I L N U O V O C I M E N T O VoL. 6 5 B , N. 1 11 S e t t e m b r e 1981

Multiple-Kink-Soliton Solutions of the Nonlinear SchrSdinger Equation.

3i . ]-{O[TI~ (]. LADDOMAI)A t/!l([ F . PI~MPINI.II.IA

l s l i t u t o d i Fi,~i~'a dell'UJ~i~'ec.~ih'~ - Le , ' ( e . I h d i a

[s t i le t to N ( l z i a ~ o l e d i Fi.~i,'~t. Y t t d e a r ~ ' - I h d i a

r i eevn to il "2 Matzo 1981)

S u m m a r y . - - In th is pape r w(, ~',,iMd(,r lhc mmlin( ,ar Schr6d inger (NLS) e q u a t i o n in the r( ,pulMv, ('a-(, ~l~d wi t lmur any r eno rmMiza t i on line:~r t e r m in it. The Bh( ,khmd tra~l~!.,rn,a,,iml (BT) of :~ s in~h, -k ink sol i ton is i n t e r r a t e d (,x!~]i(,il] 5- yi-hlill~" tllt, doubh, kinl~-~()lilon sohl t ion of the NLS equa t ion . By a re(.urMv,, appl i ( 'a t iml of the n o n l i n e a r superpoMii(m i)rin('iph, l(~r t he N l,S equa t ion w(. o b t a i n the mult iph~-kink-sol i t (m solu- bion. The sol i ton ,-hift~ duv 1o m u t u a l inl(q'a,(.ti()ns art' cah.ulat( 'd ~md iC is ~how11 t h a t old 5" pair(,d ,',lii-i(,H~ occur.

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

I n t w o p r e v i o u s 1)~pers . h e r e a f t e r ; ' e f e r r e d t o a,s I a n d I I (1,2), ~\-e s t u d i e d

t h e exa, e t r e d u c e d f o r m ~ ~f l h e m ) n l i ~ e a r ~ e h r 6 d i n ~ ' v r ( N L S ) e q u a t i o n

(1 .1 ) iqt-!-q .... zlqi ,_q=O, ~2= 1,

~md i t s B i ~ e k l u n d t r a n s f o r m : ~ t i o n ( B T ) .

I n par t icu l~w~ we ~ '~ve a c o m p l e t e i n v e s t i~z'a!ion of t h e t r a v e l l i n g - w a v e so-

l u t i o n s a, n d t h e p e r m u t ~ b i l i i y t h e o r e m f o r l h e B T s of l h e N L S e q m ~ t i o n w h e n

:~ - - - - 1 a n d t h e p a r a m e t e r s ,d t h e B T s i n v o l v e d a r c r ea l .

(1) M. Botr a n d F. PESH'IXELLI: N u o r o (%~enlo B, 59, 40 (1980). (2) 5[. J3OITI, C. LADI)OMADA a n d F. PEMPINJALLI: X'u, ot'o C i m e n t o B , 62, 315 (1981).

248

~ U L T I P L E - K I N K - S O L I T O N SOLUTIONS OF THE ~ O N L I N E A R SCHR()DINGER EQUATION 2 4 ~

I n the present pape r we s tudy the mult iple-sol i ton solution in the case

HI~OTA (a) s tudied the mult iple-sol i ton solution of the ~ L S equa t ion in the a - ~ ~ - 1 case using a direct method , and ZAKtIAROV and SHA]3A~ (4.5) s tudied the spectral t r ans fo rm (ST) for bo th cases ~ ~ =L 1.

Solitons, in the ~ - - 1 case, have k ink shape and~ therefore, the inverse-

scat ter ing p rob lem for the m a t r i x l inear Schr6dinger equa t ion associated to the ~ L S equat ion m u s t be solved under the condit ion ]q(x, t )p--~4a 2, as

x--~ =Lc~, wi th a an a rb i t r a ry real constant . This r equ i rement makes the theory of the ST much more compl ica ted in compar ison wi th the ~ ~ ~ - 1 case, in which the solitons have bell shape and, therefore, ~(x, t ) - -~0 , as X ---~ ~- C ~ .

Moreover, in order to app ly correct ly the ST me thod as proposed b y ZAKI~AROV and S~ABAT, the NLS equat ion mus t be renormal ized b y adding a l inear t e r m (6.~o). Precisely, eq. (1.1) takes the fo rm

(1.2) + ( Iq l - q = 0 .

