IL NUOVO CIMENTO VOL. 112 B, N. 7 Luglio 1997
The explicit expressions for the phase shifts of N-soliton solution
of the KdV equation
ZHOU GUANGHUI, FANGMAOFAand DUANYIWU
CCAST (World Laboratory) - P.O. Box 8730, Beijing 10080, PRC
Department of Physics, Hunan Normal University - Changsha 410081, PRC Yun Mengya, Department of Applied Physics
National Defence University of Science and Technology - Changsha 410073, PRC (ricevuto il 16 Dicembre 1993; approvato il 4 Marzo 1997)
Summary. — The collisions among pure N solitons of the KdV equation are
described by the solitons’ phase shifts, which are dependent on the scattering data of the reflectionless potential 2N(N11) sech2x . In the present work the scattering
data in the case of arbitrary positive integer N are evaluated by direct scattering and the explicit expressions for the solitons’ phase shifts are obtained according to the Inverse Scattering Transformation (IST).
PACS 03.65.Ge – Solutions of wave equations: bound states. PACS 02.90 – Other topics in mathematical methods in physics.
1. – Introduction
It is well known that the solution of the KdV equation
Ut2 6 UUx1 Uxxx4 0
(1)
with the reflectionless initial condition 2N(N11) sech2
x (N 41, 2, 3, R) will split
into N solitons in the limit of NtNKQ. This fact shows that the N solitons are stable and resume both their form and velocity after collisions. These collisions can be analyzed by the phase shifts of the N solitons so that we may identify each soliton in the particle sense. However, the expressions for the phase shifts of N solitons are dependent on the reflectionless potential scattering data which are still undetermined with arbitrary positive integer N of the reflectionless potential parameter (i.e. the number of solitons) [ 1 , 2 ]. In this work we will first get the reflectionless potential scattering data in the N-dependence form by solving the associated direct scattering equation, then evaluate the explicit expressions for the phase shifts of the N solitons by means of the IST method, and finally give an analytical function of (x , t) for the 3-soliton interactions as an application of the explicit expressions for the phase shifts of the N solitons.
ZHOU GUANGHUI,FANG MAOFAandDUAN YIWU
948
2. – The reflectionless potential scattering data
Under the variable substitution [3] of y 4tanh x, the direct scattering equation with the reflectionless potential
2d 2C dx2 2 N(N 1 1 ) sech 2 xC 4lC (2)
can be transformed into the following: ( 1 2y2) d 2 C dy2 2 2 y dC dy 1
k
N(N 11)1 l 1 2y2l
C 40 , (3)where l and C are the eigenvalue and eigenfunction, respectively. Equation (3) is the well-known associated-Legendre equation. Therefore, the bound eigenvalues and the orthogonal normalized eigenfunctions will be as follows [3]:
ln4 2n2 (n 41, 2, R, N) , (4) Cn4
o
(N 2n)! n (N 1n)! sechnx N! 2n dN 1n dyN 1n(y 2 2 1 )N. (5)The N 1n order differentiation factor in eq. (5) can be written as dN 1n dyN 1n(y 2 2 1 ) 4 d N 1n dyN 1n[ (y 11) N (y 21) ]N 4 (6) 4
!
i 40 N 1n (N 1n)! (N 1n2i)! i [ (y 21)N](N 1n2i).It is obvious that only the terms of n GiGN will remain after the evaluation of the differentiation in eq. (6). Then we have
dN 1n dyN 1n(y 2 2 1 )N4
!
i 4n N (N 1n)! (N! )2 (N 1n2i)! i!(N2i)!(i2n)!(y 11) N 2i(y 21)i 2n. (7)The substitution of eq. (7) into eq. (5) with y 4tanh x will yield Cn(x) 4
k
(N 1n)! (N2n)! n N! 2N sech n x!
i 4n N ( tanh x 11)N 2i( tanh x 21)i 2n(N 1n2i)! (N2i)! (i2n) i! . (8)
As z KQ ( tanh xK1), the summation in eq. (8) remains one term of i4n only. So that both N- and n-dependent reflectionless potential scattering data will be
kn4kNlnN 4 n (n 41, 2, R, N) , (9) Cn4 lim x KQCne knx 4 1 n!
