On the Classical and Deformed Korteweg-de Vries Equation

Abstract

Given the general nonlinear partial differential equations and the importance of the Korteweg-de Vries equation (KdV) in physics, this chapter presents a basic survey of the two-dimensional Korteweg-de Vries model. We begin by examining various symmetries of systems, and then explore the concept of integrability through two different methods: the Hamiltonian formalism and the existence of conserved quantities. By introducing the concept of q-deformation, we construct the corresponding q-deformation integrable model and the integrability of the resulting system is guaranteed by the existence of Lax pairs. We also present the KdV equation in the Moyal space of moments in its noncommutative version, we present the algebraic structure of the system and we study the integrability using the notion of Lax pair.

1. Introduction

The axis of nonlinear integrable systems is one of the interesting topics that has been recently examined from different points of view [1, 2, 3]. They are soluble models which present a very rich structure and are used in many fields of modern sciences such as mathematics, physics, chemistry and engineering sciences [3, 4, 5]. From a physical point of view, integrable systems play an important role in describing physical phenomena such as condensed matter, fluid dynamics, plasma physics, high energy physics and nonlinear optics [4, 6].

Nonlinear integrable systems involve nonlinear differential equations that can sometimes be solved exactly, but this is generally not a trivial task. However, we introduce some rigorous background such as the theory of pseudo-differential operators, Lie algebras, the inverse scattering method and some physical methods such as the related Lax formulation is needed [7, 8, 9]. The particularity of two-dimensional integrable systems lies in their revolutionary properties worthy of nonlinear differential equations such as Burgers, KP, Boussinesq, KdV [5].

In this work, we investigate some properties related to this prototype nonlinear differential equation, based on the classical Q-deformation version and the interesting system KdV of the Moyal momentum deformation, and show how to reinforce its central role [10, 11].

The composition of this work is as follows. Section 2 gives some interesting definitions of the 2d-KdV formula. Determining the Hamiltonian structure of the KdV equation, he guarantees the integrability of the KdV equation by the existence of infinitely many independent conserved quantities in the involution. in seconds In 3, we present some results related to the Lax representation of the q-deformed version. Using conventional notation and Q-analysis, we perform consistent algebraic computations based on pseudo-differential analysis and derive explicit loose pairing operators for KdV systems in the q-deformation framework. Section 4 presents some results related to his Lax representation of the Moyal $-\theta$ integrable hierarchy. Using the notation and analysis previously presented and developed in [12, 13], we perform coherent algebraic computations based on Moyal momentum analysis and integrable systems for some of the $\theta -$ deformation frames. We derive an explicit Lax pair operator for. Finally, Section 5 summarizes our research.

2. Classical 2d KdV equation

Note that the KdV equation is formulated without considering the integrability of this equation. It is a nonlinear dispersive partial differential equation for a function u of two real variables x and t, and its form is given by [2]:

$$\frac{\partial u}{\partial t}+\mathit{au}\frac{\partial u}{\partial x}+b\frac{{\partial }^{3}u}{\partial x^3}=0$$E1

where $u\left(x,t\right)$ is a dynamic variable that plays the role of basin wave height.

The eq. (1) contains two arbitrary constants “a” and “b” which can be eliminated after a series of transformations namely:

• Transformation on spatial variables $x$ (scale transformation):

  $$x\to b^{1/3}x$$E2

• Transformation on the time variable t (time reversal):

  $$t\to -t$$E3

• Transformation on the dynamic variable

$$u\to a^{-1}{b}^{1/3}u$$E4

After performing these transformations, the KdV equation can be reformulated in this way.

$$\frac{\partial u}{\partial t}=u\frac{\partial u}{\partial x}+\frac{{\partial }^{3}u}{\partial x^3}$$E5

This equation is the most used in KdV theory. However to find the solutions of this equation and to classify them, we must know the symmetries of the equation. In fact, the KdV equation then has these four symmetries

$$\begin{array}{l}i)t\to d_1+t,\\ \mathit{ii})x\to d_2+x,\\ \mathit{iii})x\to \mathit{\alpha x},t\to {\alpha }^{3}t\text{and}u\to {\alpha }^{-2}u\\ \mathit{iv})x\to x+\mathit{vt},t\to t\text{and}u\to u+v\end{array}$$E6

All these transformations are symmetries of the KdV equation. Note that transformation iii) is used for the classification of the conserved quantities of the system and that transformation iv) is considered as the Galilean invariance of such equations. However, the KdV equation does not have Lorentz invariance because of the separable treatment of the time and space variables.

