Inverse Scattering Transform (IST) analysis of KdV-burgers' equation

Pergamon

Inr. J. Non-Linear Mechanics. Vol. 30, No. 5, pp. 617427, 1995 Copyright 0 1995 Elsevicr Science Ltd Printed in Grsat Britain. All rights reserved 0020-7462/95 S9.M + 0.M)

0020-7462(%)00028-3

INVERSE SCATTERING TRANSFORM (IST) ANALYSIS OF KdV-BURGERS’ EQUATION R. Shailaja and M. J. Vedan Department

of Mathematics

and Statistics, Cochin University Cochin 682 022, India

of Science and Technology,

(Received 14 April 1994) Abstract--Inverse Scattering Transform (IST) method is applied to study a Korteweg-de V&-Burgers’ (KdV-Burgers’) type. equation due to Pramod and Vedan [lnt. J. Non-Linear Mech. 27, 197-201 (1992)]. This equation represents long wave propagation in water where the depth changes, forming a shelf. The problem is formulated in terms of a Zakharov-Shabat eigenvalue system [Sou. Phys. JETP 34, 62-69 (1972)]. This study shows the excitation of a continuous spectrum and the evolution of new solitons. As an example the excitation of a continuous spectrum by one soliton as it passes the shelf is discussed.

1. INTRODUCTION

The celebrated KdV equation derived by Korteweg and de Vries [l] is a model for unidirectional long waves of small amplitude travelling over a constant depth. In order to compute the deformation of a solitary wave climbing a beach, Peregrine [2] used a finitedifference scheme and has given the quantitative results. He derived a long wave equation in water of variable depth, which corresponds to the Boussinesq equation for water of constant depth. Madsen and Mei [3] have investigated the problem of a solitary wave propagating over a mild slope onto a shelf of constant depth. They have observed that while propagating over a shelf, the wave is disintegrating into a train of solitary waves of decreasing amplitude. If the depth of water is changing slowly the resulting equation is a perturbed KdV equation. Several successful attempts were made to find an asymptotic solution to this problem [4-61. A remarkable development in this direction is the derivation of such an equation by Kakutani [7] and Johnson [8]. Johnson [9] proved that if the depth decreases to form a shelf, then for particular new depths a solitary wave breaks up into a finite number of solitons asymptotically far along the shelf. The KdV equation with variable coefficients appears as a model for wave propagation in an inhomogeneous medium. In the case of water waves the inhomogeneity may be due to change in depth or cross-section. Nirmala et al. [ 10, 1l] have considered the integrability of a particular class of KdV equations with variable coefficients. Other works are due to Shen and Zhong [12], Zhou [13,14] and Sobezyk [15]. Johnson’s 193 equation belongs to a class of perturbed KdV equation. Kaup and Newell Cl67 used the Inverse Scattering Transform (IST) theory for the exact KdV equation as a basis for a perturbation scheme, and found that the zero wave number mode of the continuous spectrum was excited by the solitary wave interacting with the perturbation. This excitation is manifested by the creation of a shelf in the wake of a solitary wave extending between the rear of the Gave and the point to which the largest linear disturbances would have travelled. The role of the shelf is to provide a balance in the mass flux relation. Kaup and Newell also showed that the shelf acts as a precursor to the formation of new solitons. Their study involves all perturbations of equations which are exactly integrable by using the inverse scattering transforms associated with the Zakharov-Shabat eigenvalue problem [ 171 or the Schrodinger equations. An exact expression for the solution in terms of the scattering data and squared eigenfunctions was used to avoid the inverse procedure given by the Marchenko equations. The Zakharov-Shabat method has another Contributed

by W. F. Ames. 617

618

R. Shailaja and M. J. Vedan

advantage in that the unperturbed system is integrable and this system can be used as the basis for analysing the perturbed system. The fixed parameters (the action variables) of the unperturbed system will take on a slowly varying behaviour in the perturbed system. Knickerbocker and Newell [ 181 conducted numerical studies of the analytical results of Kaup and Newell [ 161 concerning the effect of a perturbation on a solitary wave of the KdV equation. Kalyakin [19], Byatt-Smith [20] and Grimshaw [21] have also studied solutions of perturbed KdV equations.

