]L NUOVO CIMENTO VOL. 68 A, 1Y. 3 1 Aprile 1982

Nonlinear Schr~dinger Equation, Potential Nonlinear Schr6dinger Equation and Soliton Solutions.

M. BOITI, C. ~ADDOMADA and :F. I:)EMPIiNELLI

I s t i tu to d i ~ i s i c a dell 'Universit~t . Lecce, I t a l i a I s t i tu to 2tazionale d i ~ i s i c a Nuc lea te . I t a l i a

(ricevuto i! 5 0ttobre 1981)

Summary. - - It is shown that the equivalent real form of the nonlinear Sehr6dinger equation (NLS), the potential nonlinear Schr6dinger equa- tion (PNLS), introduced in a previous paper by the same authors, has a B/icklund transformation (BT) which satisfies the pcrmutability theorem. By taking advantage of the analyticity properties of the PNLS equation, a method of generating new solutions of the NLS equation is proposed. In this way a three-parameter simple-kink soliton solution is obtained, in the repulsive case, which is called (~ smooth ~) kink in comparison with the two-parameter simple-kink soliton solution which is called (( pointed ~) kink. Moreover, a four-parameter solution describing the collision of two (~ smooth ~> kinks is explicitly given.

1 . - I n t r o d u c t i o n .

I n two previous papers (1,2), hereafter referred to as I and I I , we introduced

an equivalent real form of the nonlinear SchrSdinger equation (lqLS), which

we called potential nonlinear SchrSdinger equation (PI~LS).

We showed tha t the use of the PlqLS equation is crucial in order to control,

in a simple way, the complicated algebraic structure of the I~LS equation and

its B/~cklund t ransformation (BT).

We succeded in giving a straightforward derivation of the double BT for

(1) M. Bola'I, C. LADDOMADA and F. PEMPIZqELLI: N~tOVO Cimento B , 62, 315 (1981). (2) 1~. BOITI, C. LADDOMADA and F. PEMPI~V.LLI: Phys . Left . A , 83, 188 (1981).

236

NONLINEAR SCIIRODINGER EQUATION ETC. 237

the IgLS equation and consequent ly a s t ra ightforward proof of the permu- tabi l i ty theorem. The double BT repor ted in paper I I generalizes t h a t al- ready obtained by ESTABROOK and WAm~QVlST (3) wi th more laborious ge-

ometric methods. This occurs because our formula includes correct ly also the case in which the k-parameter of Es tabrook and Wahlquis t is pure imaginary. In the paper of ref. (4), hereaf ter referred to as I I I , this re levant special case has been used to get multiple-soliton solutions of the I~LS equat ion in the repulsive case.

Moreover, recently, the PlgLS equation has been shown useful and phys- ically significant in the s tudy of the equation of motion for the Heisenberg spin chain (5).

The aim of the present paper is twofold. First , we show th a t the NLS and the P NLS equations are, in some sense, dual to each other, because bo th have a BT and satisfy a permutabi l i ty theorem (s).

Second, we take advantage of the fact t h a t the P N LS equation, in contras t with the I~LS equation, is analyt ic and we propose a me thod of generat ing new solutions of the IgLS equat ion both in the repulsive and in the a t t rac t ive case (e).

By this me thod and the use of the very convenient representat ion of the BT for the :NLS equat ion in te rms of the pseudopotential , proposed by ESTABROOK and WAHLQUIST, we find, in the repulsive case, a simple-kink soliton solution with three free parameters and a double-kink soliton solution with four parameters .

The ampli tude and the phase of the three-parameter kink propagate with the same velocity, bu t the phase is distorted by an addit ive t e rm tha t grows l inearly with t ime.

Moreover, the ampli tude of this kink has a smooth behaviour to be com- pared with the pointed behaviour at the minimum of the ampli tude of the two-parameter kink repor ted in paper I I I .

For the sake of brevi ty we call these solutions, respectively, (~ smooth )~ and (~ pointed ~> kink.

The multiple-kink soliton solution obtained in paper I I I exhibits interact ing smooth kinks, bu t always accompanied by a pointed kink.