Because we are in teres ted in the soliton solutions of the nonrenormalized NLS equat ion, we t r y an a l te rna t ive approach to the ST me thod using the BT of the :NLS equat ion.

The ana ly t ic and algebraic s t ruc ture of the BT, owing to the presence of a radical , is ve ry complicated, but , in the g ~ - 1 case, when the free para- me te r in the BT is chosen to be real, the BT becomes much more simple.

Fu r the rmore , we have a t our disposal the pe rmutab i l i t y theorem and the nonl inear superposi t ion formula repor ted in pape r I I , which allow us to gen- era te the mul t ip le-kink-sol i ton solution b y a recursive appl ica t ion of the BT.

I n sect. 2 we app ly the BT to a single-kink-soli ton solution and discover the unique choice of the in tegra t ion constants for which the BT yields a regular solution. This solution, which we give explici t ly, is the double-kink-sol i ton solution. The two kinks in it have independent velocities and ampl i tudes re la ted b y the requi rements t h a t they mee t cont inuously the a sympto t i c condit ion l~ (x , t) l ~ - ~ 4 ~ , a s x - ~ : k ~ .

(3) R. HIROTA: J. Math. Phys. (N. /z.), 14, 805 (1973). (4) V . E . ZAKm~ROV and A. B. S~ABAT: Sov. Phys. JETP, 34, 62 (1972). (5) V. E. ZAKKAI~OV and A. B. SHABAT: Sov. Phys. JETP, 37, 823 (1973) (s) P . P . KULISH, S. V. MANAKOV and L. D. FADDEEV: Theor. Math. Phys. (USSR), 28, 615 (1976). (7) T. ]~AWATA and H. INONE: J. Phys. Soc. Jpn., 44, 1722 (1979). (s) V. S. GE~DJIKOV and P. P. KULISH: Buly. J. Phys., 5, 337 (1978) (in Russian). (9) J. LEON: J. Math. Phys. (_Y. Y~.), 21, 2572 (1980). (lo) F. CALOGERO and A. Dv.~ASPE~IS: J. Math. Phys. (N. Y.), 22, 23 (1981).

2 5 0 3[. P, OITI , C. LADDO3[ADA ,?tlld F. PE3IPINELLI

I~t sect . 3 we i t e r a t e t i le BT u s i n g l h e pe rnmta , b i l i t y t h e o r e m p r o v e d ill

p a p e r I I . I n th i s w a y we gel: a r ecu r s ive rel-~tion for t h e X-~old k i n k so l i t on

s o l u l i o n a n d i t is ,shown t h a t o n l y p a i r e d col l i s ions occur . The sh i f t s of t h e

c e n t r e of t he two colliding" ,~olitons are g iven e x p l i c i t l y .

2. - D o u b l e - k i n k - s o l i t o n so lu t ion .

The B T of t he ~NLS e q u a t i o n can be r e p r e s e n l e d s ehema t i ( . a l l y as fo l lows :

('2.1) BY: q :-~ ~,

w h e r e ~ is t i le (~ o ld ~> so lu t i on of t he .NLS e q u a t i o n a n d ?/ t i le ((new ~ s o l u t i o n

o b t a i n e d b y so lv ing the s y s t e m of D~vtial dift'erentia,1 e q u a t i o n s which def ine

t i le BT, ~ = ~ + ia is t he c o m p l e x p a r a m e t e r of the BT.

The g e n e r a l f o r m of t he B T is r e p o r t e d in p a p e r I . W e :ire he re i a t e r e s t e d

ht t i le ~ = - - 1 case when the ~ - p a r a m e t e r is chosen to be rea l .

I t is c o n v e n i e n t to w r i t e e x p l M t l y t he a m p l i t u d e s a n d the phase s for t h e

f u n c t i o n s i n v o l v e d :

(2.2) q(x, t) = 2+U(x, t) e x p [iO(x, t)],

(2.3) ~(x, t) = 2~ ('(.c, t) exp [ir t ) ] ,

(2A) q(x, t) - - c~(x, t) = 2~7%,', t) e x p [i0(.c. t) - - ioJ(a', t ) ] .

T h e n t h e BT, for :~ = - - 1 a n d ~ - ~ , r eads (see p a p e r I I )

(2.5) ~ ' t = 2 1 2 ~ ( T - - U coso~) + U~ sino~ -L 0~ U cos o~]g* q- 2U~P'z s inoo,

(2.6) ~v~ = ( g / - - 2 U cos o ) ) T ,

(2.7) ( O - - ~ ) t = 2 ( O ~ - - 2 ~ x ) U s i n o ~ - - 2 U , . c o s o ~ - - 4 ~ - - 2 U - ' ,

(2.8) (0 - - o~)~ = - - 2 ~ - - 2 g sin o).