2. PERTURBATION

METHOD

We consider the KdV-Burgers’ equation [22] rj*+ (1 - 5 EH(X - x0) + A E/I&(X - X0))7jX + ct IJ!+ 2 saH(x - xo))rt%C+ 3 @(x

- x&C,

(1)

+ (6s - 3sBH(x - x0))%,, = 0. Equation (1) can be written as = &((i H(x - xc) - A jIS’(x - xo))qx ?t + ?X + f a??, + &BtlXXX 4

aH(x -

~0)~~

-

3 B&x

-

XO)V,,

+

3 BHb

-

~0)~~~~).

(2)

By the definition of a Dirac delta function we have exp ik(x - x0) dk

6(x - x0) = +I s and

& H(x

- x0) = 6(x - x0).

Then, equation (2) can be written as (3) where

F=

r

Y

rk-~Blk

Equation (3) is a perturbed equation (3)]

~.~(x-xo)-~a~qrl,6(x-xo)

KdV equation.

We consider the integrable system [in (5)

Let q = Jux

+ u2,

where J=.$ 1

28 Fit.

If U is a solution of U,+ Ux+~aU2Ux+~j3Uxxx=0, then q = J U, + U2 is a solution of equation (5).

(6)

IST analysis of KdV-Burgers’

619

equation

In equation (6) we take q to be known; this then corresponds to a Riccati equation for V and can be linearised by the transformation

u= yielding

J

K

(7)

7,

v,, - qv = 0,

(ga)

where

+L.

VW

J

This is the time-independent Schrodinger equation; however, it is missing the energy level term. In IST method we consider the time-independent Schrodinger equation v,,-(f-C2)V=0,

(9)

where f is the potential and C2s are the energy levels, where { = 5 + if, and V is the wave function.

3. ZAKHAROV-SHABAT

EIGENVALUE

Equation (9) can be written as the Zakharov-Shabat

PROBLEM

(ZS) eigenvalue problem [17]

VI, - irv1 = -fV,

(IOa)

vzx+ i5v,

(lob)

= - VI.

Following Kaup and Newell [16] and Newell [23], we define, for real [, two pairs of linearly independent solutions. The solutions are b,(x, t, i) and 6(x, t, <), where 4 =(&r, -&), and $(x, t, 0 and 6(x, t, [), where li; = ($,*, -$:) and * denotes the complex conjugate, which have the following asymptotic properties: 4-+e f#~ +

I(l+e $

+

-i[x

t)e+

a([, is*

a([,

x+--co

,

,

(114

+ b(5, t)eiix,

x+co

(1 lb)

x+co

(llc)

t)eirX - b( -[, t)e-iC”,

x+-co.

(lid)

Here l/a is the transmission coefficient and b/a is the reflection coefficient. The solutions are interrelated by $4~ t, 5) = 4Lt) 4(x, c -0

+(x, t, -5) + U, t) ICI@,t, 0

= 4 -C, t) W, t>0 + b( 4,

t) $4~9 t, 4)

Wa) (12b)

and the inverse relations can be found by using the fact that 4,

t) 4 -L t) -ML t) b( 4

t) = 1.

The reality of f(x, t) implies that 4-T)

= a*(r*)

b(-c)

= b*(c*).

The zeros of a({, t), when Im(c) > 0, are the discrete eigenvalues of equation (9). Assume that these zeros are simple. The linear integral equation which allows one to reconstruct the potential f(x, t) from the scattering data uses the following combinations: S+; S-;

{ML W4L t),

C real;

(L Y& k = 1,. . . , N)

(134

(b*K, t)/a(L t),

l real;

(6, Bd,

(W

k = 1,. . . , N).