The solution t ha t we repor t in this paper describes the collision of two smooth kinks wi thout any pointed kink and, therefore, generalizes the result presented in paper I I I .

(a) F . B . ESTABROOK and H. D. WAHLQUIST: J. Math. Phys. (N. Y.), 17, 1293 (1976). (4) 1~. BOlTI, C. LADDOMADA and F. P]~MI'INELLI: 2VUOVO Cimento B, 65, 248 (1981). (5) G. R. W. QuISe]~L and H. W. CAe]~L: Equation o/ motion/or the Heisenbcrg spin chain, two preprints of June and July 1981 (Leiden). (6) IV[. BOlTI, C. LADDOMADA and F. P]~MPIN]~LLI: NLS, potential I~LS equation and soliton solutions, poster presented a~ the International Con/erence on Mathematical Physics, Berlin, 1981, and lecture given at the Workshop on Nonlinear Evolution Equa- tions, Solutions and Spectral Methods, Trieste, 1981 (unpublished).

238 M. BOITI, C. LADDOI~IADA a n d F. PEMPINELLI

Finally, we indicate the way b y which f rom the pe rmutab i l i t y theorem for the P N L S equat ion one can get a recursive relat ion which generates N-k ink

solutions wi th or wi thout one pointed k ink in them.

2. - NLS and potential ~TLS equations.

For subsequent use it is convenient to briefly recast the equivalence be- tween the NLS and the P N L S equat ions (',~).

The NLS equat ion

(2.1) iq, -F q~ -b o~lql~q = o , o~ = 1 ,

describes the t ime evolution of a complex funct ion q(x, t) in one space var iab le

x and belongs to the class of the nonlinear evolution equations solvable b y the use of the spectral t r ans fo rm (7).

Wi th one sign (~ ~ ~- 1) of the nonlinear t e r m the equat ion is called at- t rac t ive NLS, while wi th the other sign (a ~ - - 1) the equat ion is called re- pulsive NLS.

I n eq. (2.1) the ampl i tude U(x, t) and the phase O(x, t) of q(x, t), which we

define as follows:

(2.2) q(x, t) = 2~ U(x, t) exp [iO(x, t)],

are coupled in a complicated way. However , they can be decoupled b y intro-

ducing a real po ten t ia l funct ion ~(x, t) such t h a t

(2.3) U 2 ~- 2F~.

I t is easy to ver i fy direct ly f rom eq. (2.1) t h a t the po ten t ia l funct ion

can be fixed, up to a constant , b y requir ing t h a t

(2.4) 0~ = - - .F~/2F~,

and, consequently,

16owD/4~%. (2.5) 0~ = ( 2 ~ ' y ~ - ~ - F ~, + ~

(7) V.E. ZAKHAROV and A. B. S~ABAT : Soy. Phys. JETP, 34, 62 (1972) ; 37, 823 (1973); 1Yr. J. A~T,OWlTZ, D. J. KAVP, A. C. NV.WELL and H. SEGVR: Stud. Appl. Math., 53, 249 (1974); P. P. KULISH, S. V. I~[ANAKOV and L. D. ~ADDE:EV: Theor. Math. Phys. (USSR), 28, 615 (1976); T. KAWATA and H. INOV~: J. Phys. Soc. Jpn., 44, 1722 (1979); V. S. GERDJIKOV and P. P. KULISH: Bulg. J. Phys., 5, 337 (1978) (in Russian); J. L]~o~: J. Math. Phys. (N. Y.), 21, 2572 (1980); F. CALOG]~RO and A. D~GASP~RIS: J. Math. Phys. (N. Y.), 22, 23 (1981); T. KAWATA, J. SAXAI and N. KOBAYASHI: J. Phys. Soc. Jpn., 48, 1371 (1980).

