Myint Tyn U., Debnath L. Linear Partial Differential Equations for Scientists and Engineers

Подождите немного. Документ загружается.

13.13 The Lax Pair and the Zakharov and Shabat Scheme 591

or, equivalently,

∂L

∂t

ψ = λ

ψ +(ML − LM) ψ. (13.13.5)

Thus, eigenvalues are constant for nonzero eigenfunctions if and only if

∂L

∂t

= −(LM −ML) ψ = −[L, M ] ψ, (13.13.6)

where [L, M]=(LM − ML) is called the commutator of the operators L

and M, and the derivative on the left-hand side of (13.13.6) is to be in-

terpreted as the time derivative of the operator alone. Equation (13.13.6)

is called the Lax equation and the operators L and M are called the Lax

pair . It is the Heisenberg picture of the KdV equation. The problem, of

course, is how to determine these operators for a given evolution equation.

There is no systematic method of solution for this p rob lem. For a negative

integrable hierarchy, Qiao (1995) and Qiao and Strampp (2002) suggest a

general approach to generate integrable equations; they also devise strate-

gies for ﬁnding a Lax pair from a given spectral problem.

We consider the initial-value problem for u (x, t) which satisﬁes t he non-

linear evolution equation system

= N (u) (13.13.7)

u (x , 0) = f (x) , (13.13.8)

where u ∈ Y for all t, Y is a suitable fun ction space, and N : Y → Y is a

nonlinear operator that is independent of t but may involve x or derivatives

with respect to x.

We must assume that the evolution equation (13.13.7) can be expressed

in the Lax form

+(LM − ML)=L

+[L, M]=0, (13.13.9)

where L and M are linear operators in x on a Hilbert space H and depend

on u and L

= u

is a scalar operator. We also assume that L is self-adjoint

so that (Lφ, ψ)=(φ, Lψ) for all φ and ψ ∈ H with (·, ·) as an inner product.

We now formulate the eigenvalue problem for ψ ∈ H:

Lψ = λ (t) ψ, t ≥ 0,x∈ R. (13.13.10)

Diﬀerentiating with respect to t and making use of (13.13.9), we obtain

ψ =(L − λ)(ψ

− Mψ) . (13.13.11)

The inner product of ψ with this equation yields

(ψ, ψ) λ

=((L − λ) ψ, λ

− Mψ) , (13.13.12)

592 13 Nonlinear Partial Diﬀerential Equations with Applications

which, since L − λ is self-adjoint, is given by

(ψ, ψ) λ

=(0,ψ

− Mψ)=0.

Hence, λ

= 0, conﬁrming that each eigenvalue of L is a constant. Conse-

quently, (13.13.11) becomes

L (ψ

− Mψ)=λ (ψ

− Mψ) . (13.13.13)

This shows that (ψ

− Mψ) is an eigenfunction of the operator L with the

eigenvalue λ.ItisalwayspossibletoredeﬁneM by adding the product

of the identity operator and a suitable fun ction of t, so that the original

equation (13.13.9) remains unchanged. This leads to the time evolution

equation for ψ as

= Mψ, t ≥ 0. (13. 13.14)

Thus, we have the following.

Theorem 13.13.1. If the evolution equation (13.13.7) can be expressed as

the Lax equation

+[L, M] = 0 (13.13.15)

and if (13.13.10) holds, then λ

=0,andψ satisﬁes (13.13.14).

It is not yet clear how to ﬁnd the operators L and M that satisfy the pre-

ceding conditions. To illustrate the Lax method, we choose the Schr¨odinger

operator L in the form

L ≡−

∂

∂x

+ u, (13.13.16)

so that Lψ = λψ becomes the Stur m–Liouvil le problem for the self-adjoint

operator L. With this given L, the problem is to ﬁnd the operator M. Based

on the theory of a linear unitary operator on a Hilbert space H, the linear

operator M can be chosen as antisymmetric, so that (Mφ,ψ)=−(φ, Mψ)

for all φ and ψ ∈ H. So, a suitable linear combination of odd derivatives in

x is a natural choice for M. It follows from the inner product that

(Mφ,ψ)=



∞

−∞

∂

∂x

ψdx = −



∞

−∞

∂

∂x

dx = −(φ, M ψ) ,(13.13.17)

