Lax Representations for Matrix Short Pulse Equations

TL;DR

This paper develops Lax representations for various matrix generalizations of Short Pulse Equations, including four-component systems, their reductions, and associated bi-Hamiltonian structures, advancing the mathematical understanding of these integrable models.

Contribution

It introduces new four-component Lax representations for matrix SPE generalizations and explores their reductions and bi-Hamiltonian structures, extending prior two-component models.

Findings

Four-dimensional Lax representations for four-component equations obtained.

Reductions to two-component and original equations demonstrated.

Bi-Hamiltonian structure identified for a parameter family of the four-component SPE.

Abstract

The Lax representation for different matrix generalizations of Short Pulse Equations (SPE) is considered. The four-dimensional Lax representations of four-component Matsuno, Feng and Dimakis-M\"{u}ller-Hoissen-Matsuno equations is obtained. The four-component Feng system is defined by generalization of the two-dimensional Lax representation to the four-component case. This system reduces to the original Feng equation or to the two-component Matsuno equation or to the Yao-Zang equation. The three component version of Feng equation is presented. The four-component version of Matsuno equation with its Lax representation is given . This equation reduces the new two-component Feng system. The two-component Dimakis-M\"{u}ller-Hoissen-Matsuno equations are generalized to the four parameter family of the four-component SPE. The bi-Hamiltonian structure of this generalization, for special values of parameters, is defined. This four-component SPE in special case reduces to the new two-component SPE.

Equations

$$u_{x,t}=u+\frac{1}{6}(u^3{)}_{xx},$$

$$u_{xt}$$

$$u_{i,xt}=u_i+\frac{1}{2}(F{u}_{i,x}{)}_x,F=\frac{1}{2}\sum _{1\le j<k\le n}{c}_{j,k}{u}_j{u}_k,$$

$$u_{i,xt}=u_i+(F{u}_{i,x}{)}_x-\frac{1}{2}G{u}_i,G=\sum _{1\le j,k\le n}{c}_{j,k}{u}_{j,x}{u}_{k,x}.$$

$$u_{x,t}=u+\frac{1}{2}v(u^2{)}_{xx},v_{x,t}=v+\frac{1}{2}u(v^2{)}_{xx}.$$

$$=$$

$$U_{xt}=U+\frac{1}{2}(U^2{U}_x{)}_x,$$

$$u_t={\partial }^{-1}u+\frac{1}{2}uv{u}_x,v_t={\partial }^{-1}v+\frac{1}{2}uv{v}_x.$$

$$\left(\begin{array}[]{c}u\\ v\end{array}\right)_{t}=K\left(\begin{array}[]{c}H_{1,u}\\ H_{1,v}\end{array}\right)=J\left(\begin{array}[]{c}H_{0,u}\\ H_{0,v}\end{array}\right),$$

$$H_1$$

$$K$$

$$\left(\begin{array}[]{cc}\psi_{1}\\ \psi_{2}\end{array}\right)_{x}=\lambda\left(\begin{array}[]{cc}1&u_{x}\\ v_{x}&-1\end{array}\right),~{}~{}~{}\left(\begin{array}[]{cc}\psi_{1}\\ \psi_{2}\end{array}\right)_{t}=\left(\begin{array}[]{cc}\frac{\lambda}{2}uv+\frac{1}{4\lambda}&\frac{\lambda}{2}uvu_{x}-\frac{1}{2}u\\ \frac{\lambda}{2}uvv_{x}+\frac{1}{2}v&-\frac{\lambda}{2}uv-\frac{1}{4\lambda}\end{array}\right).$$

$$u_{x,t}=u+u{u}_x^2+\frac{1}{2}(u^2+v^2)u_{xx},v_{x,t}=v+v{v}_x^2+\frac{1}{2}(u^2+v^2)v_{xx}.$$