NONLIN]~AR SCHR~DING~R ]~QUATION ETC. 239

The integrabil i ty condition for eqs. (2.4) and (2.5) yields a nonlinear par t ia l differential equat ion for F

(2.6) ~ F ~ § ~ / ~ - - 2/~y~F~ ~- F~F .... - 2FY~Y~ +

which we call potent ia l NLS equat ion (PNLS). A solution ~ of the PNLS equation, if ~ > 0, supplies a solution q of the

:NLS equat ion via the integrat ion of the differential system of eqs. (2.4) and (2.5) for the phase 0. I f / ~ < 0, _~ -~ -- F is a solution of the P N LS equat ion wi th

changed into -- :r and furnishes a solution ~ of the NLS equat ion with ~ changed

into -- a. Therefore, we can say t ha t the PNLS and NLS equations are equivalent . The main advantage of the PNLS equat ion is t h a t its solutions 2', in con-

t ras t with the corresponding q's of the NLS equation~ can be cont inued ana- lyt ical ly in its parameters . I f a new solution _P obta ined by this procedure is real, i t can be used to generate a new solution ~ of the ~ L S equat ion with

or - - ~ as the case m a y be.

3. - The Bfieklund transformation of the potential NLS equation.

For any solution q(x, t) of the NLS equat ion a new solution ~(x, t) is ob- ta ined by the Bi~eklund t ransformat ion (BT) (8-~o)

(q - - ~t)~ : - - 2i~(q - - ~t) - - (q -~ ~ t )R , (3.1)

(3.2)

where

(3.3) R = ~ / 4 ~ - � 8 9 ~ ,

and ~, a are real free parameters . The BT can be schematically represented as follows:

(3.4) BT: q ~'~ > ~.

I t is convenient to write explicit ly the ampli tudes and the phases for the

(8) It. H. CH]~N: Relation between Bdeklund trans]ormations and inverse scattering problems, in Lecture Notes in Mathematics, No. 515, edited by R. M. NIIV~A (Berlin, 1976). (9) 1~. BoI~I and F. P~MPI~ELLI: Nuovo Cimento B, 59, 40 (1980). (lo) G. L. LAMB: J . Math. Phys . (N. Y.), 15, 2157 (1974).

240

c o m p l e x func t ions invo lved .

(3.5)

(3.6)

(3.7)

M. BOITI~ (3. LADDOMADA ~ I l d :F. P:EMPIIW:ELLI

W e set

q(x, t) : 2~ U(x , t) exp [iO(x, t ) ] ,

~(x, t) = 2~ ~;(x, t) exp [iO(x, t)],

q(x, t) - - ~(x, t) = 2t~V(x, t) exp [iO(x, t) - - ir t ) ] .

T h e n t h e rea l a n d i m a g i n a r y c o m p o n e n t s of t h e B T in eqs. (3.1) a n d (3.2)

can be e q u a t e d to o b t a i n

(3.8) R~ = - - ~ ( ~ - - 2 U cos co),

(3.9) Re ~ - - 2a~[(O~ - - 2~) U cos co -[- U~ sin co + R U sin ~o + 2 ~ ] ,

(3.10) }P(0 - - w)~ ---- - - 2 ~ T - - 2 R U sin to,

(3.11) }P(O - - ro)~ ~ 2R[(O~ - - 2~) U sin co - - U~ cos co] -[-

+ ~ ( 4 a 2 - - 4~ ~ + 2 a U ~) - - 8a ~ U cos co.

This se t of d i f ferent ia l equa t ions can be expl ic i t ly i n t e g r a t e d in t e r m s of

t h e p o t e n t i a l func t ions ~ a n d _P re la t ive to q and q, r e spec t ive ly . E q u a t i o n s (3.8) a n d (3.9) can be a l t e r n a t i v e l y w r i t t e n as

(3.12) R~ = ~ ( u s _ t ~ ) ,

(3.13) Re = 2 a ( ~ 6 ~ - - U20~) ,

a n d exp l i c i t ly i n t e g r a t e d to ge t

(3.14) R ~ %/4a ~ - ~ T ~ = 2 a ( 2 7 - / ~ ) ,

where t h e difference b e t w e e n the u n d e t e r m i n e d c o n s t a n t s up to wh ich ~v a n d _P are def ined has been f ixed in such a w a y as to cancel t h e e v e n t u a l add i t i ve

cons t an t . F r o m eqs. (3.8) a n d (3.14) one ge ts ( / v ~ 0)

(3.15) T c o s to :

