Open access peer-reviewed chapter

On the Classical and Deformed Korteweg-de Vries Equation

Written By

Abderrahman El Boukili, Hicham Lekbich, Tahir Toghrai, Najim Mansour and Moulay Brahim Sedra

Reviewed: 15 December 2022 Published: 16 January 2023

DOI: 10.5772/intechopen.109541

From the Edited Volume

Optimization Algorithms - Classics and Recent Advances

Edited by Mykhaylo Andriychuk and Ali Sadollah

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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.

Keywords

  • Korteweg-de Vries equation
  • integrability
  • Lax generating technique
  • Q-deformation
  • Moyal momentum space

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 θ 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 θ deformation frames. We derive an explicit Lax pair operator for. Finally, Section 5 summarizes our research.

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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]:

ut+auux+b3ux3=0E1

where uxt 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):

    xb1/3xE2

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

    ttE3

  • Transformation on the dynamic variable

ua1b1/3uE4

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

ut=uux+3ux3E5

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

i)td1+t,ii)xd2+x,iii)xαx,tα3tanduα2uiv)xx+vt,ttanduu+vE6

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:

ut=uxtHE7

Here .. represents the Poisson brackets and H is the Hamiltonian of our system. In fact, rewriting the KdV equation in the form

ut=x12u2+x2ux2=x12u2+2ux2E8

we can then choose the following Hamiltonian:

Hu=+dx13!u312ux2E9

Based on the functional derivative defined by

δFuxδux=limε01εFux+εδxyFuxE10

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

δHδux=12u2+2ux2E11

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

ut=xδHδuxE12

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

uxH=xδHδuxE13

that is to say

+dyδHδuxuxuy=xδHδuxE14

in this case, the Poisson bracket of ux and uy is given by

uxuy=xδxyE15

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

uxuy2=13xux+uxx+3x3δxyE16

and a second Hamiltonian operator

H2u=+dx12u2xE17

with this choice, the equation of KdV becomes

ut=uxtH22=+dyδH2δuxuxuy2=uux+3ux3E18

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

LKdV=12+dxdyuxεxyuyt+dx13!u3x12uxx2E19

The Euler Lagrange equations obtained are

+dyεxyuyt=12u2x+u2xx2E20

this means that

ut=uux+3ux3E21

so we have recovered the eq. (5). The existence of the term εxy in the Lagrangian implies the nonlocality of our system.

2.2 Integratability through the conserved quantities

In general, a quantity Qu is said to be conserved if for a bracket and an Hamiltonian H, we have th following equation

dQudt=QuH=0E22

if moreover we take this expression

Qu=+dxρuxtE23

we have

dQudt=+dxρuxtt=0E24

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

ρuxtt+juxtx=0E25

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

ρ0uxt=uxtj0=12u2+2ux2E26

consequently

Q0=H0=+dxρ0uxt=+dxuxtE27

is a constant of movement. In the following, we redefine ρ0 and H0 in this way

ρ0uxt=3uxtH0=+dxρ0uxt=3+dxuxtE28

and the KdV equation becomes

t12u2=x13u312ux2+u2ux2E29

let us pose

ρ1uxt=12u2j1=13u312ux2+u2ux2E30

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

H1=+dx12uxt2E31

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

HKdV=+dx13!u312ux2E32

which define a constant of movement, namely

dHKdVdt=HKdVHKdV=0E33

then we can identify the two Hamiltonians H2 and HKdV.

From the Poisson bracket defined by the eq. (15) we can also identify the Hamiltonian H2 with the generators defining the translations in the time

uxtH2=uxttE34

and H1 with the generator defining the translations in the space

uxtH1=uxtxE35

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].

ρn=31nv2nE36

and

Hn=+dxρn=31n+dxv2nE37

with

v0=u,vn+ivn1x+16m=0n2vnm2vm=0E38

we notice that the first 3 values of ρn and Hn 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 H0, H1 and H2 satisfy the functional relation

Dx3+13Dxu+uDxδHn1δux=DxδHnδuxE39

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

H1=0Dx3+13Dxu+uDxδHn1δux=DxδHnδuxE40

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

HnHm1=+dxδHnδuxDxδHmδux=+dxδHn1δuxDxδHm+1δux=Hn1Hm+11==H1Hm+n+11=0E41

by an iteration relation we can show that

HnHm1=HmHn1E42

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

HnHm2=0E43

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

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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 α derivative, defined in the form

αfg=αfdαg+dαfg,E44

with f and g are two polynomial functions in an indeterminate x and its inverse x1 and the parameter α is a linear map. As an example of the derivation α, we present Jackson’s q-differential operator q, defined by [9].

qf=fqxfxq1xE45

in this formalism the eq. (44) becomes

qfg=ηqf.qg+qf.g,E46

where the operator ηq named q-shift operator which is given by

ηqfx=fqx.E47

we can define the commutation relation as follows

[f,g]=fggf,E48

where is an internal composition law which is given by

qf=ηqfq+qf,q1f=k01kqkk+1/2ηqk1qkfqk1E49

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

q1qf=qq1f=f.E50

where q1 is the formal inverse q-derivative of the q-derivative q. Note that the shift operator ηq does not commute with q because we have

qηqf=qηqqf,qηqkf=qkηqkqf,kZ.qmηqkf=qk+mηqkqmf,k,mZ.E51

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

qnf=k0nkqηqnkqkfqnk,E52

where the q binomials take the form [15].