$$X$$

$$T$$

$$u_{x,t}=u+\frac{1}{6}(u^3{)}_{xx},v_{x,t}=v+\frac{1}{2}(u^2{v}_x{)}_x$$

$${\Pi }_x$$

$$=$$

$$U_{t,x}$$

$$V_{t,x}$$

$$U$$

$$V$$

$$v_{0,t,x}$$

$$v_{1,t,x}$$

$$v_{2,t,x}$$

$$v_{3,t,x}$$

$$f_{t,x}$$

$$g_{t,x}$$

$${\Pi }_x$$

$$=$$

$$U_{t,x}$$

$$V_{t,x}$$

Full text

Institute of Theoretical Physics, University of Wrocław,

Wrocław pl. M. Borna 9, 50-205 Wrocław Poland, ziemek@ift.uni.wroc.pl

1 Introduction

The short pulse equation (SPE)

[TABLE]

derived by Schäfer and Wayne [1] as a model of ultra-short optical pulses in nonlinear media has attracted considerable attention. For the first time the SPE has appeared in an attempt to construct integrable differential equations associated with pseudo spherical surfaces [2]. The associated linear scattering problem appeared for the first time in differential geometry [2, 3].

The integrability of the SPE have been studied from various points of view [4, 5, 6]. More specifically it was shown that SPE equation admits a Lax pair and possess a bi-Hamiltonian formulation.

The problem of mathematical description of the propagation ultra-short pulses has been considered from different points of view. For example, the SPE was generalized in different manners to the 2-component systems. The first generalization have been considered by Pietrzyk Kanattsikov and Bandelow in the form [7].

[TABLE]

Sakovich [8] presented another 2-component generalization

[TABLE]

As was shown, these generalizations are integrable and possess Lax representation and bi-Hamiltonian formulation.

Recently Matsuno [9, 10] generalized, in two different manners, the SPE to the n-component case as

[TABLE]

where $c_{j,k}$ are arbitrary constants such as $c_{j,k}=c_{k,j}$.

For these equations Matsuno applied the bilinear method and presented multisolitons solutions in the parametric form. Moreover, Matsuno found the Lax representation, local and nonlocal conservation laws for the equation 5, for special case $n=2,u_{1}=u,u_{2}=v$ and $c_{11}=c_{22}=0,c_{12}=1$

[TABLE]

Matsuno defined the zero curvature condition for the equation 6 as $\textbf{X}_{t}-\textbf{Z}_{x}+[\textbf{X},\textbf{Z}]=0$  where

[TABLE]

Dimakis and Müller-Hoissen [11] studied matrix generalization of the SPE in the form

[TABLE]

where $U$ is matrix valued function. Assuming that $U^{2}$ has to be a scalar times the identity matrix, Dimakis and Müller-Hoissen were able to construct the Lax pair. When $U^{2}=(u^{2}+v^{2})I$, where $I$ is an identity matrix, the equation (13) reduces, after the transformations of variables $(u,v)\rightarrow((u+v)/2,(u-v)/2i)$, to

[TABLE]

This equation, as it was shown by Matsuno [9], is a special case of the system (4) when $n=2,c_{1,1}=c_{2,2}=0,c_{1,2}=1,u_{1}=u,u_{2}=v$.

For this equation, Brunelli and Sakovich found [12] the bi-Hamiltonian formulation

[TABLE]

where

[TABLE]

and Matsuno [9] discovered the Lax representation

[TABLE]

Quite different generalization of two-component SPE was proposed by Feng [13]

[TABLE]

Brunelli and Sakovich in [12] obtained the zero curvature condition $X_{t}-T_{t}+[X,T]=0$ and the bi-Hamiltonian structure for this equation, where

[TABLE]

The last two-component generalizations of SPE were given by Yao-Zeng [14]

[TABLE]

for which Brunelli and Sakovich found, using the four-dimensional matrices, the Lax representation in [12].

In this paper we would like to study several different generalizations of the two-component SPE to the matrix and then to the four-component case. Therefore we investigate the equations obtained from the matrix generalization of the Lax representations eqs. (9,22,26).