E q u a t i o n s (2.5), (2.6) a n d (2.8) can be c o m b i n e d to f o u n d a c o n s e r v a t i o n l aw

(2.9) ~r/t =- (4~17 s -~ 2 I+lP ̀ sinej)~,

w h i c h wi l l b e u se fu l in t h e fo l lowing .

L e t now q be a s i n g l e - k i n k - s o l i t o n s o l u t i o n w i t h e n v e l o p e

(2.10) U(x, t) = 2~a t g h [2~arT],

M U L T I P L E - K I N K - S O L I T O N SOLUTIONS OF THE N O N L I N E A R SCHR(~DINGER ~ Q U A T I O N 251

where

(2.11) ~ = x -- x~ + 4~t,

and with carrier

(2.12) O(x, t) -~ - - 2~x - - 4(a ~ ~- ~2)t + 0o.

The speed v of the envelope and the speed v of the carrier are parametrized as suggested in paper I I :

(2.13)

(2.14)

with a and ~ real parameters.

,J = - - 2 o ' + ~

The arbi t rary real constants xo and Oo fix the location of the travelling wave at the initial time.

I f these U and 0 are inserted in the BT~ the two equations (2.7) and (2.8) can be explicitly integrated to

(2.15)

where

(2.16)

(2.17)

~1 = x - - Xol + 2(~ + ~1) t ,

2 a ~ = 2 a ~ - (~--~1) ~

and e is an arbi t rary constant. The Riccati equations (2.5) and (2.6) yield~ respectively~ the t-dependence

of ~ a t fixed x

(2.18) ~(x, t) = ~(x, t.) exp [-- A ( x , t)]

1 - - ~(x, to);(4~l -~ 2 U sin m) exp [-- A ( x , t ')]dt ' t*

where

(2.19) t

A ( x , t) -~ 2f[2(~ A- ~1) U coseo - - (2a ~ - U s) sin e)] dr ' , ta

exp [- -B(z , t)]

(2.20) ~(x, t) --~ ~(xo, $)1 - - kr~(xo, $);exp [ - - B ( x ' , $)] dx' ' xa

and the x-dependence of ~ at fixed t

252 3~. BOITI, C. LADDOMADA and F. PEMPINELLI

where

(2.21) B(,r, t) = 2 f u cos dx'. x0

By direct i n spec t ion of eqs. (2.18) a n d (2.20) we can p rove t ha t , for al real,

i] the cow, slant c is finite, the a m p l i t u d e ~u blows up. This s l a t e m e n t is eas i ly

p roved ab absurdo b y suppos ing t h a t the a m p l i t u d e T is f ini te in a l l the (x, t)-

p lane , i n c l u d i n g the po in t at in f in i ty . T h e n one can see, j u s t looking a t eqs. (2.18)

a n d (2.20), t h a t 7 ' has a lways the same sign, because o therwise it wou ld have

a zero a n d wou ld be i d e n t i c a l l y zero.

By t a k i n g in to accoun t t h a t Ucose ) goes to a c o n s t a n t as t ~-* ~cx~, the

a s y m p t o t i c behav iou r s of 7, a t t = • can be easi ly e v a l u a t e d f rom eq. (2.20). Because U coso) w i th c f ini te has oppos i te l imi t s a t l = - - c ~ a n d a t

t = q- 0% it resu l t s t h a t the d e n o m i n a t o r in eq. (2.20) vanishes , i.e. T (x , t) diverges, a t some f ini te .r, as t --* T c~ or as t -+ - - cxD accord ing to the s ign of ~ .

Moreover , if one takes in to accoun t t h a t , when c is f inite, U cos o~ has op-

posi te l imi t s a t ;r - - - c x ~ a n d a t x = q- oo a n d Us ino9 has the same l imi t

a t x = - - oo a n d at x = + oo a n d if one eva lua t e s the a s y m p t o t i c b e h a v i o u r

in x of ~ f rom eq. (2.18), one can show b y the saute p rocedure t h a t the am-

p l i t u d e 7 j b lows up a t some f ini te t as x --* + oo or as x ~ - - oo.

By s imi lar m e t h o d s one can show t h a t the a m p l i t u d e ~ is u n b o u n d e d

also for (h i m a g i n a r y a n d a n y c.