B y c o m b i n i n g eq. (3.15) a n d i ts de r iva t ive wi th respec t to x w i th eq. (3.10)

i t fol lows t h a t (F, ~ 4 ~ . )

(3.16) ~ s i n co = 2 ~ t _ 4~xW ~ ( F - - i ~ ) ~ -t- 4~(-P -- / '~)(-P -[- /0)~ -

_ 1 [ ~ + 4 ~ ( F - - ~ V ) f . ] [ 2 ~ - .o~(2~ - _P), + (F - i~).] / . 2F~ �9

NONLINEAR SCHR(~DINGER ]EQUATION ETC. 241

By inserting eqs. (3.14)-(3.16) into eq. (3.9) we obtain (2' .~ 0, @~ 0)

(317) (_~ + _P)= = 2 a ~ - 2~(~-_~)~ + 2fo(~95 + 06) ~ 2 § e 2

and from the compatibility condition cos2 co ~ sin2 co ---~ 1

(3.18)

where

(3.19)

(3.20)

@*+6' ~ '+q"

Q ~ 2~ -- 4~V~,

and q5 and ~ are the corresponding expressions obtained from ~ and ~ by the exchange ~" ++ _P.

The BT ~ of q can then be expressed directly in terms of q, 2 and P in the following simple and very convenient form:

95§ (3.21) q -- q ~ § ~q"

By a direct but laborious computation one can verify that this ~ satisfies the differential equations which define the BT'q --> ~ with parameters ~ and a i f / 0 satisfies the system of differential equations (3.17) and (3.18) (*).

Therefore, we can say that the differential equations (3.17) and (3.18) define the BT of the PlgLS equation.

I t is worthwhile noting that the structure of this BT for the PIgLS equa- tion is very different from the already known BTs of the sine-Gordon, KdV, NLS and other integrable nonlinear evolution equations, because the integ- rability condition does not give the PlgLS for the transformed/~ but a com- plicated differential equation which couples E and /~. This result suggests that, in general, one must look for the BTs of the integrable nonlinear evolution equations in a more general class than one has so far.

4. - The theorem of permutability for the potential I~LS equation.

Let 2'0, zv~, ~2 and ~3 be four solutions of the PiNLS equation related by the BTs schematically represented in the Lamb diagram of fig. 1.

(*} The condition Q ~ 0 can be removed, but then ~ ~ 0.

16 - I I Nc~ovo C l m e n t o A .

242 M. BOITI~ C. LADDOMADA and F. ~]~MPINELLI

The permutabil i ty theorem says tha t for any 2'0, 91 and 2,3 there exists a 2,8 for which the diagram commutes. Moreover, 93 can be expressed alge- braically by means of 2,o, 2,~ and 2,2. ~

Fig. 1.

I n pape r I I we have a l ready p roved the re la ted pe rmutab i l i t y theorem for the NLS equat ion wi th a n y choice of the pa ramete r s ~ and a~ (i = 1, 2).

This result can be s t ra ight forwardly extended to the PiNLS equat ion b y means of the same equations obta ined in paper I I .

The double B~cklund t rans format ion for the P N L S equat ion looks simpler t h a n t h a t for the NLS equat ion and, precisely, i t results t h a t

(4.1) 1 8

By specializing eq. (3.21) to the two BTs which define, respectively, ql and q~, the modulus [ql-- q~]2 can be expressed in terms of ~o, 2,1, 2,~ and their derivatives with respect to x and t.

5. - S o l i t o n s o l u t i o n s .

In paper I I I we derived a N-kink soliton solution of the repulsive :NLS equation with ~V ~ - I free parameters.

The solution was obtained by a reeursive application of the BT of eqs. (3.8)- (3.11), w i t h a = O, to the single-kink soliton solution with two real parameters

ao, ~o

(5.1)

where

(5.2)

(5.3)

q = 2ao tgh [2~ao~] exp [iO],

V = X - - Xo -~ 4~o t ,

o = - 2~oX - 4 ( , ~ + ~ ) t - Oo,

NONLINEAR SCHR6DING]~I~ EQUATION ]~TC. 243

and where the real constants Xo, Oo fix the location of the travel l ing wave at the init ial t ime.