The paper is organized as follows. In the following two chapters we study the matrix version of the Lax representation of Matsuno and Feng equations. The four-component Matsuno equation is discussed in the first chapter too, and its reduction to the new version of the two-component Feng equation is given. In the third section, we study the four-component version of Feng equation and its different reductions. This generalization describes the interaction between the Feng, Matsuno and Yao-Zeng equations and reduces to the original Feng equation or to the two-component Matsuno equation or to the Yao-Zeng system. Also, the three-component version of Feng equation is obtained as a result of reduction. In the last section we study the matrix Lax representation of the two-component Dimakis-Müller-Hoissen - Matsuno equation. This representation in the case of four-dimensional matrices produces the four parameters family of four-component SPE. For special values of free parameters we defined the bi-Hamiltonian structure for this equation. As a result of reduction of this four parameter equations, we obtained a new two-component version of SPE. The last section is a conclusion. The paper contains two appendixes.

2 Lax Representation of Matrix Matsuno Equation

Let us consider the Lax representation for matrix Matsuno equation

[TABLE]

where $I$ is identity matrix, $U$ and $V$ are arbitrary dimensional matrices such that $UV$ and $U_{x}V_{x}$ are scalar functions multiplied by identity matrix.

From the integrability condition $\Pi_{x,t}=\Pi_{t,x}$ we obtained the following matrix equations

[TABLE]

These equations reduce to the Matsuno eq.(6) when $U,V$ are scalar functions.

The four-component version of the Matsuno equation are concluded from eq.(40) assuming that

[TABLE]

where $\sigma_{i}$ are Pauli matrices and $I$ is identity matrix.

[TABLE]

The system eq.(42) allows interesting reduction to the two-component case when $v_{1}=v_{0}=\frac{1}{2}(f+g),~{}~{}~{}v_{3}=v_{2}=\frac{1}{2}(f-g)$

[TABLE]

and describes the Feng fields plus additional interaction.

3 Lax Representation of Matrix Feng Equation

We postulate the Lax representation for the matrix Feng equation as

[TABLE]

where $I$ is identity matrix, $U$ and $V$ are arbitrary matrices such that $UV$ and $U_{x}V_{x}$ are scalar functions multiplied by the identity matrix.

The matrix Feng equation is obtained from the integrability condition $\Pi_{x,t}=\Pi_{t,x}$ and reads

[TABLE]

When $U$ and $V$ are scalar function, then eq.(53) reduces to the Feng equation (23).

In the next section we show that this system, in the case where we consider the two dimensional matrices $U$ and $V$, contains the Feng, Matsuno and Yao-Zheng equations.

4 Unification of Feng, Matsuno and Yao - Zeng equations.

The four-component Feng equation is concluded from eq.(53) assuming that

[TABLE]

where $\sigma_{i}$ are Pauli matrices and $I$ is identity matrix.

[TABLE]

We see that this system of equations describes the interaction between Feng fields $\{v_{0},v_{1}\}$ and Matsuno fields $\{v_{2},v_{3}\}$. For this reasons the system of equations could be considered as the interacting system of Feng - Matsuno type.

The Matrices X and T in the Lax representation eq.(46) are rewritten in terms of the 16 dimensional Lie algebra, spanned by the generators $\{e_{1},e_{2},\dots,e_{16}\}$ as

[TABLE]

where $\gamma=\frac{1}{2}(v_{0}^{2}+v_{1}^{2}+2v_{2}v_{3})$,

The explicit representation of the Lie algebra $e_{e},e_{2},\dots e_{16}$ is given in the appendix A.

The system of equations (55) allows three different reductions to the two-component case.

When $v_{2}=v_{0}=\frac{1}{2}(a+b),~{}~{}~{}v_{3}=v_{1}=\frac{1}{2}(a-b)$ then eq.(55) reduces to the Yao-Zeng eq.(30) in the variables $a,b$.

For this type of reduction the matrices $\textbf{X},\textbf{T}$ are

[TABLE]

This representation is different than the one given in [12].

When $v_{1}=v_{0}=\frac{1}{2}(a+b),v_{3}=v_{2}=\frac{1}{2}(a-b)$ then eq.(55) reduces to the Feng eq.(23).