Because we are i n t e r e s t ed o n l y in b o u n d e d so lu t ions of the N LS e qua t i on ,

we choose a~ real , c = c~ a n d set

(2.22) (~ - - i t ) c t g �89 ~ 2 ~ a y - - 2~0"1 ,

where

(2.23) y - - tgh [2t a~] .

I n order to give exp l i c i t l y ~P(x, t), i n s t e ad of c o m p u t i n g the in tegra l s i n

eqs. (2.18) a n d (2.20), i t is more c o n v e n i e n t to solve b y e l e m e n t a r y m e t h o d s

the n o n h o m o g e n e o u s l inear p a r t i a l d i f ferent ia l e q u a t i o n (2.9), b y 11sing as

i n d e p e n d e n t var iab les ~ a n d ~1.

I t r e su l t s t h a t

where G is a n a r b i t r a r y func t i on , t h a t m u s t be d e t e r m i n e d b y i n s e r t i n g the

o b t a i n e d ~P of th is e q u a t i o n i n to eq. (2.6).

MULTIFLE-KINK-SOLITON SOLUTIONS OF TH~ NONLII~]~AR SCHR(}DINGER EQU&TION 253

l~inally we get

(2.25) ~ r 2�89 1 -r y 2 + 2((6/(~)y

(~/~r)y + 1 + d exp [2J(~l~TX ] '

where d is an a rb i t ra ry real constant t ha t mus t be not less t han zero in order t ha t T might be finite.

As t - ~ • the solution ~ breaks up into two individual solitons:

(2.26)

(2.27)

where

(2.2s)

and

Iq(x, t)]~ ~ 4q, tgh' [2ta v + 6-] +

-4- 4a~ tgh ~ [2�89 + 5~-] - - 4a~,

]~(x, t)] ~ --~ 4a 2 tgh 2 [2ta~ + (~+] +

+ 4~2 tgh 2 [ 2 � 8 9 + (~+] - - 4a~,

~ = 0 [T a~(~ - - ~1)] ar tgh (~1/~)

(2.29) 0T : ~ log 1 T sgn (~--#l)(~l/la[)

t - - -~- - oo~

t - - ~ + o o ,

with O(x) the Heaviside step funct ion. We have, therefore, succeeded in obtaining an explici t solution ~ of the

I~LS equat ion which describes two solitons of k ink shape which are separated in the remote past, successively, because of their different speeds, in teract and finally separate again in the remote fu tu re reversing their order.

The only memory of the interact ion is a shift of thei r centres.

L

x

Fig. 1. - Collision of two kinks~

2 5 4 3 I . B O I T I , C . L A D D O M A D A a n d 1". P E M F I N E L L I

Figure 1 shows the t ime evolution of the ampli tude of the double-kink-

soliton solution ~nd fig. 2 the real par t of this solution, before, during ~nd ~fter collision.

I t is worthwhile not ing tha t the << overtaking >> of the two soli~ons appears

as an exchange of identi ty with the larger soliton becoming smMler and vice v c r 8 ( t .

1_t 11.

[t-II-IItIIltIf i .ll)tt_ltllt.ltt , " ,v!i.l[lll.v v: ll_ll.l l_ll_t

t~tJlf~lAli], At!ll[h-: ,,d~ hj~ tl J ' : ; , i : ~ . l ' �9 " . , " : i . , z' ~ ' ~: ;!. : , Jl i' ,? : ! : ' ~ , i : ' ,~: .! ' :

. i , . . . . . . . , , . . . , . I . . . . . ,~, I I , , " " v , , "1 q ' " , I

. i : ' ! ! , l , ' . . ' ' ' , ; , . . ' . . , ",;; i. ,!;':.i: :~..~' '~'. :..':,'!. !: : . " .i~_~ml]J].~ ",~l.~]tw ,~ ~,~-~]_l.~_l_li_d_[

/ili/iiii i l!ilil-l,l-l!-Itlili, ,iii/l;il-l x + ( 3 ~ + ~ ) t

Fig. 2. - Successive f rame~ showing ~ w~ve trMn modulated by two colliding kinks.

3 . - M u l t i p l e - k i n k - s o l i t o n s o l u t i o n .

In paper I I we proved the permutabi l i ty theorem for the BTs of the ~TLS

equation with c~ = - ] , when the parameters of the BTs involved are real. A remarkable consequence of this theorem is a nonlinear superposition prin- ciple, which states tha t the last solution in the hierarchy of solutions obtained

by an i terated application of a BT can be reached MgebricMly if the preceding solutions in the hierarchy are known.