provided M = ∂

φ/∂x

for odd n,andφ, ψ, and their derivatives with

respect to x tend to zero, as |x|→∞. Moreover, we require that M has

suﬃcient freedom in any unknown constants or functions to make L

[L, M ] a multiplicative operator, that is, of degree zero. For n =1,the

simplest choice for M is M = c (∂/∂x), where c is a constant. It then follows

that [L, M]=−cu

, which is automatically a multiplicative op erator. Thus,

the Lax equation is

13.13 The Lax Pair and the Zakharov and Shabat Scheme 593

+[L, M]=u

− cu

=0, (13.13.18)

and hence, the one-dimensional wave equation

− cu

= 0 (13.13.19)

has an associated eigenvalue problem with the eigenvalues that are con-

stants of motion.

Thenextnaturalchoiceis

M = −a

∂

∂x

+ A

∂

∂x

∂

∂x

A + B, (13.13.20)

where a is a constant, A = A (x, t), and B = B (x, t), and the third term

on the right-hand side of (13.13.20) can be dropp ed, but we retain it for

convenience. It follows from an algebraic calculation that

[L, M ]=au

xxx

− A

xxx

− B

− 2 u

+(3au

− 4A

− 2B

)

∂

∂x

+(3au

− 4A

)

∂

∂x

This would be a multiplicative operator if A =

au and B = B (t). Conse-

quently, the Lax equation (13.13.15) becomes

−

auu

xxx

=0. (13.13.21)

This i s the standard KdV equation if a = 4. The operator M deﬁned by

(13.13.20) reduces to the form

M = −4

∂

∂x



∂

∂x

∂

∂x



+ B (t) . (13.13.22)

Hence, the time evolution equation for ψ can be simpliﬁed by using the

Sturm–Liouville equation, ψ

− (u − λ) ψ =0,to

=4(λψ − uψ)

+3ψ

+3(uψ)

+ Bψ

=2(u +2λ) ψ

− u

ψ + Bψ. (13.13.23)

We close this section by adding several comments. First, any evolution

equations solvable by the inverse scattering transform (IST), like the KdV

equation, can be expressed in Lax form. However, the main diﬃculty is

that there is no completely systematic method for determining whether or

not a given partial diﬀerential equation produces a Lax equation and, if

so, how to ﬁn d the Lax pair L and M . Indeed, Lax proved that there is

an inﬁnite number of operators, M, one associated with each odd order of

∂/∂x, and hence, an inﬁnite family of ﬂows u

under which the spectrum

594 13 Nonlinear Partial Diﬀerential Equations with Applications

of L is preserved. Second, it is possible to study other spectral equations

by choosing alternative forms for L. Third, the restriction that L and M

should b e limited to the class of scalar operators could be removed. In fact,

L and M could be matrix operators. The Lax formulation has already been

extended to such operators. Fourth, Zakharov and Shabat (1972, 1974)

published a series of notable papers in this ﬁeld extending the nonlinear

Schr¨odinger (NLS) equation and other evolution equations. For the ﬁrst

time, th ey have generalized the Lax formalism for equations with more

than one spatial variable. This extension is known as the Zakharov and

Shabat (ZS) scheme, which, essentially, follows the Lax method and recasts

it in a matrix form, leading to a matrix Marchenko equation.

Finally, we b rieﬂy discuss the ZS scheme for nonself-adjoint operators

to obtain N-soliton solutions for the NLS equation. Zakharov and Shabat

introduced an ingenious method for any nonlinear evolution equation

= Nu, (13.13.24)

that represents the equation in the form

∂L

∂t

= i [L, M ]=i (LM − ML) , (13.13.25)

where L and M are linear diﬀerential operators that include the function

u in the coeﬃcients, and L refers to diﬀerentiating u with respect to t.

We consider the eigenvalue problem

Lφ = λφ. (13.13.26)

Diﬀerentiation of (13.13.26) with respect to t gives

iφ



dλ



=(L − λ)(iφ

− Mφ) . (13.13.27)

If φ satisﬁes (13.13.26) initially and changes in such a manner that

iφ

= Mφ, (13.13.28)

then φ always satisﬁes (13.13.26). Equations (13.13.26) and (13.13.28) are

the pair of equations coupling the function u (x, t) in the coeﬃcients with

a scattering problem. Indeed, the nature of φ determines the scattering

potential i n (13.13.26), and the evolution of φ in time is given by (13.13.28).