620

R. Shailaja and M. J. Vedan

Here yk = bk/ak, ,?k = - l/b,a;, and bk(t) iS defined by the relation 4(&, t) = bk(t) $([k, t); b,(t) is the analytic extension of b([, t) to & if there is one. We will work with S_ . We make the substitution f = q/J’ in equations (3)-(5). Then, equation (5) is the integrable system (14) In equation (14), PO([) is an entire function (real for [ recall) and M, is the operator M,(Ic1’) = C2ti2, where P&2) = -4[2

(154

and

The scattering data for the above integrable system is ikt

=

0

(b*/a), = 2ic PO(c’) b*/a . Now we can solve

(17) where F =

_A- 212 j3.k’ )

(2ik

By mapping q(x, t) into the scattering functions associated with equation (9), then bkrkt

=

2iiBa;i k

j

m -m

(184

F$,Zdx

F

W-W

j -CC mFij’dx.

(184

w(k~=2i~kp~(~~)Bk+~~~ kk

-m

for k = 1,2, . . . , N and for real c and

@*/ah= WW2) b*la + $

For the unperturbed system the multi-soliton state, following methods similar to that of Zakharov and Shabat [17], is given by

Wb)

IST analysis of KdV-Burgers’

equation

621

where Jlj, j = 1,2, _ . . , N, are found from the equations $j+

i

Ykex~~~i+)iz’x$I=expi(i,x),wherej=l,...,N.

(19c)

J

k=l

Using the property of the delta function, equations (18) become

2ix exp (2&)x

’

1 -

1

(b

Z=l

+ -

b)x.

ck)

(6.k

(ck

I=1

+

+ 6,) (cl

+

+

&I2

-

exp(i~k)x 2ck

II

1

ykh$k!hexpi(~k k=l

Yk@k

k=l

Yk~Ztikhexpi(~~k k=l

N c

2

&

x=x,,

(2W

11


c)

X=.X0

W’b)

’

It has been pointed out by Kaup and Newell [16] that the perturbed KdV equation cannot conserve mass flux. While conservation of energy is consistent with the fact that the eigenvalue of the Schrodinger operator for real potentials adjusts its value in an adiabatic way to change in depth, the growth or decay of a soliton cannot correspond to the supply of water across the shelf to keep a constant flux. Now, in order to study the propagation of a soliton past the shelf, we consider the contribution of the continuous spectrum. For this we calculate the reflection coefficient b/a from equation (2Oc) with PO = -4c2 and (b/a),+, = 0. From equation (20~)

.

exp (2icx)

N

1- 2 1 k=l

Yk+k

(CC

YkYl$klC/Iexpi(~k k=l

1=1

(ck

+

c)(cl

exp

+ +

c)

(Ykx) +

ldx

l)

IIX=X0 ’

(21)

622

R. Shailaja and M. J. Vedan

Integrating we obtain with (rft = 0 to first order)

. (exp (Sir3 t) - 1).

From equation (22) we find that b(<, t) is not defined for < = 0. In fact we have to solve the perturbed equation in an iterative manner treating Eas a small parameter and using well-known ideas from singular perturbation theory. Our interest is to obtain an asymptotic expansion for q(x, t) which is uniformly valid for times c = O(E-‘) for some r > 0. Uniformly valid asymptotic expansions in scattering space result in a uniform expansion in the physical space. However, when the asymptotic expansion in scattering space is non-uniform we resolve the question by demanding that the resulting asymptotic expansion [16,23] for f(x, t) in physical space in terms of the scattering data and squared eigenfunctions (23) is uniformly valid. Here the summation corresponds to the contribution of the discrete spectrum and the integration corresponds to the contribution of the continuous spectrum. In equation (19) If N = 1, ii = iii, and yi = 2irje2’, where then

e-@

*1 = 1 + exp(-2fx

- =iexp(-8) + 20) 2

sech 6.

Therefore equation (19a) becomes $(&j) = eitx

1 - &&).