MULTIPI,E-KINK-SOLITON SOLUTIONS OF THE NONLINEAR SCHRODING]gR ]~QUATIO/~ 255

The commuta t iv i ty p rope r ty of the BTs can be schematical ly represented

as in fig. 3.

/ qo\ /

q2+

Fig. 3.

The nonlinear superposit ion formula corresponding to this scheme is re- por ted in paper I I . Fo r subsequent use it is more convenient to rewri te i t in the following way:

(3.1) q3= ql + q,--qo-~-

+ . i (~ - - ~,)[~o~(qo-- q~) - - ~o,(qo-- ql)] - - (2qo-- q t - - q,)~o~ ~o~ sin ' ((o~o~-- (Oo,)/2) ( ~ - - 2 , ) ' - - ~o, ~o~ s in ' ((o~ol-- me,)/2)

where the ampli tudes ~ and the phases eo are defined as in eq. (2.4) with indices

corresponding to the q~'s involved in the BTs. I f qo, ~ and q, are differentiable and bounded solutions of the I~LS equat ion

at a ny x and t, including the points a t infinity, also q3 is differentiable and bounded.

In fact , the denominator of eq. (3.1) can even tua l ly vanish only when ~ox and ~o~ have the same sign. B u t in this case the funct ion

(3.2) q} ~ ~Fot ~02 sin g o91-- ~o, 2

in the denominator reaches a max imum at some x and t, which satisfy the necessary condit ion ~ - - - - 0 . Using the differential equat ions (2.5)-(2.8) for the BTs involved, this condizion can be wr i t ten as

t o o l - - ~o2 __ 0 (3.3) ~ol-}- ~o2-~ 2 ( ~ , - - ~ ) ctg 2

and the m a x imum of �9 is given, a t some x and t, b y the following expression:

(3.4) era,= = 4(~--~)~§ (~o~+ ~'o~)" '

which is always less t han ( ~ - - ~ ) 2 . Therefore, the denominator never vanishes.

256 3I. BOITI, C. LAI)DO3iADA and F. PE3IPINELLI

Moreover , we note tha t , in the region of the (x, t ) -plane in which 7*o, (7to2)

goes to zero, q3 -* q~ (q3 --* ql) or equ iva l en t l y 7t23 --* 0 (Tz~ --* 0).

I n the p e r m u t a b i l i t y t h e o r e m as r ep resen ted in fig. 3, let now qo be a single-

k ink-so l i ton solut ion and ql, q~ the two double-k ink-so l i ton solut ions ob t a ined

f r o m q0 and let us choose, for the sake of definiteness, ~ 1 < 0, (r.2 > 0 a n d

> ~ > $2- As t ~ - - o% the three in te rva ls on the x-axis, I~I < M, I~] < M and ]~]] < M wi th M a rb i t r a r i l y large are dis joint a nd o rdered f r o m left to r ight .

I n the first and second in te rva l ~ 2 -* q- oo and, therefore , 7t0 ., -* 0 and con-

s e q u e n t l y q 3 - * q t , while in the t h i rd in te rva l ~1~ , -* + oo and , therefore~ ~ 0 1 - + 0 and consequen t ly q3--* q,.

The a s y m p t o t i c behav iou r at t = - - oo of the enve lope displays th ree k i n k sol i tons :

(3.5) U~ - - 2a- tgh 2 [2~ ~1 + 5-] + 2~[ t~h ~ [2= ~ , + (57] _ 2~2, +

q- 2 ~ tgh 2 [2~2~0 q- 67] - - 2a~,

where the phases c5 are those r epo r t ed in eqs. (2.28), (2.29).

F igure 4 shows the t ime evo lu t ion of the a m p l i t u d e of tile t r ip le -k ink- sol i ton solut ion.

t Uj

Fig. 4. - Collision of three kinks.

Before c o m p u t i n g expl ic i t ly the shifts of soli tons due to collision, let us

show t h a t the reeurs ive use of the non l inea r superpos i t ion fo rmu la genera tes a solut ion which conta ins n solitons.