Although this formulation is quite general, the crucial step is to factor

L according to (13.13.25). Zakharov and Shabat (1972) introduce 2 × 2

matrices associated with (13.13.25) as follows:

L = i

⎡

⎣

1+α 0

01− α

⎤

⎦

∂

∂x

⎡

⎣

0 u

∗

u 0

⎤

⎦

, (13.13.29)

M = −α

⎡

⎣

⎤

⎦

∂

∂x

⎡

⎢

⎣

|u|

1+α

∗

−iu

−|u|

1−α

⎤

⎥

⎦

, (13.13.30)

13.14 Exercises 595

and the NLS equation for complex u (x, t)isgivenby

+ u

+ γ |u|

u =0, (13.13.31)

where

γ =2/



1 − α



Thus, the eigenvalue problem (13.13.26) and the evolution equation

(13.13.28) complete the inverse scattering problem. The initial-value prob-

lem for u (x, t) can be solved for a given initial condition u (x, 0). It seems

clear that the signiﬁcant contribution would come from the point spectrum

for large times (t →∞). Physically, the disturbance tends to disintegrate

into a series of solitary waves. The mathematical analysis is limited to the

asymptotic solutions so that |u|→0as|x|→∞, but a series of solitary

waves is expected to be the end result of the instability of wavetrains to

modulations.

13.14 Exercises

1. For the ﬂow density relation q = vρ(1 − ρ/ρ

), ﬁnd the solution of the

traﬃc ﬂow problem with the initial condition ρ (x, 0) = f (x) for all x.

Let

f (x)=

⎧

⎪

⎨

⎪

⎩

,x≤ 0

x, 0 ≤ x ≤ 1

,x≥ 1.

Show that

(i) ρ =

along the characteristic lines ct =3(x − x

), x

≤ 0,

(ii) ρ =

along the characteristic lines ct =2(x

− x), x

≥ 1,

(iii) ρ =

along ct



− x



(x − x

), 0 ≤ x

≤ 1.

Discuss what happens at the intersection of the two lines ct =3x and

ct =2(1− x). Draw the characteristic lines ct versus x.

2. A mountain of height h (x, t) is vulnerable to erosion if its slope h

very large. If h

and h

satisfy the functional relation h

= −Q (h

show that u = h

is governed by the nonlinear wave equation

+ c (u) u

=0,c(u)=Q

′

(u) .

3. Consider the ﬂow of water in a river carrying some particles through

the solid bed. During the sedimentation process, some particles will

596 13 Nonlinear Partial Diﬀerential Equations with Applications

be deposited in the bed. Assuming that v is the constant velocity of

water and that ρ = ρ

+ ρ

is the density, where ρ

is the density of

the particles carried in the ﬂuid and ρ

is the density of the material

deposited on the solid bed, the conservation law is

∂ρ

∂t

∂q

∂x

=0,q= vρ

(a) Show that ρ

satisﬁes the equation

∂ρ

∂t

+ c (ρ

)

∂ρ

∂x

=0,c(ρ

1+Q

′

(ρ

)

where ρ

= Q (ρ

(b) This problem also arises in chemical engineering with a second re-

lation between ρ

and ρ

in the form

∂ρ

∂t

= k

(α − ρ

) − k

(β − ρ

) ρ

where k

, k

represent constant reaction rates and α , β are constant

values of the saturation levels of th e particles in the solid bed and ﬂuid

respectively. Show that the propagation speed c is

c =

βv

α + k

β)

provided that the densities are small.

4. Show that a steady solution u (x, t)=f (ζ), ζ = x − ct of Burgers’

equation with the boundary conditions f → u

−

∞

or u

∞

as ζ →−∞

or +∞ is

u (x , t)=



c −



−

∞

− u

∞



tanh



−

∞

− u

∞

)

4ν

(x − ct)



where u

−

∞

= c +



+2A



, u

∞

= c −



− 2A



are two roots of

− 2cf − 2A = 0 and A is a constant of integration.

5. Show that the transformations x → γ

x, u →−6γ

u, t → t reduce

the KdV equation u

+ uu

+ γu

xxx

= 0 into the canonical form u

−

6uu

+ u

xxx

=0.

