AKNS equations of the inverse scattering method

Volume 112A, number 5

PHYSICS LETTERS

28 October 1985

C O M P L E T E S O L I T O N S O L U T I O N S OF T H E Z S / A K N S E Q U A T I O N S OF T H E I N V E R S E SCATFERING M E T H O D Abbas A. R A N G W A L A and Jyoti A. R A O 1 Department of Physics, University of Bombay, Bombay 400 098, lndia Received 14 May 1985: accepted for publication 19 August 1985

Starting with the (n - l)-soliton solution of a non-linear evolution equation (NLEE) and the corresponding Zakharov-Shabat, Ablowitz-Kaup-Newell-Segur (ZS/AKNS) wavefunctions, we obtain the n-soliton solution of the NLEE and the corresponding ZS/AKNS wavefunctions. This is then used to obtain complete soliton solutions of the NLEE and the ZS/AKNS equations.

There has been an increasing interest in non-linear evolution equations (NLEEs) for the past twenty years. It was Gardner, Greene, Kruskal and Miura who in their pioneering paper [1], first obtained soliton solutions of the KdV equation using ideas of the inverse scattering method. Their method remained a curiosity till in 1972, Zakharov and Shabat (ZS) [2] extended this method to obtain the soliton solutions of the cubic Schr6dinger equation. In a comprehensive work, Ablowitz, Kaup, Newell and Segur (AKNS) [3], enlarged the ZS equations to encompass a large class of NLEEs. These ZS/AKNS equations are differential equations involving two-component wavefunctions. They are also eigenvalue equations. The ZS/AKNS inverse scattering problem is def'med by [3] Olx + ikv 1 = q ( x , t ) v 2,

U2x - i k v 2 = r ( x , t ) v 1,

(1)

where o = (vl) is the two-component ZS/AKNS wavefunction an~q and r are functions o f x and t and satisfy NLEEs of interest. The time evolution of Ol, 2 is governed by [3]

mined by the specific non-linear equation of interest [3]. In the application of the inverse scattering method, the main interest lies in obtaining the soliton solutions of no n-linear equations satisfied by q and r; (1) and (2) then play only an ancillary role. There are, however, situations like coherent pulse propagation [4,5], self-focussing and self-modulation of waves in non-linear media [2], perturbation about soliton solutions [6] where o itself has either direct physical relevance or utility and in such circumstances we would like to know the wavefunctions themselves. In the present letter we develop a recursive method by which we determine not only the soliton solutions of the non.linear equation of interest but also the ZS/AKNS wavefunctions. This method is not any more complicated than the other standard method and has the additional merit of giving us the wavefunctions. We introduce a convenient notation for (n - 1)-soliton and n-soliton problems. We denote the quantities referring to the (n - 1)-sohton problem by v 1 = - - o l ( n - 1),

1),

o2=v2(n-

Vlt = ~ ( k ; q , r ) u 1 + Q S ( k ; q , r ) o 2, q=-q(nv2t = ~ ( k ; q , r ) u 1 - ~ z t ( k ; q , r ) v 2.

u= ( ul o2 ) ,

1),

r~-r(n-

(3a)

(2)

The applicability of the inverse method rests on the requirement k t = O. The functions ~4 , ~ , ~? are deter1 On leave from: Department of Physics, Ramnarain Ruia College, Bombay, India.

188

1),

and those referring to the n-soliton problem by v,1 - vl(n),

0'2 - v2(n), t

q =q(n),

r-r(n),

v'=

I)2I

.

(3b)

0.375-9601/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

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u1 2 satisfy the ZS/AKNS equations (1) and (2). u; 2 saiisfy equations similar to (1) and (2) with all qua& ties replaced by the primed ones except the eigenvalue k which remains the same: viX t ikv; = q’vi,

vix - ikv; = r'v; ,

(4)

and uit = d(k;q’,r’)u;

+ q(k;q’,‘r?u;,

u;~ = E’(k;q’,r’)u;

- sQ(k;q’,r’)u;.

v;=Cv,

+Dv2,

B=blktbo,

C=clk+co,

D=dlktdo,

0,

co =

d, = WI,

+icu [r’ - r(S/ol)] ,

do = -@/+Q

+ P(t),

(9)

with a,, given by the differential equation aox = - $icu(qr - q’r’),

(10)

-+io[@/o)q~

- s,l = -K+)d+qlao

+P4’,

(5)

(6)

+r[ri - @/a)r,] = [r’ + @/cw)r]a0 - or.