2.1 Integrability through Hamiltonian structure

Let us consider the eq. (5), in order to give the Hamiltonian form of our equation, We need to calculate a a Poisson bracket structure and an Hamiltonian operator of the system [14]. In fact, we have to make, under somes assumptions, the KdV equation in the following Hamiltonian form:

$$\frac{\partial u}{\partial t}=\left\{u\left(x,t\right),H\right\}$$E7

Here $\left\{.,.\right\}$ represents the Poisson brackets and H is the Hamiltonian of our system. In fact, rewriting the KdV equation in the form

$$\begin{array}{l}\frac{\partial u}{\partial t}=\frac{\partial }{\partial x}\left(\frac{1}{2}{u}^2\right)+\frac{\partial }{\partial x}\left(\frac{{\partial }^{2}u}{\partial x^2}\right)\\ =\frac{\partial }{\partial x}\left(\frac{1}{2}{u}^2+\frac{{\partial }^{2}u}{\partial x^2}\right)\end{array}$$E8

we can then choose the following Hamiltonian:

$$H\left[u\right]={\int }_{-\infty }^{+\infty }\mathit{dx}\left(\frac{1}{3!}{u}^3-\frac{1}{2}{\left(\frac{\partial u}{\partial x}\right)}^2\right)$$E9

Based on the functional derivative defined by

$$\frac{\mathit{\delta F}\left(u\left(x\right)\right)}{\mathit{\delta u}\left(x\right)}=\underset{\epsilon \to 0}{lim}\frac{1}{\epsilon }\left(F\left[u\left(x\right)+\mathit{\epsilon \delta }\left(x-y\right)\right]-F\left[u\left(x\right)\right]\right)$$E10

we can calculate the variation of H with respect to the dynamic variable $u$ by

$$\frac{\mathit{\delta H}}{\mathit{\delta u}\left(x\right)}=\frac{1}{2}{u}^2+\frac{{\partial }^{2}u}{\partial x^2}$$E11

with this choice of H, the eq. (5) becomes

$$\frac{\partial u}{\partial t}=\frac{\partial }{\partial x}\frac{\mathit{\delta H}}{\mathit{\delta u}\left(x\right)}$$E12

by comparing the eqs. (7) and (12), we then find

$$\left\{u\left(x\right),H\right\}=\frac{\partial }{\partial x}\frac{\mathit{\delta H}}{\mathit{\delta u}\left(x\right)}$$E13

that is to say

$${\int }_{-\infty }^{+\infty }\mathit{dy}\frac{\mathit{\delta H}}{\mathit{\delta u}\left(x\right)}\left\{u\left(x\right),u\left(y\right)\right\}=\frac{\partial }{\partial x}\frac{\mathit{\delta H}}{\mathit{\delta u}\left(x\right)}$$E14

in this case, the Poisson bracket of $u\left(x\right)$ and $u\left(y\right)$ is given by

$$\left\{u\left(x\right),u\left(y\right)\right\}=\frac{\partial }{\partial x}\delta \left(x-y\right)$$E15

This defines the Poisson structure associated to our KdV equation, however this bracket is antisymmetric and it verifies the Jacobi identity [2].

We can also define a second Hamiltonian structure of the KdV equation, for this we can choose a second structure of the Poisson bracket namely

$${\left\{u\left(x\right),u\left(y\right)\right\}}_2=\left(\frac{1}{3}\left(\frac{\partial }{\partial x}u\left(x\right)+u\left(x\right)\frac{\partial }{\partial x}\right)+\frac{{\partial }^3}{\partial x^3}\right)\delta \left(x-y\right)$$E16

and a second Hamiltonian operator

$$H_2\left[u\right]={\int }_{-\infty }^{+\infty }\mathit{dx}\frac{1}{2}{u}^2\left(x\right)$$E17

with this choice, the equation of KdV becomes

$$\begin{array}{ll}\frac{\partial u}{\partial t}& ={\left\{u\left(x,t\right),H_2\right\}}_2\\ & ={\int }_{-\infty }^{+\infty }\mathit{dy}\frac{\delta H_2}{\mathit{\delta u}\left(x\right)}{\left\{u\left(x\right),u\left(y\right)\right\}}_2\\ & =u\frac{\partial u}{\partial x}+\frac{{\partial }^{3}u}{\partial x^3}\end{array}$$E18

Remark

• This second Poison structure is also antisymmetric and it verifies the Jacobi identity, thus we can identify it with the Virasoro algebra with a specific charge [6].

• Based on the Lagrangian formalism, we can re-establish the KdV equation, indeed

$$\begin{array}{l}{L}_{\mathit{KdV}}=\frac{1}{2}{\int }_{-\infty }^{+\infty }\mathit{\text{dxdyu}}\left(x\right)\epsilon \left(x-y\right)\frac{\partial u\left(y\right)}{\partial t}\\ -{\int }_{-\infty }^{+\infty }\mathit{dx}\left(\frac{1}{3!}{u}^3\left(x\right)-\frac{1}{2}{\left(\frac{\partial u\left(x\right)}{\partial x}\right)}^2\right)\end{array}$$E19

The Euler Lagrange equations obtained are

$${\int }_{-\infty }^{+\infty }\mathit{dy\epsilon }\left(x-y\right)\frac{\partial u\left(y\right)}{\partial t}=\frac{1}{2}{u}^2\left(x\right)+\frac{\partial u^2\left(x\right)}{\partial x^2}$$E20

this means that

$$\frac{\partial u}{\partial t}=u\frac{\partial u}{\partial x}+\frac{{\partial }^{3}u}{\partial x^3}$$E21

so we have recovered the eq. (5). The existence of the term $\epsilon \left(x-y\right)$ in the Lagrangian implies the nonlocality of our system.