Because the van i sh ing of ~ff in eq. (3.1) implies the van i sh ing of 7 ~ in the successive non l inea r superpos i t ion fo rm u la ob ta ined b y a recurs ive use of the

MULTIPLE-KINK-SOLITON SOLUTIONS OF THE NONLINEAR SCHRODINGER EQUATION ~ 7

pe rmu tab i l i t y theorem, the previous procedure can be easi ly i t e ra ted in the

case a t<O, a~>O (i ~ 2, ..., n - - l ) and ~ > ~ i > ~ (i = 2, ..., n - -1 ) . The a s y m p t o t i c behav iour a t t : - - oo of the envelope of the n-fold k i n k

soli ton solution is a t r iv ia l general izat ion of eq. (3.5). When the signs of the ampl i tudes a are different f rom those considered

above, i t is convenient to use the arbi t rar iness of the ini t ial phases in the 7 var iables in order t h a t dur ing the t ime evolut ion f r o m t ~-- - - co to t = + co

never more t h a n two in tervals ]7;] < M (i -~ 0, 1, ..., n - - 1 ; 7o = 7) super- impose. This choice does no t change the ana ly t i c fo rm of the a s y m p t o t i c be-

hav iour and, therefore, the phys ica l con ten t of the solution. There is a lways a t ime t for which a n y in te rva l [~/i[ < M is disjoint wi th

a n y o ther in te rva l and has in it ajT~ -~ + co for some j . Thus in the snperposi t ion formula }/1 vanishes a n d the solution reduces

a t t ime t in the in te rva l ]7~] < M to a single soliton. We conclude t h a t the solution obta ined b y a reeursive use of the permu-

t ab i l i ty t heo rem contains n k ink solitons. Each soli ton dur ing i ts t ime evolut ion f rom t = - - cx~ to t ---- + oo collides

once wi th each other soliton. The only m e m o r y of the in terac t ion is a shif t

of i ts centre. The phase shif t ~o~ of the soliton of ampl i tude a due to i ts in terac t ion wi th

a soliton of amp l i t ude a~ can be deduced f r o m eqs. (2.28) and (2.29) of the previous section. I t resul ts t h a t

(3.6) ~o, - - sgn [a,($- $,)] a r tgh (a,/a).

The in terac t ion induces also a shif t ~o of the centre of the soli ton of am- p l i tude ~ , which is g iven b y

(3.7) ~,o ---- sgn [a(~, - - ~)] a r tgh (~d~).

I n order to compu te the phase shift (~, of the solitons of ampl i tudes a~ colliding wi th the soliton of ampl i tude a~, i t is necessary to eva lua te expl ici t ly the a s y m p t o t i c behavioUr a t t = - co and a t t = + r of the t r ip le-kink- soli ton solution.

We obta in

(3.8) ~,j = sgn [ a j (~ , - -~ ) ] a r tgh ( ~ , _ ~)~ ~_ 2a~ + 2a]"

The same remarks , made previous ly in order to show t h a t the i t e ra ted appl ica t ion of the nonl inear superposi t ion principle gives a solution which contains n solitons, can be used also to p rove t h a t the to t a l shift of each soliton, regardless of the detai led pic tures of the interact ions, is equal to the sum of the shifts suffered b y it in individual collisions.

17 - I l Nuovo G~zr~nto B.

2 ~ 3i. BOITI, C. LADDOMADA and F. PEMPINELLI

I n t h i s w,~y, u s i n g fo rmu la t e (3.6)-(3.8) , o n e c~n e a s i l y c o m p u t e t h e ex -

p l i c i t a s y m p t o t i c b e h a v i o u r as t --+ ___c~ of t h e e n v e l o p e of t h e n - f o l d k i n k

s o l i t o n s o l u t i o n .

T h e ~ u t h o r s ~re g r ~ t e f u l to Mrs . A. I~Z)IILIO fo r t r a n s l a t i n g t h e p a p e r of

G e r d j i k o v ~ n d K u l i s h .

�9 R I A S S U N T O

In questo lavoro si csamin,~ l ' equaz ione di SchrSdinger non l ineare (NLS) nel caso repul- sivo e senza l ' ,~ggiunta di alcun te rminc l ineare di r inormal izzazione. L a t r a s fo rma ta di Biicklund (BT) di una soluzione ,~ 1 soli tone di t ipo k ink ~ in tegra ta esp l ic i tamente : in questo modo si o t t ienc la soluzione a due soli toni di t ipo k ink per l ' equaz ione NLS. Usando Fappl icazione r ipe tu t a del pr incipio di sovrapposizione non l ineare per l ' equa- zione NLS, si o t t iene la sotuzione a ~ soti toni di ~ipo kink. Sono poi cttleolati gli sfasa- ment i dei solitoni, causat i dal la loro in terazione reciproca, e si fa vedere che le colli- sioni avvengono solo a coppie.

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