nkq=nqn1q.nk+1q1q2qkq,n0q=1,E53

and the numbers nq called the q-numbers are given by

nq=qn1q1,E54

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

qf=qf+ηqfq,q2f=q2f+q+1ηqqfq+ηq2fq2,q3f=q3f+q2+q+1ηqq2fq+q2+q+1ηq2qfq2+ηq3fq3,E55

and

q1f=ηq1fq1q1ηq2qfq2+q3ηq3q2fq3q6ηq4q3fq4+1q10ηq5q4fq5++1kq1+2+3++kηqk1qkfqk1+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,tB]q=0,E57

where tt and fgq=fgfg. The eq. (57) and the associated pair of operators LB 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 sln-KdV hierarchy equation is given by

Ltk=Lk2+LqE58

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

LtBqLtkLk2+q=0E59

here the operator B is the analogue of the operator Lk2+ 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=qnLm+B˜,E60

where B˜ is another operator with the same equiangular weights as B. Second, using this approach alleviates the problem of finding operator 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

LqKdV=q2+u2E61

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

B=qL+B˜=q3+ηqu2q+qu2+B˜E62

and the associated Lax equation is given by

tBLq=0E63

after a simple calculation we find

LBq=u̇1E64

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

u̇2=u2+ηqu2qu2+q2qu2+ηqqu2E65

In the classical limit where q=1 we have ηqu2=u2 and qu2=u2, however, we find the following classical KdV equation

u̇2=u2u2+u2E66

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.

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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 fxp 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

uixtpj,E67

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

fxpgxp=s=0i=0sθss!icsixipsifxsipig,E68

where csi=s!i!si! is the combination parameter. We define the Moyal bracket as follows

fxpgxpθ=fggf2θ,E69

where θ is the parameter of the noncommutativity which is considered in our study as a constant.

4.2 Algebraic structure

Let us consider the algebra Σ̂θ of all arbitrary differential momentum operators of arbitrary conformal weight m and arbitrary degrees rs. It is obtained as a direct sum of all operators of conformal weight m and degrees rs in the following way:

Σ̂θ=rsmZΣ̂mrs.E70

it is an infinite dimensional algebra which is closed under the action of the Moyal bracket [16, 17]. The subspace Σ̂mrs is the algebra of differential operators of conformal weight m and degrees rs with rs, a simple prototype operator of this algebra is given by

Lmrsu=i=rsumixpi.E71

the special cases of this algebra is given by: Σ̂m00 which represents the ring of functions of conformal weight m and Σ̂mkk which represents the algebra of momentum operators type,

Lmkku=umkpk.E72

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

Reŝαp1=α.E73

and we find the following θ- Leibnitz rules

pnfxp=s=0nθscnsfsxppns,pnfxp=s=0sθscn+s1sfsxppns.E74

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

pnfθ=s=0nθs1cns1s2fspns,pnfθ=s=0θs1cs+n1ss12fspnsE75

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)

pfθ=fp2fθ=2fpp3fθ=3f2+θ2fE76
p1fθ=f]2θ2f4θ2kf2k+1p2k2p2fθ=2f34θ2f'''52k+2θ2kf2k+1p2k3p3fθ=3f410θ2f62k+32k+22θ2kf2k+1p2k4E77

In the case of classical noncommutativity, the standard limit is taken such that θlimit=0. Whereas in the Moyal deformation, the standard limit is shifted by (1/2) such that θlθlimit+12. This shift is due to the consistent Wθ3-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 sln-Moyal KdV hierarchy is defined as

Ltk=Lk2+Lθ.E78

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

ut1=u,ut3=32uu+θ2u,ut5=5θ2uu+12uu'''+158u2u+θ4u5,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].

LT+tθ=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

LT+tθLLk2++tkθ=0,E81

where T is an operator analogue of Lk2+ of conformal spin k [17].

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

T=pnLm+T',E82

With the previous ansatz, the problem is reduced to that of the operator T which is determined manually so that the Lax equation is a differential equation belonging to the ring Σ̂00.

4.4 Moyal KdV equation

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

L=p2+uxt,E83

where

LΣ̂202/Σ̂211,E84

for the deformed sl2-KdV θ system, the ansatz (82) can be written as follows

T=pL+T.E85

The operator T in this case (k=3), is shown to behaves as L32+ with t3t3. Simply algebraic computations give

LTθ=u22θup+u̇2θ+uu2u,E86

for T, we have to consider the following Ansatz [13]:

T=Xp+Y,E87

where X and Y are arbitrary functions on u and its derivatives. With T=XpθX+Y and by performing straightforward computations we have

LTθ=uXθ2θX+2Yp+2X2+XuθuX+uYE88

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

X=12u2+a,Y=12θu2+b,E89

with the following nonlinear differential equation

u̇2θ=32uu2u.E90

where the constants a and b are to be omitted for a matter of simplicity. The last equation is noting but the sl2 KdVθ equation. This deformed equation contains also a non linear term 32uu. We have to underline that the sl2 KdVθ equation obtained through this Lax method belongs to the same class of the KdV equation derived in [12, 16] namely

u̇=32uu2u.E91

In fact, performing the following scaling transformation t32θt3 we recover exactly the KdV equations.

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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 θ-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].

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Classification

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

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Written By

Abderrahman El Boukili, Hicham Lekbich, Tahir Toghrai, Najim Mansour and Moulay Brahim Sedra

Reviewed: 15 December 2022 Published: 16 January 2023