2.2 Integratability through the conserved quantities

In general, a quantity $Q\left[u\right]$ is said to be conserved if for a bracket $\left\{,\right\}$ and an Hamiltonian $H$, we have th following equation

$$\frac{\mathit{dQ}\left[u\right]}{\mathit{dt}}=\left\{Q\left[u\right],H\right\}=0$$E22

if moreover we take this expression

$$Q\left[u\right]={\int }_{-\infty }^{+\infty }\mathit{dx\rho }\left[u\left(x,t\right)\right]$$E23

we have

$$\frac{\mathit{dQ}\left[u\right]}{\mathit{dt}}={\int }_{-\infty }^{+\infty }\mathit{dx}\frac{\partial \rho \left[u\left(x,t\right)\right]}{\partial t}=0$$E24

this implies the existence of the continuity equation of the following form

$$\frac{\partial \rho \left[u\left(x,t\right)\right]}{\partial t}+\frac{\partial j\left[u\left(x,t\right)\right]}{\partial x}=0$$E25

In the following, we write the KdV eq. (5) as an equation of continuity (8) and compare it with the eq. (25), which then gives us the following equations

$$\begin{array}{l}{\rho }_0\left[u\left(x,t\right)\right]=u\left(x,t\right)\\ j_0=-\left(\frac{1}{2}{u}^2+\frac{{\partial }^{2}u}{\partial x^2}\right)\end{array}$$E26

consequently

$$Q_0=H_0={\int }_{-\infty }^{+\infty }\mathit{dx}{\rho }_0\left[u\left(x,t\right)\right]={\int }_{-\infty }^{+\infty }\mathit{dx}u\left(x,t\right)$$E27

is a constant of movement. In the following, we redefine ${\rho }_0$ and $H_0$ in this way

$$\begin{array}{l}{\rho }_0\left[u\left(x,t\right)\right]=3u\left(x,t\right)\\ H_0={\int }_{-\infty }^{+\infty }\mathit{dx}{\rho }_0\left[u\left(x,t\right)\right]=3{\int }_{-\infty }^{+\infty }\mathit{dx}u\left(x,t\right)\end{array}$$E28

and the KdV equation becomes

$$\frac{\partial }{\partial t}\left(\frac{1}{2}{u}^2\right)=\frac{\partial }{\partial x}\left(\frac{1}{3}{u}^3-\frac{1}{2}{\left(\frac{\partial u}{\partial x}\right)}^2+u\frac{{\partial }^{2}u}{\partial x^2}\right)$$E29

let us pose

$$\begin{array}{l}{\rho }_1\left[u\left(x,t\right)\right]=\frac{1}{2}{u}^2\\ j_1=\frac{1}{3}{u}^3-\frac{1}{2}{\left(\frac{\partial u}{\partial x}\right)}^2+u\frac{{\partial }^{2}u}{\partial x^2}\end{array}$$E30

we have a continuity equation, therefore the second constant of motion is obtained by

$$H_1={\int }_{-\infty }^{+\infty }\mathit{dx}\frac{1}{2}{\left(u\left(x,t\right)\right)}^2$$E31

In addition, note that the Hamiltonian of KdV equation is given from eq. (9) in the form

$$H_{\mathit{KdV}}={\int }_{-\infty }^{+\infty }\mathit{dx}\left(\frac{1}{3!}{u}^3-\frac{1}{2}{\left(\frac{\partial u}{\partial x}\right)}^2\right)$$E32

which define a constant of movement, namely

$$\frac{{\mathit{dH}}_{\mathit{KdV}}}{\mathit{dt}}=\left\{H_{\mathit{KdV}},H_{\mathit{KdV}}\right\}=0$$E33

then we can identify the two Hamiltonians $H_2$ and $H_{\mathit{KdV}}$.