(11)

Here o,& 6 are integration constants which may depend on t. Thus we can obtain A, B, C, D in (6) if we can solve for a0 and be able to satisfy the conditions in(l1). We now restrict our discussion to the following three of the better known NLEEs: Korteweg-de Vries (KdV) equation, sine-Gordon (sG) equation and nonlinear Schriidinger equation (NLSE). The method can, however, be applied to any equation solvable by the inverse scattering method of ZS/AKNS. I. KdV equation. The potential q (q’) satisfies the KdV equation [3] : 4, + 6qqX + qXXX= 0,

r=-1. (7)

where ai, bi, ci, di are functions of (4, r) and (q’, r’) and through them, functions of x and t. Differentiating (6) with respect to x and using (1) and (4), we get on equating the coefficients of u1,2 on both sides: B, + 2ikB = -qA t 4’0,

CX - 2ikC = r’A - rD,

D, = r’B - qC.

(8)

Using (7) in (8) and equating the coefficients of k on both sides, we obtain a set of differential equations for (a, ... , d) which are easy to solve. The solution is: al = o(t), b,=O,

=

(12)

while the potential r (r’) is given by

A=alk+ao,

A, = -rB t q%,

cl

1985

and provided the following two equations are satisfied:

It is welI known that the ZS/AKNS equation corresponding to the KdV equation is the Schrijdinger equation [3]. In the context of the Schrijdinger equation, Bargmann [7] has shown that for a potential capable of giving n bound states, the solution of the Schrodinger equation can be written in the form etixx(k, x) where x(k,x) is an nth degree polynomial ink. On the other hand, we know that the n-soliton solution of the NLEE appearing in ZS/AKNS equations (1) can be looked upon as a potential giving n bound states (see, e.g., ref. [6]). The above Bargmann result along with the relation between solitons and bound states suggests that u and u’ will differ by a linear function of k. Thus we write u; = Au, + Bv,, with

28 October

LETTERS

b,,=+[-q+q’(6/@],

(13)

The functions SQ,93 , C?in (2) are given by d(k;q,

-1) = -4ik3 t 2ikq - q,,

Q(k;q,

-1) = -q,,

e(k;q,

-1) = -4k2 + 2q.

+ 2iqXk + 4k2q - 2q2, (14)

Introducing w through q = -wX and using the space part of the Backlund transformation (BT) for the KdV equation [8-lo] : ox t 0; = -2n’2 + f(w’ - w)2,

(15)

where 9’ = qn is a real parameter characterising the nth soliton, it is easy to see that the choice (6/o) = - 1, 0 = 0 satisifes the requirements in (11) and the differential equation in (10) withau given by a,, = $cu(w’ - w). Finally if we choose the overaIl constant cysuch that io = 1 then the wavefunctions are given by 189

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PHYSICS LETTERS

t

1

t

V1 = [ - i k + ~(CO - CO)]v 1 + "5(q + q ) v 2 , s

1

t

v 2 = - v 1 + [ik + i(CO - CO)]v 2.

(16)

It is a straightforward matter to verify that the above solution satisfies the time part of the ZS/AKNS equation (5) in view of (14) and the time part of the BT [9,10] : ~(CO't - COt) = - q x x

(17)

This is a reflection of the fact that the space part of the ZS/AKNS equations, (1), and the KdV equation satisfied by the q determine the functions M , q0 and e in (2). We now come to the second part of the programme. Konno, Wadati and Sanuki [8,9] and Chen [10] have obtained the BTs for the NLEEs solvable by the inverse scattering transform method of ZS/AKNS. They obtain these by introducing a quantity: P = Vl/02 Ik=ir( .

(18)

It is important to note that o 1 , v 2 appearing in this equation are general solutions and not the f u n d a m e n tal solutions o f ( l ) corresponding to the (n - 1)-soliton problem. In the process of obtaining the BTs they show that the n-soliton solution, co', is related to the (n - 1)-soliton solution, co, by CO' = CO+ 2(P + r/').

(19)

This observation coupled with (16) then furnishes us with a recursive method by which we can calculate all the soliton solutions of the KdV equation and the corresponding ZS/AKNS wavefunctions. The procedure simply is the following. We start with the zero-soliton (n --- 0) solution, q(0), of the KdV equation and using this in (1) obtain the zero-soliton ZS/AKNS wavefunction, o(0). Using q(0) and v(0) in (19), we obtain CO(l) or q(1). These q(1), q(0) and o(0) are then used in (16) to obtain o(1). This procedure can then be continued to obtain all the higher soliton solutions and the accompanying wavefunctions. We illustrate the procedure in a little detail later in the context of the NLSE. For the present case we only remark that the results obtained by the present method reproduce the standard ones in the literature (see, e.g. ref. [6]). Thus besides providing an alternative method of determining 190

the soliton solutions of the NLEEs, which is no more involved than other methods available in the literature, it simultaneously gives the ZS/AKNS wavefunctions. 2. SG equation. The SG equation is given by [3] COtx - ~sm 26o,

(20)

where, as before, q = -cox and r is given by r = -q.

