Ref. [23]. This generalization also can be obtained by considering the untwisted algebraÂ n−1 ∼ŝl(n). These conclusions have been proposed in Ref. [21], so we continue to consider higher order SP type integrable systems. When n = 3, zero curvature representation reads which can decomposes into six independent equations: We solve this system step by step. The projection into G (−3) implies that a −3 , b −3 and c −3 are all constants. To consider furthermore, if setting a −3 = b −3 = 0, we have c −2 = constant and by calculating the G (−2) projection. Then the G (−1) projection yields Similarly,we can obtain the following equation from the G (0) projection The projection into G (2) implies that a 1 = pc 1 and b 1 = qc 1 . The G (1) projection yields the field equations plus one constraint Substituting (4.2) into the third equation of (4.3), and choosing c −2 = 0, we can obtain Fixing c −3 , we have the following nonlocal equations Introducing a new field function defined through p = −q = u xxx , we get the following model which is the higher order SP equation. Here c −3 is an arbitrary constant. The Lax pair of this model reads by setting c −3 = 1 for the convenience of writing. It is possible to combine a positive flow with a negative flow, so we consider mixed WKI-SP integrable systems by using the following zero curvature equation It can decompose into eight independent equations, We can exactly solve each grade projection starting from highest to lowest. From the G (0) to the G (4) projection, the process of operation is almost similar to the positive flows in the case n = 3. Thus, in the G (0) projection, we have From the G (4) projection, we have a 3 = pc 3 and b 3 = qc 3 . The G (3) projection gives The G (2) projection gives So we can obtain Now we let some coefficients, that were previously considered constants, to depend on time t, thus providing the non-autonomous ingredient. For individual flows those coefficients were not interesting because they come as a global factor in the final equation. From the G (−3) to G (−1) projection, the results are the same as (4.4). Therefore, we get after choosing p = −q = u xxx . Here a(t) and b(t) are arbitrary functions. This equation (4.7) just is a higher order mixed WKI-SP integrable system. Due to a(t) and b(t) the dispersion relation will have a time depend velocity and the solitons will accelerate [21]. Eq.(4.7) may be nice candidates in applications having accelerated ultra-short optical pulses [21].

Conclusions and discussions
In this paper, we have constructed a standard-form WKI type integrable hierarchy (2.12), together with a bi-Hamiltonian structure (2.14) by using the trace identity. This method can be used to other integrable hierarchies as well.
By using the higher grading construction method, we give a Kac-Moody algebraic interpretation of some special equations in this hierarchy (2.12). In fact, the higher grading construction method is general and other affine Lie algebras can be considered. we might be able to use this method to construct novel integrable models. We also have derived local and nonlocal charges from the Riccati form of the spectral problem (3.6).
We have extended this WKI type integrable hierarchy to negative flow, which yields a higher order SP type integrable system (4.4). A novel integrable non-autonomous WKI-SP equation (4.7) is also proposed, mixing a positive with a negative flow. This mixed model may have applications in nonlinear optics, specially concerning accelerated ultra-short optical pulses [21].
Here I 2 is the 2 × 2 identity matrix and q = q 1 q 2 −q * 2 q * 1 , q = q * 1 −q 2 q * 2 q 1 . The exact solutions and physical applications of these equations need to be further studied.