From the Poisson bracket defined by the eq. (15) we can also identify the Hamiltonian $H_2$ with the generators defining the translations in the time

$$\left\{u\left(x,t\right),H_2\right\}=\frac{\partial u\left(x,t\right)}{\partial t}$$E34

and $H_1$ with the generator defining the translations in the space

$$\left\{u\left(x,t\right),H_1\right\}=\frac{\partial u\left(x,t\right)}{\partial x}$$E35

These are the first conserved quantities obtained by a direct calculation. However, Miura had found 11 conserved quantities [7], while Kruskal postulated that there must be an infinite number of independent conserved quantities which are then obtained by a relation of recurrences [2, 7].

$${\rho }_n=3{\left(-1\right)}^n{v}_{2n}$$E36

and

$$H_n={\int }_{-\infty }^{+\infty }\mathit{dx}{\rho }_n=3{\left(-1\right)}^n{\int }_{-\infty }^{+\infty }\mathit{dx}{v}_{2n}$$E37

with

$$\begin{array}{l}{v}_0=u,\\ v_n+i\frac{\partial v_{n-1}}{\partial x}+\frac{1}{6}\sum _{m=0}^{n-2}{v}_{n-m-2}{v}_m=0\end{array}$$E38

we notice that the first 3 values of ${\rho }_n$ and $H_n$ correspond to the eqs. (28), (30), (32).

2.3 Integrability criterion

In the following, we ensure the integrability of the KdV equation by seeking the existence of an infinite number of independent conserved quantities which are in evolution [8]. Indeed, the conserved quantities $H_0$, $H_1$ and $H_2$ satisfy the functional relation

$$\left({D_x}^3+\frac{1}{3}\left(D_{x}u+{\mathit{uD}}_x\right)\right)\frac{\delta H_{n-1}}{\mathit{\delta u}\left(x\right)}=D_x\frac{\delta H_n}{\mathit{\delta u}\left(x\right)}$$E39

We can generalize this equation for higher orders of n, by defining the following recurrence relation [2, 5, 7].

$$\begin{array}{c}{H}_{-1}=0\\ \left({D_x}^3+\frac{1}{3}\left(D_{x}u+{\mathit{uD}}_x\right)\right)\frac{\delta H_{n-1}}{\mathit{\delta u}\left(x\right)}=D_x\frac{\delta H_n}{\mathit{\delta u}\left(x\right)}\end{array}$$E40

Equation (40) is very important to say that the conserved quantities are in evolution. Indeed

$$\begin{array}{ll}{\left\{H_n,H_m\right\}}_1& ={\int }_{-\infty }^{+\infty }\mathit{dx}\frac{\delta H_n}{\mathit{\delta u}\left(x\right)}{D}_x\frac{\delta H_m}{\mathit{\delta u}\left(x\right)}\\ & ={\int }_{-\infty }^{+\infty }\mathit{dx}\frac{\delta H_{n-1}}{\mathit{\delta u}\left(x\right)}{D}_x\frac{\delta H_{m+1}}{\mathit{\delta u}\left(x\right)}\\ & ={\left\{H_{n-1},H_{m+1}\right\}}_1=\dots \\ & ={\left\{H_{-1},H_{m+n+1}\right\}}_1=0\end{array}$$E41

by an iteration relation we can show that

$${\left\{H_n,H_m\right\}}_1={\left\{H_m,H_n\right\}}_1$$E42

Therefore this relation affirms that our conserved quantities are independent and that they are also in evolution. We can also show that

$${\left\{H_n,H_m\right\}}_2=0$$E43

To conclude, according to Liouville’s theorem [1, 2, 3, 4, 7], the KdV equation is integrable.

3. Q-deformation of KdV eq.

3.1 Q-deformed formalism

To understand the $q$-deformation, we begin the section with a general framework, the $\alpha -$ derivative, defined in the form

$${\partial }_{\alpha }\left(\mathit{fg}\right)=\alpha \left(f\right)d_{\alpha }\left(g\right)+d_{\alpha }\left(f\right)g,$$E44

with $f$ and $g$ are two polynomial functions in an indeterminate $x$ and its inverse $x^{-1}$ and the parameter $\alpha$ is a linear map. As an example of the derivation $\alpha$, we present Jackson’s $q$-differential operator ${\partial }_q$, defined by [9].

$${\partial }_q\left(f\right)=\frac{f\left(\mathit{qx}\right)-f\left(x\right)}{\left(q-1\right)x}$$E45

in this formalism the eq. (44) becomes

$${\partial }_q\left(\mathit{fg}\right)={\eta }_q\left(f\right).{\partial }_q\left(g\right)+{\partial }_q\left(f\right).g,$$E46

where the operator ${\eta }_q$ named q-shift operator which is given by

$${\eta }_q\left(f\left(x\right)\right)=f\left(\mathit{qx}\right).$$E47

we can define the commutation relation as follows

$$[f,g]=f\circ g-g\circ f,$$E48

where $\circ$ is an internal composition law which is given by

$$\begin{array}{c}{\partial }_q\circ f={\eta }_q\left(f\right){\partial }_q+{\partial }_{q}f,\\ {\partial }_q^{-1}\circ f={\sum }_{k\ge 0}{\left(-1\right)}^k{q}^{-k\left(k+1\right)/2}{\eta }_q^{-k-1}\left({\partial }_q^{k}f\right){\partial }_q^{-k-1}\end{array}$$E49