- 2q2 - qx ( c o ' - CO)

1 p + 2r/'Zq ' - ~q(CO - co) 2.

28 October 1985

(21)

Using the space part of the BT for the SG equation [8 --10[, t

COx + COx = 2~' sin(CO' - CO),

(22)

where, as before, 7/' - ~n is a real positive parameter characterising the nth SG soliton, it can easily be seen that the choice (8/a) = -1,/~ = 0 satisfies the requirements in (11) and the differential equation in (10) with a 0 now given by a 0 = io~r/' cos(CO' - CO).

(23)

Choosing, as in the KdV case, the constant a to be - i , the wavefunctions v], v~ are given by [11] t V1 = [ s

it 1

+ ~ ' COS(CO' - - CO)] V 1 + F

v 2 = - 7(q + q ) v 1 + [ik +

.F/r

}(q' + q)v2,

cos(CO' - co)] v 2. (24)

We had earlier obtained these equations following a different procedure [ 11 ]. Deffming P as in (18) and noting from refs. [ 8 - 1 0 ] that for the SG equation the relation between the n-soliton solution co' and the (n - 1)-soliton solution COis given by CO'= CO+ 2 tan 1 p,

(25)

the recursive method mentioned in the context of the KdV equation above can be repeated once again and both the soliton solutions of the SG equation and the wavefunctions of the accompanying ZS]AKNS equations are obtained. We make a few comments on the modified KdV equation (mKdV) which is given by qt + 6 q 2 q x + q x x x = 0.

(26)

It is well known [ 8 - 1 0 ] that the space part of the BT for the mKdV equation is identical to the one for the SG equation and consequently the ZS/AKNS wavefunction v' is given in this case too by (24). Moreover, the relation (25) remains valid also. The difference be-

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tween the two equations in the inverse scattering framework is that the two equations have different a f , qS, e appearing in (2). As a consequence their dispersion relations and hence the time dependences are different. For instance, the (x, t) combinations for the SG equation are in the form +ik(x - t / 4 k 2) and 2r/(x + t/4r12), whereas for the mKdV they are in the form +ik(x + 4k2t) and 2ri(x - 4r/2t).

I

28 October 1985

I

t

qt + qt = 211 (qx + qx ) _ i(q,x _ qx)(4v,2 _ jq,+ q]2)1/2 + i(q' + q)(Iq'[ 2 + Iq12).

(32)

To carry out the programme of obtaining all soliton solutions and the corresponding wavefunctions, we define, following refs. [9,10] :

3. NLSE. The evolution equation satisfied by q is given by

F = Vl/V21k=,+iv,,v,>O.

iqt +qxx + 21q12q = 0,

The n-soliton solution q' can be expressed in terms of the (n - 1)-soliton solution q and F by [9,10]

(27)

with

(33)

q ' = - q - 4v' F/(1 + IPI2).

r = -q

.

(28)

The functions s~, cB, e in (2) are here given by s~(k;q, - q * ) = ilql 2 - 2ik 2, q 6 ( k ; q , - q * ) = iqx + 2kq, C ( k ; q , - q * ) = lqx " * - 2kq*.

(29)

(34)

We now carry out the procedure of obtaining recursively the multiple-soliton solutions and the corresponding ZS/AKNS wavefunctions starting from the zero-soliton solution of NLSE. We begin by noting from (27) that the zero-soliton solution is q(0) = 0. We use this in (1) to obtain the zero-soliton ZS/AKNS wavefunctions: Ul(0) = or(t) e - i k x ,

02(0) = 13(0 e ikx,

(35)

The space part of the BT is [9,10] : I

qx + qx = -2i11'(q' + q) -- (q' - q)(4v '2 - tq' + q12) 1/2,

(30)

where 11' =/a n and v' = v n are real parameters (v n > O) characterising the nth soliton solution of the NLSE. Committing, as before, to (8/a) = - 1 , we see that the requirements in (11) as well as the differential equation in (10) are satisfied if we set 13= 2t~' and take

As before, we choose a = - i to obtain v' in the form

t

+ ~(q + q)u 2, t

•

where o~0 and 130 are constants independent of t. To obtain q(1), we use (34) which reads in the present q(1) = -4v1I'(0)/(1 + IF(0)I2),

v 1 > 0,

(37)

q(1) = - 2 v 1 e -ix1 seth ~1'

+ q )o 1

+ [ik - i 1 1 ' - ~ ( 4 v '2 - I q ' + q 1 2 ) l / 2 ] 0 2 •

o2(0) = 13oeik(x+2kO, (36)

where ~ 1 , Vl) are the two real parameters characterising the first NLSE soUton. Evaluating F(0) using (33) with k = 111 + iv1 and defining ix0/130= e - 2i(8 x+ iv1 AD, we obtain

o'1 = [ - i k + i11' - ~ ( 4 v '2 - I q ' +q12)1/2101

02 = _

o1(0) = ot0 e -ik(x+2kt),

case

a 0 = its[ill' - ½(4v '2 - Iq' + ql2)1/2].