The success of the procedure suggests t h a t one can also compute the BT, with ~ ---- 0, of the special solution of the repulsive NZS equat ion

(5.4) q ---- 2a0 exp [iO],

with 0 defined as in eq. (5.3). However , this BT, which can be explicit ly solved, is always singular. The next step is to t r y and solve the BT, with any a, for the same solutions

(5.1) and (5.4). We shall see tha t , in order to get a more general soliton solution t h an in

paper I I I , the procedure of analyt ic cont inuat ion outl ined in sect. 2 mus t be used.

The BT of eqs. (3.8), (3.11) can be al ternat ively wri t ten in the following way:

(5.5)

(5.6)

where

(5.7)

y~ = _ (q?f~ + l ~ q , + k y ) ,

Yt = - - ik (qY 2 + laq* + k Y ) - - i (q~Y ~ - ~ * ~o:~o + o:lql ~ ~ ) ,

Y ---- 2 - ~ t - l ( 2 a - R) exp [ - i(O - co)],

(5.8) k - - - 2 ( a + i ~ ) .

Equat ions (5.5) and (5.6) are the par t ia l differential equations obta ined by ESTABROOK and WAltLQUIST for the pseudopotent ia l in thei r geometric approach to the s tudy of the NLS equat ion (3).

They arc easily linearizable (3) and, therefore, the pseudopotent ia l :Y is analyt ic in the complex free parameter k.

We are interested in the repulsive case ~ ~ -- 1 and in the BT of the two- parameter simple-kink soliton solution (5.1).

For subsequent use it is convenient to write also the potent ia l funct ion relat ive to q

(5.9)

where

~ - - 2 - taoy ~ a ~ ,

(5.10) y = tgh [2~ao~].

To find the general solution of the system of differential equations (5.5) and (5.6), we compute first the pseudopotent ia l Y for k - ~ - 2i~, t hen we obtain Y for any k by a procedure of analyt ic continuation.

2 4 4 M. BOITI~ C. LADDOMADA and F. PEMPINELL1

At k ~ - - 2 i ~ , i.e. a t ~ = 0, we get

(5 .11) y = - 2-~ exp [ - i ( O - ~o)],

where the funct ion too(x, t)~- to(x, t; a----O, ~) can be easily eva lua ted by solv-

ing explicit ly the sys tem of differential equations (3.10) and (3.11) (4).

I t results t h a t

~ c t g (�89 = 2 ~ o y -- 2~# tgh [2~6~ ~ 6], (5.12)

where

(5.13)

(5.14)

(5 .15)

~ = ~ o - ~,

2,~ ~ = 2a] - ~ ,

= x + 2(~o § ~ ) t ,

and ~ is an a rb i t r a ry constant . B y insert ing eq. (5.12) into eq. (5.11) and cont inuing analy t ica l ly in k, we

obta in s t ra ight forwardly the pseudopotent ia l :Y for any a and ~ and conse-

quent ly f rom eq. (5.7) the radical R. F r o m eq. (3.14)~ in the case 2a~-- [~ > 0, we get finally the poten t ia l func-

t i o n / ~ relat ive to ~, i.e. the BT of the k ink q,

(5.16) 2 ( P - - ~ ) = 2- ta[(~ ~ ~- a s ~- 2agy~)(cosh 2z ~- cos 2w) +

~- 2(a ~ -]- b~)(cosh 2z - - cos 2w) - - 4a(~oy sinh 2z ~- 4b(~oy sin 2w].

�9 [aaoy(cosh 2z ~- cos 2w) -[- (~b -- aa) sinh 2z + (~a ~- (~b) sin 2w] -1 ,

where, for convenience, a real cons tant b, restr ic ted by the condition ~ - - - - 2 ~ < 2b 2 < ~ , parametr izes a as follows:

(5.17)

and where

(5.18)

(5.19)

(5.20)

: - - 2t b[2al/(~ 2 - 2b 2) - - 1]�89

a = 2 - t ~[2a~/(~ 2 - - 2b 2) - - 1] ~ ,

z : 2t[a~ ~ 2bat ~- c],

w : 2t[b~ - - 2aat ~ d],

with c and d real a rb i t r a ry constants . One can ver i fy t h a t P does not blow up a t some x and t if and only if

c ~ c~. I n this case /P is the poten t ia l funct ion of a s imple-kink soliton so-

NONLINEAR 8CHR~DING]~R EQUATION ]~TC. ~

lut ion ~ with ampli tude and phase shifted with respect to the original kink soliton solution q.