to obtain the second relation, we used the following unitarity relation

$${\partial }_q^{-1}\circ {\partial }_q\circ f={\partial }_q\circ {\partial }_q^{-1}\circ f=f.$$E50

where ${\partial }_q^{-1}$ is the formal inverse q-derivative of the q-derivative ${\partial }_q$. Note that the shift operator ${\eta }_q$ does not commute with ${\partial }_q$ because we have

$$\begin{array}{l}{\partial }_q\left({\eta }_q\left(f\right)\right)=q{\eta }_q\left({\partial }_{q}f\right),\\ {\partial }_q\left({\eta }_q^k\left(f\right)\right)=q^k{\eta }_q^k\left({\partial }_{q}f\right),k\in \mathbb{Z}.\\ {\partial }_q^m\left({\eta }_q^k\left(f\right)\right)=q^{k+m}{\eta }_q^k\left({\partial }_q^{m}f\right),k,m\in \mathbb{Z}.\end{array}$$E51

However, we can generalize eq. (49) for all $n$ as follows

$${\partial }_q^n\circ f=\sum _{k\ge 0}{\left(\begin{array}{c}n\\ k\end{array}\right)}_q{\eta }_q^{n-k}\left({\partial }_q^{k}f\right){\partial }_q^{n-k},$$E52

where the $q-$ binomials take the form [15].

$${\left(\begin{array}{c}n\\ k\end{array}\right)}_q=\frac{{\left(n\right)}_q{\left(n-1\right)}_q.\dots {\left(n-k+1\right)}_q}{{\left(1\right)}_q{\left(2\right)}_q\dots {\left(k\right)}_q},{\left(\begin{array}{c}n\\ 0\end{array}\right)}_q=1,$$E53

and the numbers ${\left(n\right)}_q$ called the q-numbers are given by

$${\left(n\right)}_q=\frac{q^n-1}{q-1},$$E54

Explicitly, we can write the eq. (49) for positive and negative degrees as follows

$$\begin{array}{l}{\partial }_q\circ f=\left({\partial }_{q}f\right)+{\eta }_q\left(f\right){\partial }_q,\\ {\partial }_q^2\circ f=\left({\partial }_q^{2}f\right)+\left(q+1\right){\eta }_q\left({\partial }_{q}f\right){\partial }_q+{\eta }_q^2\left(f\right){\partial }_q^2,\\ {\partial }_q^3\circ f=\left({\partial }_q^{3}f\right)+\left(q^2+q+1\right){\eta }_q\left({\partial }_q^{2}f\right){\partial }_q\\ +\left(q^2+q+1\right){\eta }_q^2\left({\partial }_{q}f\right){\partial }_q^2+{\eta }_q^3\left(f\right){\partial }_q^3,\end{array}$$E55

and

$$\begin{array}{l}{\partial }_q^{-1}\circ f={\eta }_q^{-1}\left(f\right){\partial }_q^{-1}-q^{-1}{\eta }_q^{-2}\left({\partial }_{q}f\right){\partial }_q^{-2}+q^{-3}{\eta }_q^{-3}\left({\partial }_q^{2}f\right){\partial }_q^{-3}\\ -q^{-6}{\eta }_q^{-4}\left({\partial }_q^{3}f\right){\partial }_q^{-4}+\frac{1}{q^{10}}{\eta }_q^{-5}\left({\partial }_q^{4}f\right){\partial }_q^{-5}+\cdots \\ +{\left(-1\right)}^k{q}^{-\left(1+2+3+\cdots +k\right)}{\eta }_q^{-k-1}\left({\partial }_q^{k}f\right){\partial }_q^{-k-1}+\cdots \end{array}$$E56

3.2 Integrability through Lax pair technique

The purpose of this section is to investigate the integrability of some nonlinear models in the q-deformed version based on the existence of Lax pairs. Such studies are compatible with those already established in the literature by assuming the classical limit $q=1$ [8, 14].

The basic idea is to consider a nonlinear system with the Lax representation [10].

$$L,{\partial }_t-B]{}_q=0,$$E57

where ${\partial }_t\equiv \frac{\partial }{\partial t}$ and ${\left[f,g\right]}_q=f\circ g-f\circ g.$ The eq. (57) and the associated pair of operators $\left(L,B\right)$ are called the Lax-q differential equation and the Lax pair operator, respectively.

In this technique, we must fix from the start the q-differential operator $L$ and the q-deformation of the ${\mathit{sl}}_n$-KdV hierarchy equation is given by

$$\frac{\partial L}{\partial t_k}={\left[{\left(L^{\frac{k}{2}}\right)}_{+},L\right]}_q$$E58

the eq. (57) is then equivalent to the following equation

$${\left[L,{\partial }_t-B\right]}_q\equiv {\left[L,{\partial }_{t_k}-{\left(L^{\frac{k}{2}}\right)}_{+}\right]}_q=0$$E59

here the operator B is the analogue of the operator ${\left(L^{\frac{k}{2}}\right)}_{+}$ describing a $q$-deformation operator of conformal spin $k$.