1

where ix, 13are constants of integrations which depend on t. We determine the t-dependence by employing (2), with sff, q~, e obtained from (29) by setting q = 0. This leads to the time dependence of zero-soliton wavefunctions as

(31)

These wavefunctions can be shown to satisfy the time part of Z S / A K N S equations, (5), in view of the time part of the BT for the NLSE [10,9] :

X n = 2[lanX + 20a 2 - v 2 ) t +CSn],

~n = 2vn(x + 411nt + An),

n = 1, 2, 3 .....

(38)

Using q(1), q(0) = 0, and u(0) in (31), we get

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PHYSICS LETTERS

- i k + i#l - v 1 tanh ~1) o(1) = al

e_ik(x+2kt)

v 1 e ix~ sech ~1

+~ [-v le 1[

lxa sech ~1 ik - i/.t1 - t) 1 tanh ~1 )

eik(x+2kt)'

(39)

where Otl,/31 are new constants characterising the onesoliton wavefunctions. We use (38) and (39) in (34) to obtain after straightforward simplifications, q(2)

=N/O,

N = 2vle-~Xx sech ~1

- 2v 2 e -lx2 sech ~2 X [ - ( a u ) 2 + 2 i v 1 ( a # ) tanh ~1 + v2 - v 2 ] ' D = (d~/a)2 + @2 + v 2) _ 2VlV 2 tanh ~1 tanh ~2 (40)

where (A/~) = #2 -- #1 and (#2, v2) are the parameters of the second soliton with v 2 > 0. This process can be continued and higher soliton solutions and wavefunctions determined recursively. We end with two observations. It is clear that the present method will apply to any non-linear equation solvable by the ZS]AKNS inverse method, once the BTs have been worked out for that case. Secondly since the BTs can be applied to generate not only the soliton solutions but soliton-antisoliton and breather solutions as well [12,13], we conjecture that the above

192

method can be applied to generate these solutions of NLEEs as well as their corresponding ZS/AKNS wavefunctions. Thus we conjecture that all the "solitonlike" solutions and their wavefunctions can be obtained by the present method. This and related matter will be reported elsewhere. One of us, AAR, would like to thank the Board o f Research in Nuclear Studies, Department of Atomic Energy, India for a grant. The other, JAR, would like to thank University Grants Commission (India) for a fellowship and the authorities of Ramnarain Ruia College, Bombay for a grant of study leave.

References

X [(dx#) 2 + 2iv2(•#) tanh ~2 + v2 - v~]

- 4VlV 2 seeh~l sech~2 cos(x 2 - X1),

28 October 1985

[ 1 ] C.S. Gardner, J .M. Greene, M.D. Kruskal and R.M. Miura, Phys. Rev. Lett. 19 (1967) 1095. [2] V.B. Zakharov and A.B. Shabat, Sov, Phys. JETP 34 (1972) 62. [3] M.J. Ablowitz, D.J. Kaup, A.C. Newell and H. Segur, Stud. Appl. Math• 53 (1974) 249. [4] G.L. Lamb Jr., Rev. Mod. Phys. 43 (1971) 99. [5] M.J. Ablowitz, O.J. Kaup and A.C. Newell, J. Math. Phys. 15 (1974) 1852. [6] V.I. Karpman, Phys. Scr. 20 (1979) 462. [7] V. Bargmann, Rev. Mod. Phys. 21 (1949) 488. [8] M. Wadati, H. Sanuki and K. Konno, Prog. Theor. Phys. 53 (1975) 419. [9] K. Konno and M. Wadati, Prog. Theor. Phys. 53 (1975) 1652• [10] H.H. Chan, Phys. Rev. Lett. 33 (1974) 925. [tl] J.A. Rao and A.A. Ran~gwala,to be published in J. Math. Phys. (1985)• [12] D.W. McLaughlin and A.C. Scott, J. Math. Phys. 14 (1973) 1817• [13] G.L. Lamb Jr., Elements of soliton theory (Wiley, New York, 1980)•