Therefore, the only way to generate N-kinks by an i terat ive application of the BT to the simple k ink is to set a ~ 0 in all the BTs involved~ as in paper I I I .

However, we can generalize the results obta ined in paper I I I using the procedure of analyt ic cont inuat ion outl ined in sect. 2.

The potent ia l funct ion _P is analyt ic in b in a neighbourhood of the origin and consequently can be cont inued analytical ly to the point i b . We get, choosing convenient ly e and d with respect to which s is meromorphic, a new solution of the P H L S equation.

For the sake of simplicity, we use for this new solution and its parameters the same nota t ion as for the old one.

We obtain

(5.21) : a~/-]- 2-t[ao(a + ~)(a -~ b ) y -}- c t (a 2 - ~ ) tgh u --

-- 2a(a 2 -- b 2) tgh v - ~ Cro(a - - ( ) ( a - - b ) y tgh u tgh v].

�9 [(a -- ~)(a -]- b) -- 2gaoy tgh u ~- (a -]- ~)(a -- b) tgh u tgh v] -~ ,

where y, ~ and ~ are defined as in eqs. (5.10), (5.13) and (5.15), whereas t h e r e a l constant b, restr icted by the condition 2b ~ < 2a~ -- (~, parametr izes a as follows:

(5'.22)

and.

(5.23)

(5.2~)

(5.25)

(~ : - - 2 � 8 9 ~ 2 r- 2b 2) -- 1]�89

a ---- 2-�89 ~ + 2b 2) -- 17~,

u : 2�89 + b)[~/-- 2at + c],

v : 2�89 - - b)[~ -~ 2 ~ ~- d] ,

with c and d real a rb i t ra ry constants.

We can obtain another interest ing solution of the P ~ L S equat ion with the same procedure bu t s tar t ing from the solution (5.4) of the repulsive ~ILS equation.

In contras t with the case a --~ 0, one can achieve directly the final result pu t t ing y ~ 1 in eqs. (5.16) and (5.21).

In bo th cases, when y - - - - t g h [ 2 ~ o ~ ] and y ~ 1 , the _P defined by eq. (5.21) is a solution of the BT defined by eqs. (3.17) and (3.18) with a ~-- -- 1 and a2 changed into -- a S and with a defined as in eq. (5.22). I t results f rom eq. (3.18) t ha t P~ > 0 and, therefore, the funct ion _P is the potent ia l funct ion of a solution ~ of the repulsive NLS equation.

2 ~ 6 1~I. BOITI , C. LADDOMADA and F. P:EMPIN]~LLI

To get a regular solution the parameter b must be chosen such tha t

(5.26) as < ~s, a~ < 0 .

However , this is not an actual restriction, because a finite solution can be recovered in bo th the cases 1) a ~ < ~s, a ~ > 0 and 2) a s > ~s by cont inuing analyt ical ly in _P tgh u into ctgh u, tgh v into ctgh v in case :1) and y into l/y, tgh u into ctgh u in case 2).

Le t us first examine, in the case in which y --~ 1, the special solution which is obta ined by requiring t ha t a ~ b.

The ampl i tude ~ s ~ 2P~ reduces to

(5.27)

where

~s : 2a~ -- 8b s + 8b 2 tgh s [2tb(x -- vt + c')],

(5.28) v : -- 2(~0 ~- ~) -- 4b~/~,

and c' is a real a rb i t ra ry constant . The corresponding ~ can be easily computed by means of eq. (3.21). I t

results t h a t the phase 0 of ~ is the sum of a distortionless t e rm which travels at the same speed v of U and a t e rm which grows linearly with t ime.

The ampli tude U in eq. (5.27) has constant 2tao behaviour at infinity and describes a t ravel l ing wave which looks like a depression 2tb deep.