Our goal is to find an operator B that satisfies the eq. (57), to simplify the task, we will make constraints on the operator $B$ called Ansatz, namely:

$$B={\partial }_q^n\circ L^m+\tilde{B},$$E60

where $\tilde{B}$ is another operator with the same equiangular weights as $B$. Second, using this approach alleviates the problem of finding operator $\tilde{B}$. Below we perform an application to the q-modified KdV equation.

3.3 Q-KdV equation

The discovery of Lax pairs is the first development major in modern integrable systems theory. By building a pair of Lax from one integrable Hamiltonian system, we obtain the solution of the system. Because the Lax equation is the equivalent of the dynamic system equation [10, 11].

We choose an example of $q$-deformed operator of conformal weight 2 through the KdV system, the associated Lax operator is given by

$$L_{q-\mathit{KdV}}={\partial }_q^2+u_2$$E61

We follow the same method of the previous example, the Ansatz for operator B is as follows

$$B={\partial }_q\circ L+\tilde{B}={\partial }_q^3+{\eta }_q\left(u_2\right){\partial }_q+{\partial }_q\left(u_2\right)+\tilde{B}$$E62

and the associated Lax equation is given by

$${\left[{\partial }_t-B,L\right]}_q=0$$E63

after a simple calculation we find

$${\left[L,B\right]}_q=-{\dot{u}}_1$$E64

after simplification of the Lax equation, we find the following KdV equation in its q-deformed version

$${\dot{u}}_2=\left[u_2+{\eta }_q\left(u_2\right)\right]{\partial }_q\left(u_2\right)+{\partial }_q^2\left[{\partial }_q\left(u_2\right)+{\eta }_q\left({\partial }_q{u}_2\right)\right]$$E65

In the classical limit where $q=1$ we have ${\eta }_q\left(u_2\right)=u_2$ and ${\partial }_q\left(u_2\right)=u_2^{\prime }$, however, we find the following classical KdV equation

$${\dot{u}}_2=u_2{u}_2^{\prime }+u_2^{‴}$$E66

The same work can be done for a nonlinear system with equal weight $3$ in the Boussinesq system and a nonlinear system with equal weight $1$ in the Burgers system.

4. Moyal deformation

4.1 Fundamental notions

In this section, we treat the Moyal deformation defined on the two-dimensional phase space. Such a deformation is seen as a non-commutative formulation of the functions $f\left(x,p\right)$ where the variable x represents the coordinate of the space while $p$ describes the coordinate of the momentum [12].

The simple vision of this deformation is given by the following main object

$$u_i\left(x,t\right)\star p^j,$$E67

which is an object of conformal weight $\left(i+j\right)$ in the noncommutative space characterized by the noncommutativity parameter $\theta$. The star product law $\star$, defining the internal composition law of the elements in the noncommutative phase space (multiplication), which is given by the following expression

$$f\left(x,p\right)\star g\left(x,p\right)=\sum _{s=0}^{\infty }\sum _{i=0}^s\frac{{\theta }^s}{s!}{\left(-\right)}^i{c}_s^i\left({\partial }_x^i{\partial }_p^{s-i}f\right)\left({\partial }_x^{s-i}{\partial }_p^{i}g\right),$$E68

where $c_s^i=\frac{s!}{i!\left(s-i\right)!}$ is the combination parameter. We define the Moyal bracket as follows

$${\left\{f\left(x,p\right),g\left(x,p\right)\right\}}_{\theta }=\frac{f\star g-g\star f}{2\theta },$$E69

where $\theta$ is the parameter of the noncommutativity which is considered in our study as a constant.

4.2 Algebraic structure

Let us consider the algebra $\widehat{\mathrm{\Sigma }}\left(\theta \right)$ of all arbitrary differential momentum operators of arbitrary conformal weight m and arbitrary degrees $\left(r,s\right)$. It is obtained as a direct sum of all operators of conformal weight m and degrees $\left(r,s\right)$ in the following way:

$$\widehat{\mathrm{\Sigma }}\left(\theta \right)={\oplus }_{r\le s}{\oplus }_{m\in Z}{\widehat{\mathrm{\Sigma }}}_m^{\left(r,s\right)}.$$E70

it is an infinite dimensional algebra which is closed under the action of the Moyal bracket [16, 17]. The subspace ${\widehat{\mathrm{\Sigma }}}_m^{\left(r,s\right)}$ is the algebra of differential operators of conformal weight $m$ and degrees $\left(r,s\right)$ with $r\le s$, a simple prototype operator of this algebra is given by

$${\mathcal{L}}_m^{\left(r,s\right)}\left(u\right)=\sum _{i=r}^s{u}_{m-i}\left(x\right)\star p^i.$$E71