Hence or otherwise, prove that the solution of the canonical equation

with the boundary conditions that u (x, t) and its derivatives tend to

zero as |x|→∞is

u (x , t)=−

sech

(



x − a



)

13.14 Exercises 597

6. Verify that the Riccati transformation u = v

+ v

transforms the KdV

equation so that v satisﬁes the associated KdV equation

− 6v

+ v

xxx

=0.

For a given u, show that the Riccati equation v

+ v

= u becomes the

linear Schr¨odinger equation, ψ

= uψ (without the energy-level term)

under the transformation v =(ψ

/ψ).

7. Apply the metho d of characteristics to solve the pair of equations

∂u

∂t

∂v

∂x

=0,

∂v

∂t

∂u

∂x

=0,

with the initial data

u (x , 0) = e

,v(x, 0) = e

−x

Show that the Riemann invariants are

2 r (α) = 2 cosh α, 2 s (β)=2sinhβ,

Also, show that solutions are u =cosh(x − t)+sinh(x + t)andv =

cosh (x − t) − sinh (x + t).

8. For an isentropic ﬂow, the Euler equations are

+(ρu)

=0,

+ uu

=0,

+ uS

=0,

where ρ is the density, u is the velocity in the x direction, p is the

pressure, and S is the entropy.

Show that this system has three families of characteristics.

= u, Γ

= u +

where c



∂p

∂ρ



= constant.

Hence derive the following full set of characteristic equations

=0, on Γ

ρc

=0, on Γ

In particular, when the ﬂow is isentropic (S = constant everywhere),

show that the characteristic equations are



c (ρ)

dρ +

u = constant on Γ

and Γ

= u +

598 13 Nonlinear Partial Diﬀerential Equations with Applications

9. (a) Using equations (13.8.32)–(13.8.36), derive the second-order equa-

tion

+ φ (r + s)(t

+ t

)=0,φ(r + s)=



dρ



(b) For a polytropic gas, show that

φ (r + s)=

γ +1

F (ρ)

,α≡



γ +1

γ − 1



and

r + s

+ t

)=0.

γ−1

= r + s and u = r −s, show that the diﬀerential

equation in 9(b) reduces to the Euler–Poisson–Darboux equation

−



γ − 1





2α



=0,

where u and c are indep endent variables.

10. Show that the KdV equation

− 6uu

+ u

xxx

satisﬁes the conservation law in the form U

+ V

= 0 when (a) U = u,

V = −3u

+ u

, and (b) U =

, V = −2u

+ uu

−

11. Show that the KdV equation

+6uu

+ u

xxx

satisﬁes the conservation law

+ V

=0,

where

(a) U = u, V =3u

+ u

(b) U =

, V =2u

+ uu

−

, V =

+3u

+ u

12. Show that the conservation laws for the associated KdV equation

− 6v

+ v

xxx

=0,

are

(a) (v )



+ v



=0,

(b)







+ vv

−



=0.

13.14 Exercises 599

13. For the nonlinear S chr¨odinger equation (13.12.13), prove that



∞

−∞

Tdx= constant,

where (i) T ≡ i



ψψ

− ψ ψ



and (ii) T ≡|ψ

−

γ |ψ|

14. Show that the conservation law for Burgers’ equation (13.10.20) is

(u)



+ νu



=0.

15. Show that the conservation laws for the equation

− uu

− u

xxt

are of the forms

(a) (u)

−



+ u



=0,

(b)



+ u



−



+ uu



=0.

16. Show that the linear Schr¨odin ger system

iψ

+ ψ

=0, −∞ <x<∞,t>0,

ψ → 0, |x|→∞,

ψ (x, 0) = ψ (x)with



∞

−∞

|ψ|

dx =1.

has the conservation law



i |ψ|



+(ψ

∗

− ψψ

∗

)

and the energy integral



∞

−∞

|ψ|

dx =1.

17. Seek a dispersive wave solution of the telegraph equation (see problem

14, 3.9 Exercises)

− c

+ ac

+ bc

u =0

in the form

u (x , t)=A exp [i (kx − ωt)] .

(a) Show that ω = −



iac





4b − c



(b) If 4b = a

, show that the solution

u (x , t)=A exp



−



exp [ik (x +

ct)]

represents nondispersive waves with attenuation.