(24)

(

Now we determine the contribution of continuous spectrum iif(x, t) of this term to q(x, t), which can be computed from equation (23). We obtain

(25) Clearly this has a non-vanishing contribution

4. ONE-SOLITON

in the neighbourhood

of r = 0.

CASE

As a particular example we consider the motion of a soliton f(x, t) = 2~~ sech’ ~(x - X)

(26)

623

IST analysis of KdV-Burgers’ equation

after it has passed the position x = x0 at which the depth changes. We assume that the soliton reaches the position at t = 0. We obtain the equations of motion for the amplitude p(t) and position Z(t) of the soliton. Then equation (17a) becomes

F=(&-$j?ik).

-4$

+ $$sech”p(x

sech2 p(x - X) tanhp(x - X) 6(x - x0)

- x)tanhp(x

-:(8p’sech’p(x

- X)6(x -x0)

- X) - 12p4sech4p(x - X))6(x -x0)

P +3ik(-16$sech2p(x-x)tanhp(x-x) + 48,~~sech4p(x - X) tanh p(x - x)) 6(x - x0).

(27)

For the scattering functions associated with equation (9) we obtain

pktkt

=

sech2 ~(x {[(&-ibk)---4pl

&

+ $p’sech’p(x

-38~~

- x)tanhp(x

X) - tanh ~(x - X)

- X)

sech2p(x - 3) - 12p4sech4 ~(x - 2))

+$(-16$sech’p(x-i)tanhp(x-5)

+ 48~~ sech4 ~(x - X) tanh ~(x - X))

‘exp

N 1- 2 c

(2&x)

[

YkYl +s k=l

fikt+

$++ (

Yk$kexpirkx

21 k

k=l

$ktilexp 2ik(ik

1=1

1

i(ck +

+ id

b)x

(284 11

x=x0

fik
.

-4~~ sech2 ~(x - 2) tanh ~(x - X) + $

pL5sech4 ~(x - X) tanh ~(x - X)

-s (8,~’ sech2 ~(x - X) - 12~~ sech4 ~(x - X))

+&(-16/c5sech2a(x-x)tanhp(x-i)

+ 48~’ sech4 ~(x - X) tanh ~(x - X))

1

624

R. Shailaja and M. J. Vedan

Ykwbk#JeXP k=l

I=1

k=l

I=1

i(
2ck(ck

+

+

cdx

id

?k+k;;;;;iikx) k

k=l

YkYl+k$leXPi(3~k

+

(il

+2 ; 1

tdx

ik)

-

and

@*/ah= NIP, m*/a +g-g{[(A-$j?ik).-4p’sech’p(x-i)tanhp(x-2) + T$

sech4p(x - X) tanh(x - X)

-z(Sp4 sech’ p(x B + 3ik

X) - 12p4 sech4 p(x - x))

- 16~’ sech2p(x - X) tanhp(x

N

2k&

Yktik

exp(ilkX) ([

N k=l

I=1

YkYlrC/klC/IeXPi(~k (ik

+

+

i)

k

[

+lic

- X)

p(x- X)1

+ 48~’ sech4 p(x - X) tanh

- exp(2Xx) 1 -

1

+ <)(b

+

&)x

11

i)

cw

x=x0’

We calculate the reflection coefficient b/a from equation (28~) with PO = -4c2 (blc),=o = 0. Then the equation becomes

{[(&$ik)- -4~~ sech’ ~(x -

@*la), + w3(b*/a)= &

+ $

and

X) tanhp(x - X)

I_L’ sech4 p(x - X) tanh p(x - X)

-$3/14sech2/1(x

- X) -12p4sech4&

- 2))

1

+& [ -16$sech2p(x-x)tanhp(x--3)

1

+ 48~’ sech4p(x - X) tanhp(x - X) N 1- 2 1

exp(2icx) [ N +tc k=l

I=1

Yk$kexp(i~k)x

k=l

Ykh+kl(/leXP (ik

(ck

%k +

i)(il

+

+ +

i)

t;)

idx

11 x=x0

(29)

IST analysis of KdV-Burgers’ equation

625

Integrating (with pt = 0 to first order), equation (29) becomes

b*(r, t) = exp(-8ic3t)

-

dt

a

+*oa xp sech4&x 5

B -$8p4sech2p(x

-4~’ sech’ ~(x - 2) tanh ~(x - 2)

- x) tanhp(x

- x’)

- X) -12~4sech4~(x

-2))

B +z(-16$sech’&x--f)tanhp(x-2)

+ 48~’ sech” ~(x - 2) tanh ~(x

+cc

’

YkYlh~leXP

if&

+

l(exp(8it3t)


- I).