the special cases of this algebra is given by: ${\widehat{\mathrm{\Sigma }}}_m^{\left(0,0\right)}$ which represents the ring of functions of conformal weight $m$ and ${\widehat{\mathrm{\Sigma }}}_m^{\left(k,k\right)}$ which represents the algebra of momentum operators type,

$${\mathcal{L}}_m^{\left(k,k\right)}\left(u\right)=u_{m-k}\star p^k.$$E72

In this algebra, we define the Residual operation ($\theta$-Residue) by the following formula

$$\widehat{\mathit{Res}}\left(\alpha \star p^{-1}\right)=\alpha .$$E73

and we find the following $\theta$- Leibnitz rules

$$\begin{array}{c}{p}^n\star f\left(x,p\right)=\sum _{s=0}^n{\theta }^s{c}_n^s{f}^{\left(s\right)}\left(x,p\right)p^{n-s},\\ p^{-n}\star f\left(x,p\right)=\sum _{s=0}^{\infty }{\left(-\right)}^s{\theta }^s{c}_{n+s-1}^s{f}^{\left(s\right)}\left(x,p\right)p^{-n-s}.\end{array}$$E74

However, we can establish the following expressions for the Moyal bracket:

$$\begin{array}{l}{\left\{p^n,f\right\}}_{\theta }=\sum _{s=0}^n{\theta }^{s-1}{c}_n^s\left\{\frac{1-{\left(-\right)}^s}{2}\right\}{f}^s{p}^{n-s},\\ {\left\{p^{-n},f\right\}}_{\theta }=\sum _{s=0}^{\infty }{\theta }^{s-1}{c}_{s+n-1}^s\left\{\frac{{\left(-\right)}^s-1}{2}\right\}{f}^s{p}^{-n-s}\end{array}$$E75

here we present some examples of this Moyal bracket for positive values of degree (where the development is finite) and for negative values of degree (where the development is infinite)

$$\begin{array}{l}{\left\{p,f\right\}}_{\theta }=f^{\prime }\\ {\left\{p^2,f\right\}}_{\theta }=2f^{\prime }p\\ {\left\{p^3,f\right\}}_{\theta }=3{f^{\prime }}^2+{\theta }^2{f}^{‴}\end{array}$$E76

$$\begin{array}{l}{\left\{p^{-1},f\right\}}_{\theta }=-{f^{\prime }}^{]-2}-{\theta }^2{f^{‴}}^{-4}-\dots -{\theta }^{2k}{f}^{\left(2k+1\right)}{p}^{-2k-2}-\dots \\ {\left\{p^{-2},f\right\}}_{\theta }=-2{f^{\prime }}^{-3}-4{\theta }^2{f}^{\text{'}\text{'}\text{'}-5}-\dots -\left(2k+2\right){\theta }^{2k}{f}^{\left(2k+1\right)}{p}^{-2k-3}-\dots \\ {\left\{p^{-3},f\right\}}_{\theta }=-3{f^{\prime }}^{-4}-10{\theta }^2{f^{‴}}^{-6}-\dots -\frac{\left(2k+3\right)\left(2k+2\right)}{2}{\theta }^{2k}{f}^{\left(2k+1\right)}{p}^{-2k-4}-\dots \end{array}$$E77

In the case of classical noncommutativity, the standard limit is taken such that ${\theta }_{\mathit{\text{limit}}}=0$. Whereas in the Moyal deformation, the standard limit is shifted by ($1/2$) such that ${\theta }_l\to {\theta }_{\mathit{\text{limit}}}+\frac{1}{2}$. This shift is due to the consistent $W_{\theta }^3$-Zamolodchikov algebra construction [12, 18].

4.3 Moyal deformation of Lax representation

First, we define the hierarchical structures of the Lax representation of the KdV equation in the deformed phase space, such equation named ${\mathit{sl}}_n$-Moyal KdV hierarchy is defined as

$$\frac{\partial \mathcal{L}}{\partial t_k}={\left\{{\left({\mathcal{L}}^{\frac{k}{2}}\right)}_{+},\mathcal{L}\right\}}_{\theta }.$$E78

Explicit calculations related to these hierarchies give [12, 17, 18].

$$\begin{array}{l}{u}_{t_1}=u^{\prime },\\ u_{t_3}=\frac{3}{2}u{u}^{\prime }+{\theta }^2{u}^{‴},\\ u_{t_5}=5{\theta }^2\left(u^{\prime }{u}^{″}+\frac{1}{2}{\mathit{uu}}^{\text{'}\text{'}\text{'}}\right)+\frac{15}{8}{u}^2{u}^{\prime }+{\theta }^4{u}^{\left(5\right)},\\ \dots \end{array}$$E79

In fact, these results are calculated for the case $n=2$, they can be generalized for higher orders.