For 4b ~ -~ a~ the funct ion ~ reduces continuously to a two-parameter simple- k ink soliton solution with ampli tude U given by

(5.29) -~ 2ta0ltgh [2~qo~]] �9

The ampli tude U of eq. (5 .29)has a pointed depression of depth 2�89 to be compared with the smooth behaviour of the three-parameter ampli tude of eq. (5.27).

For the sake of brevi ty we will call the three-parameter solution of eq. (5.27) and the two-parameter solution of eq. (5.29) (( smooth ~) k ink and (( pointed )> kink, respectively.

When y----1, if no special choices of the four free parameters are made, the potent ia l funct ion P of eq. (5.21) furnishes a solution ~ of the repulsive

:NLS equation which describes the collision of two smooth kinks. In fact , the ampli tude ~2 = 2P~ at t - + • c~ breaks up into two smooth

kinks

(5.30) ~s __> 2a~ -- 2(a + b) 2 ~- 2(a + b) s tgh s (u ~- ~u • --

-- 2(a -- b) s -~ 2(a -- b) ~ tgh 2 (v -~ ~v~),

N0:NLII~EAR SCHI~6DINGER :EQUATION ETC. 247

The constants ~u • and 3v • indicate t h a t the two kinks are differently shifted a t t-----~ c~ and a t t = - 0+.

Figure 2 shows the complete t ime evolut ion of the collision,

Fig. 2.

The x-shift ~x~+ of the i - th k ink of dep th D~ and speed ~ due to its inter- action with the j - t h k ink of depth Dj and speed ~+ is given b y

(5.31) 3x~j = sgu [Dj(~j - - ~)] a r tgh 8D~Dj

When y ~ tgh2+a0~ the ampl i tude ~ 2 ~ 2P~ breaks up a t t - - - - • into two smooth kinks and one poin ted k ink

(5.32) /~2 _+ 2a~ tgh 2 [2tao~/+ 5 • -- 2(a -~- b) 2 ~- 2(a -~- b) 2 tgh 2 [u ~- 3u • - -

- - 2(a - - b) ~ + 2(a - - b) 2 tgh 2 Iv ~- 3v• t - + • c~.

B y consideration similar to those repor ted in paper I I I one can show t h a t

only paired collisions between kinks occur.

The soliton shifts due to m u t u a l interact ions are given also in this case b y eq. (5.31).

The ampl i tude U has the same a sympto t i c bchaviour wi th the same shifts of the t r ip le-kink soliton solution given in paper I I I .

We have, therefore, recovered the same solution.

248 M. BOITI, C. LADDOMADA and F. PEMPINELLI

B y us ing recurs ive ly t he modif ied fo rm (4.1) of t he p e r m u t a b i l i t y t heo rem,

o b t a i n e d b y a n a l y t i c a l c o n t i n u a t i o n , n- fo ld k i n k sol i ton solut ions, w i th or

w i t h o u t a p o i n t e d k i n k i n t h e m , can be ob ta ined .

The au tho r s are g ra te fu l to Dr. NI. LI~c~A~o for ass i s tance w i t h t he com-

p u t e r plots .

�9 R I A S S U N T O

Si dimostra the la forma reale equivalente dell 'equazione di SehrSdinger non lineare (NLS), eio~ l 'equaziono potenziale di SchrSdinger non lineare (PNLS), int rodot ta dagli stessi autori in un preeedente lavoro, ha una trasformazione di B~cklund (BT) che sod- disfa il teorema di permutabilit~. Usuffuendo delle propriet~ di analitieit~ dell'equa- zione PNLS, si formula un metodo per generare nuove soluzioni dell 'equazione NLS. In questo modo si ottiene, nel caso repulsivo, una soluzione, deserivente un solitone di tipo kink con 3 parametri , che ~ chiamata , smooth ~) kink rispetto alla soluzione deserivente un solitone di tipo kink con 2 parametri, che ~ chiamata (~ pointed ~) kink. Inoltre si d~ esplicitamente una soluzione a 4 parametri, ehe deserive la eoIlisione di 2 kink (( smooth ~).

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