The reflection coefficient b(<, t) becomes b(&t) = ;exp(8ir3t)

l

dt &exp(-8i<3t)

-4p3 sech’ p(x - x) tanh p(x - 2)

x{[(&&Bik)*

$p5sech4p(x

+

- 1)

- x)tanhp(x

- X)

B + $8~’ sech2 ~(x - 2) + 12p4sech4p(x - 2)) B +G(-16$sech’p(x

- Z)tanhp(x

- 2)

1

+ 48~~ sech4 ~(x - X) tanh ~(x - X) Yk$kexp(-itkX) (ck +

‘t)

(30) From equation (30) we find that b(& t) is not defined for < = 0. Now we determine the contribution of the continuous spectrum &(x, t) of this term to q(x,t), which can be computed from equation (23). We obtain

dt*(exp(-8it3t)

l

- 1)

s -4p3sech2p(x + $jA?ech4p(x

- 2). tanhp(x - 2) - 2) tanh ~(x - 2)

626

R. Shailaja and M. J. Vedan

+

:(8p4sech2p(x

- 2) -12p4sech4p(x

+&(-16pssech2p(x

- Z)tanhp(x

- 2)

+ 48~’ sech4 ~1(x - 2) tanh p (x - X)

1- 2 C [

k=l

1

N Yk$kexp-(iSkx)

.exp(-2irx)

k=l

N

Ykhtikhexp

I=1

(
+?I

- X))

(5k

+

+

W

-i(tk +

t-)(t,

Clearly this has a non-vanishing contribution

+

5)

0

IIX=.X0

(31)

’

in the neighbourhood

of l = 0.

5. DISCUSSION

The equations of Kakutani [73 and Johnson [9] have led to the study of a perturbed KdV equation as a model for diverse physical systems. Some of the major contributions in this field are due to Kaup and Newell [16], Knickerbocker and Newell [18] and Newell [23], especially in the context of water waves. The classical KdV equation is known to have an infinite number of conserved quantities. In the context of water waves this includes the conservation laws of mass and energy. The asymptotic solutions of the perturbed KdV equation show that mass is not conserved, which indicates that the solution is non-uniform. To circumvent this difficulty, Kaup and Newell studied this problem using a different method. They exploited the fact that the classical KdV equation is exactly integrable, that is, an infinite dimensional Hamiltonian system. The IST transform is a canonical transformation which carries the old coordinates (wavefunction) to the scattering data of the corresponding Schrodinger equation. Here the bound state eigenvalues are the action variables which prescribe the constant amplitude, shape and speed of the soliton; the normalization constant corresponds to angle variable and defines its position. The reflection coefficient measures the degree to which the continuous spectrum is excited. In the case of the unperturbed system, the reflection coefficient is identically zero. When the system is perturbed it is no longer exactly separable. The normal modes become mixed so that an initial state consisting only of solitons can stimulate radiation and conversely the radiation can result in the creation of new solitons. Thus the reflection coefficient is no longer identically zero and the system does not have an infinite number of conservation laws. The physical argument given by Kaup and Newell is as follows: the KdV soliton has only one parameter-its amplitude-to adjust as it faces a perturbation. Therefore, it cannot simultaneously satisfy conservation of mass and energy flux. It chooses to satisfy the latter. The failure to satisfy conservation of mass flux means that another solution component must be excited to preserve the mass balance. The physical manifestation of this is the creation of a shelf and mathematically it means that the reflection coefficient develops a Dirac delta function behaviour. Our study here shows the excitation of a continuous spectrum and the evolution of new solitons.