When $v_{3}=v_{0}=\frac{1}{2}(a+b),v_{2}=v_{1}=\frac{1}{2}(a-b)$ then eq.(55) reduces to the Yao-Zeng eq.(30) again but now we have different representations of the matrices $\textbf{X},\textbf{T}$

[TABLE]

The system (55) allows the reduction to the three component case as well. For example, assuming that

[TABLE]

then equations (55) reads as

[TABLE]

For this type of reduction the matrices $\textbf{X},\textbf{T}$ are

[TABLE]

5 Lax representation of Matrix Dimakis-Müller-Hoissen-Matsuno Equation

Now let us consider the matrix generalization of Lax pair representation eq.(22)

[TABLE]

where $I$ is an $N$ dimensional identity matrix and $U(x,t),V(x,t)$ are at the moment arbitrary $N$ and dimensional matrices.

The zero-curvature condition \Omega_{t}-P_{x}+\big{[}\Omega,P\big{]}=0 produces the following equations on $U,V$

[TABLE]

with the constraints on $U,V$

[TABLE]

The equations (71) with the constraints 72 are our matrix generalization of two-component Dimakis-Müller-Hoissen-Matsuno equation different than the one proposed in [11].

Now we consider the case where $U$ and $V$ are two-dimensional matrices. We parametrize $U$ by two functions $u_{1},u_{2}$ and $V$ by two functions $u_{3},u_{4}$ as

[TABLE]

where $s_{i},z_{i},i=1,2,3,4$ are an arbitrary constants. 111In general it is possible to parametrize the two dimensional matrix by two arbitrary functions in six different manners. However, all these parametrizations could be obtained from eq.(73) using the linear transformations of these functions.

Substituting $U,V$ defined by eq.(73) to the equations (72), we find the connection between $z_{i}$ and $s_{i}$

[TABLE]

Now the Lax representation generates four parameters family of equations

[TABLE]

These equations are integrable in the sense that they possess the Lax representation and allow several different reduction to the one or two-component SPE equation case.

When $u_{4}=u_{3}=u_{2}=u_{1}=u,s_{1}=-s_{2},s_{4}=-s_{3}-1$ the equations (5) reduce to the original SPE equation (1).

For $u_{1}=u_{2}=u,u_{3}=u_{4}=v,s_{1}=1-s_{2},s_{4}=s_{3}-1$ the equations (5) reduce to the two-component Dimakis-Müller-Hoissen-Matsuno (14).

When $u_{1}=u_{3}=u,u_{2}=u_{4}=v,s_{1}=s_{4}=0,s_{3}=1$ and $s_{2}=\pm 1,0$ we obtained three generalizations of the SPE equation considered by Pietrzyk at.al.

On the other hand for $u_{1}=u_{3}=u,u_{2}=u_{4}=v,s_{1}=0,s_{2}=1,s_{3}=1,s_{4}=-2$ we obtained new two-component generalization of the SPE equation

[TABLE]

This equation was missing in the classification of Pietrzyk at.all [7]. For this equation the Lax representation is exactly as in the paper [7]

[TABLE]

but now

[TABLE]

We found the bi-Hamiltonian formulations for the equations (5) for $s_{1}=0,s_{2}=1$.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

The proof that this operator satisfies the Jacobi identity is postponed to the appendix A.

Moreover we showed that the Hamiltonians operators $J_{0}$ and $J_{1}$ are compatible. It means that $\mu J_{0}+J_{1}$ is also the Hamiltonian operator, where $\mu$ is an arbitrary constant.

[TABLE]

The equation (5) for $s_{4}=-2,s_{3}=1$ possesses very nice bi-Hamiltonian structure

[TABLE]

[TABLE]

[TABLE]

[TABLE]

6 Conclusion

In this paper we studied several different generalizations of the two-component SPE to the matrix case and then to the four-component case. In particular we studied four-component version of Feng, Matsuno and Dimakis-Müller-Hoissen-Matsuno equations with its different reductions. The four-component Feng equation is very reticular, because it contains the Feng, Matsuno and Yao-Zeng equations and thus unifies these equations.