To realize the Lax formulation, we need to consider a noncommutative integrable system which has the Lax representation defined by the Moyal deformation of the Lax equation given by [6, 19].

$${\left\{L,T+{\partial }_t\right\}}_{\theta }=0,$$E80

The differential operator $L$ defines the integrable system studied, consequently, it must be fixed from the start.

Note that the Lax eq. (80) is equivalent to (78) namely

$${\left\{L,T+{\partial }_t\right\}}_{\theta }\equiv {\left\{L,{\left({\mathcal{L}}^{\frac{k}{2}}\right)}_{+}+{\partial }_{t_k}\right\}}_{\theta }=0,$$E81

where $T$ is an operator analogue of ${\left({\mathcal{L}}^{\frac{k}{2}}\right)}_{+}$ of conformal spin $k$ [17].

In order to apply the $\theta$-deformation Lax-pair generating technique, one have to make some constraints on the operator $T$ namely:

$$T=p^n\star L^m+T^{\text{'}},$$E82

With the previous ansatz, the problem is reduced to that of the operator $T^{\prime }$ which is determined manually so that the Lax equation is a differential equation belonging to the ring ${\widehat{\mathrm{\Sigma }}}^{\left(0,0\right)}$.

4.4 Moyal KdV equation

For the KdV system, The $L$-operator for the noncommutative equation is given by

$$L=p^2+u\left(x,t\right),$$E83

where

$$L\in {\widehat{\mathrm{\Sigma }}}_2^{\left(0,2\right)}/{\widehat{\mathrm{\Sigma }}}_2^{\left(1,1\right)},$$E84

for the deformed ${\mathit{sl}}_2$-KdV ${}_{\theta }$ system, the ansatz (82) can be written as follows

$$T=p\star L+T^{\prime }.$$E85

The operator $T$ in this case ($k=3$), is shown to behaves as ${\left({\mathcal{L}}^{\frac{3}{2}}\right)}_{+}$ with ${\partial }_{t_3}\equiv \frac{\partial }{\partial t_3}$. Simply algebraic computations give

$${\left\{L,T^{\prime }\right\}}_{\theta }={u^{\prime }}^2-2\theta u^{″}p+\frac{\dot{u}}{2\theta }+\left(u{u^{\prime }}^2{u}^{‴}\right),$$E86

for $T^{\prime },$ we have to consider the following Ansatz [13]:

$$T^{\prime }=X\star p+Y,$$E87

where $X$ and $Y$ are arbitrary functions on $u$ and its derivatives. With $T^{\prime }=\mathit{Xp}-\theta X^{\prime }+Y$ and by performing straightforward computations we have

$$\begin{array}{c}{\left\{L,T^{\prime }\right\}}_{\theta }=\left({\left\{u,X\right\}}_{\theta }-2\theta X^{″}+2Y^{\prime }\right)p+\\ 2{X^{\prime }}^2+\left(-X{u}^{\prime }-\theta \left\{u,X^{\prime }\right\}+\left\{u,Y\right\}\right)\end{array}$$E88

Identifying (86) and (88), leads to the following constrains on the functions $X$ and $Y$

$$X=\frac{1}{2}{u}_2+a,Y=-\frac{1}{2}\theta u_2^{\prime }+b,$$E89

with the following nonlinear differential equation

$$-\frac{\dot{u}}{2\theta }=\frac{3}{2}u{u^{\prime }}^2{u}^{‴}.$$E90

where the constants $a$ and $b$ are to be omitted for a matter of simplicity. The last equation is noting but the ${\mathit{sl}}_2$ KdV_θ equation. This deformed equation contains also a non linear term $\frac{3}{2}u{u}^{\prime }$. We have to underline that the ${\mathit{sl}}_2$ KdV_θ equation obtained through this Lax method belongs to the same class of the KdV equation derived in [12, 16] namely

$$\dot{u}=\frac{3}{2}u{u^{\prime }}^2{u}^{‴}.$$E91

In fact, performing the following scaling transformation ${\partial }_{t_3}\to -2\theta {\partial }_{t_3}$ we recover exactly the KdV equations.

5. Conclusion

The Korteweg-de Vries equation was formulated to explain the solitary water waves observed by J. Scott Russell in the Edinburgh Glasgow Canal. It represents a nonlinear equation in a (1 + 1) space–time. As we show, It has soliton solutions. But at the time of its formulation, nothing was known about the integrability of the equation.

We set up the KdV equation as a Hamiltonian system and we analyzed the integrability of this system. In order to generalize the KdV equation, we have discussed the solutions of the KdV equation in the q-deformed case through the Lax pair method.

The $\theta$-deformation of the KdV system shows, among others, the consistency of the Moyal momentum formulation in describing integrable systems and the associated Lax pair generating technics in the same way as the successful GD formulation [17, 18, 19].

Classification

PACS number(s): 35Q53; 37 K10; 81R60; 81 T75; 05A30