REFERENCES 1. D. G. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel and on a new type of long stationary waves. Phil. Mag. 39,422-443 (1895). 2. D. H. Peregrine, Long waves on a beach. J. Fluid Mech. 27,815-827 (1967). 3. 0. S. Madsen and C. C. Mei, The transformation of a solitary wave over an uneven bottom. J. Fluid Me& 39, 781-791(1969). 4. R. Grimshaw, The solitary wave in water of variable depth. J. Fluid Mech. 42, 6399656 (1970). 5. R. Grimshaw, The solitary wave in water of variable depth. Part 2. J. Fluid Mech. 46, 611622 (1971).

IST analysis of KdV-Burgers’ equation

627

6. S. Leibovich and J. D. Randall, Amplification and decay of long nonlinear waves. J. Fluid Mech. 58,481-493 (1973). 7. T. Kakutani, Effect of an uneven bottom on gravity waves. J. Pbys. Sot. Japan 30,272-276 (1971). 8. R. S. Johnson. Some numerical solutions of a variable coefficient Kortewe.g-de Vries equation (with applica_tions to solitary wave development of a shelf). J. Fluid Mech. 54, 81-91 (i972). _ 9. R. S. Johnson, On the development of solitary wave moving over an uneven bottom. Proc. Camb. Phil. Sot. 73, 183-203 (1973). 10. N. Nirmala, M. J. Vedan and B. V. Baby, Auto-Backlund transformation, Lax pairs and Painleve property of a variable coefficient Korteweg-de Vries equation I. J. Math. Phys. 27, 264&2643 (1986). 11. N. Nirmala, M. J. Vedan and B. V. Baby, A variable coeficient Korteweg-de Vries equation: similarity analysis and exact solution. J. Math. Phys. 27, 26442646 (1986). 12. M. C. Shen and X. C. Zhong, Derivation of Korteweg-de Vries equations for water waves in a channel with variable cross-section. J. Mecanique 20, 789-801 (1981). 13. X. C. Zhou, The solitary waves in a gradually varying channel of arbitrary cross-section. Appl. Math. Mech. 2, 429-440 (1981). 14. X. C. Zhou, Nonlinear periodic wave and fission of a solitary wave in a slowly varying channel with arbitrary cross-section. Sci. Sinica A 26, 626-636 (1983). 15. K. Sobezyk, KdV solitons in a randomly varying medium. Int. J. Non-Linear Mech. 27, l-8 (1992). 16. D. J. Kaup and A. C. Newell, Solitons as particles, oscillators and in slowly changing media: a singular perturbation theory. Proc. R. Sot. Land. A361,413-446 (1978). 17. V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media. Sov. Phys. JETP 34, 62-69 (1972). 18. C. J. Knickerbocker and A. C. Newell, Shelves and the Korteweg-de Vries equation. J. Fluid Mech. 98, 803-818 (1980). 19. L. A. Kalyakin, Asymptotics of an integral that arises in the perturbation of KdV solitons. Math. Notes 50, 1114-1122 (1991). 20. J. G. B. Byatt-Smith, Solutions of the perturbed Korteweg-de Vries equation. In Nonlinear Dispersive Wave Systems, Lokenath Debnath (ed.), pp. 157-179. World Science, Singapore (1992). 21. R. Grimshaw, Nonlinear waves in fluids-the KdV paradigm. In Nonlinear Dynamics and Chaos, Canberra, 1991, pp. 175-198. World Science, River Edge, New Jersey (1992). 22. K. V. Pramod and M. J. Vedan, Long wave propagation in water with bottom discontinuity. Int. J. Non-Linear Me& 27, 197-201 (1992). 23. A. C. Newell, The Inverse Scattering Transform. In Topics in Modern Physics: Solitons, R. Bullough and

P. Caudrey (eds). Springer, Berlin (1980).