The four parameters of four-component Dimakis-Müller-Hoissen-Matsuno equations were discussed. For special values of free parameters, we obtained the bi-Hamiltonian structure and presented new two-component SPE (76).

We obtained several new integrable equations in the sense that these possess the Lax representation and this open perspective for further investigations. For example the problem of finding the bi-Hamiltonian structures for other generalized SPE is still an open issue, similarly as the problem of presentation of their solutions.

7 Appendix A. Representation of $e_{1},e_{2}$,…$e_{16}$

[TABLE]

[TABLE]

8 Appendix B. Hamiltonian operators and verification of the Jacobi identity

.

We assume the most general form on the second Hamiltonian operator as $J_{1}=J_{0}+\hat{J}$ where

[TABLE]

where $c_{j,k,s,l}$ are arbitrary constants $j,k,s,l=1\dots 4$ for $j\neq k$ and $c_{j,j,s,k}=c_{j,j,k,s}$.

Now we will investigate the equations obtained from the second Hamiltonian structure

[TABLE]

and compare the solutions with the solutions obtained from Lax representation eq. (66). This fixes all coefficients $c_{i,j,k,l}$ and moreover restricts $s_{1}=0,s_{2}=1$.

We use traditional manner to verify the Jacobi identity [15]. In order to prove that the operator $J_{1}$, defined in (87), satisfies the Jacobi identity, we utilize the standard form of the Jacobi identity

[TABLE]

where $A,B$ and $C$ are the test vector functions, for example $A=(a_{1},a_{2},a_{3}.a_{4})$ while $\star$ denotes the Gato derivative along the vector ${\cal{L}}(B)$. We check this identity utilizing the computer algebra Reduce and package Susy2 [16]. We will, briefly explain our procedures applied for the verification of the Jacobi identity (235).

In the first stage, we remove the derivatives from the test functions $a_{i,x},b_{i,x}$ and $c_{i,x}$ in the Jacobi identity, using the rule

[TABLE]

where $f$ is an arbitrary function.

Because of this, the Jacobi identity can be split into three segments. The first and second segments contain terms in which the integral operator $\partial^{-1}$ appears twice and once respectively. The last segment does not contain any integral operators. We consider each segment separately.

The first segment is a combination of the following expressions:

[TABLE]

where $n_{a}$ denotes $n_{j}a_{i}$ or $n_{j,x}a_{i}$ or $n_{j,xx}a_{i},~{}~{}~{}i,j=1,2,3,4$ and similarly for $n_{b},n_{c},$ $\tilde{n}_{a},\tilde{n}_{b},\tilde{n}_{c}$.

Here $\partial^{-1}$ is an integral operator, and, therefore, each ingredient could be rewritten as, for example,

[TABLE]

Now, we replace $n_{a}$ in the last formula by

[TABLE]

Hence, the expression (237) transforms to

[TABLE]

Now repeating this procedure for $n_{a}$ and $\tilde{n}_{a}$ in the first segment, it appears that this segment reduces to zero.

The second segment is constructed from the combinations of the following terms:

[TABLE]

where $\Lambda_{a}$ takes values in $\{n_{j}a_{i},~{}n_{j,x}a_{i},~{}n_{j,xx}a_{i},~{}n_{j}a_{i,x},~{}n_{j}a_{i,xx}\},i,j=1,2,3,4$. In a similar manner the $\Lambda_{b},\Lambda_{c},\tilde{\Lambda}_{a},\tilde{\Lambda}_{b},\tilde{\Lambda}_{c}$ are defined. These terms are rewritten as

[TABLE]

Next, using rule (236), we replace once more the derivatives in $a_{k,x}$ and $b_{k,x}$ in the second segment. After this replacement, it appears that the second segment contains no term with an the integral operator. Therefore, we add this segment to the third segment.

Now, it is easy to check that this last segment vanishes. Indeed, it is enough to use the rule (236) in order to remove the derivatives from $a_{k,x}$ in the last segment.

This finishes the proof.