o
(N 1n)! n (N 2n)! . (10)THE EXPLICIT EXPRESSIONS FOR THE PHASE SHIFTS OFN-SOLITON SOLUTION ETC. 949
3. – The explicit expressions for the phase shifts
Following the steps of the IST method [ 1 , 2 ] with subsequent calculations, we have the asymptotic form of the N-soliton solution of the KdV equation with the reflectionless initial condition as follows:
U(x , t) ````Kt K6Q jnfixed 22
!
i 41 N k2 nsech (knjn1 d6n) , (11) where jn4 x 2 4 kn2t , (12) d1 n4 Lny
Cn k2 kn»
i 41 n 21 k n2 ki kn1 kiz
, (13) d2 n4 Lny
Cn k2 kn»
i 4n11 N k i2 kn ki1 knz
. (14)The n-dependence constants d1n and d2n were defined as the phase shifts [ 1 , 2 ] of
the N solitons after their collisions. After the substitution of the reflectionless potential scattering data, eqs. (9), (10), into eq. (13) and eq. (14) with some algebraical calculations, we can finally obtain the explicit expressions for the phase shifts of the N solitons of the KdV equation as follows:
d1 n4 Ln
y
(n 21)! ( 2 n 21)!o
(N 1n)! 2(N 2n)!z
, (15) d2 n4 Lny
( 2 n) ! (n) !o
(N 2n)! 2(N 1n)!z
, (16)with the special case of
d1 N4 Ln
y
(N 21)! ( 2 N 21)!o
( 2 N) ! 2z
G 0 , (17) d2 N4 Lny
1 N!o
( 2 N) ! 2z
F 0 . (18)The sign of equality corresponds to the special case of the soliton number N 41. It means that there exists no collisions for only one soliton, so the phase shift of one soliton is zero.
From eqs. (15), (16), the relative phase shifts of the N solitons can be calculated as
dn4 d1n2 d2n4 Ln
y
(N 1n)! (N2n)! n! ( 2 n) ! ( 2 n 21)! (N2n)!
z
, (19)with which the conservation of the total phase shifts can be verified directly as
!
n 41 N dn4 Lny
»
n 41 N (N 1n)! (N21)! n! ( 2 n) ! ( 2 n 21)! (N2n)!z
4 0 ,ZHOU GUANGHUI,FANG MAOFAandDUAN YIWU 950 or, equivalently,
!
n 41 N d1 n4!
n 41 N d2 n . (21)It is worthy to note that eq. (21) is true only in the asymptotic case. However, since the solution of the KdV equation splits into solitons rapidly as NtNKQ, the conservation of the total phase shifts may be a useful notion in the study of the KdV equation.
4. – The analytical formula of 3-soliton interactions
For N 43, as an application of both our N- and n-dependent expressions of the phase shifts, eqs. (15), (16) and the scattering data, eqs. (9), (10), using the IST method with some tedious calculations we can get the analytical apparent function of (x , t) form for 3-soliton interactions of the KdV equation as
U(x , t) 4224[ cosh (10x2280t)110 cosh (8x2216t)115 cosh (6x272t)1
(22)
130 cosh ( 6 x 2 216 t) 1 80 cosh ( 4 x 2 64 t) 1 50 cosh ( 2 x 2 8 t) 1 1135 cosh ( 2 x 2 56 t) 1 25 cosh ( 2 x 2 152 t) 1 40 cosh ( 2 x 2 208 t) 1
1126 ] /[ cosh ( 6 x 2 144 ) 1 6 cosh ( 4 x 2 136 t) 1 15 cosh ( 2 x 2 80 t) 1 10 cosh 72 t]2, which reduce exactly to 212 sech2x at t 40.
The corresponding asymptotic form of eq. (22) is
U(x , t) ````Kt K6Q jnfixed 22 sech2
g
j16 1 2 ln 6h
2 8 sech 2g
2 j 26 1 2 ln 5 3h
2 (23) 218 sech2g
3 j3Z 1 2 ln 10h
, where jn4 x 2 4 n2t (n 41, 2, 3) . (24)Obviously, the analytical formula of 3-soliton interactions is much longer than that of the well-known 2-soliton interactions in ref. [ 1 , 2 ]. From this we see the complexity of nonlinear interactions and the difference between nonlinear and linear interactions.
R E F E R E N C E S
[1] WADATIM. and TODAM., J. Phys. Soc. Jpn., 32 (1972) 1403.
[2] GRADNERG. S., GREENEJ. E., KRASKALM. D. and MIURAR. M., Commun. Pure Appl. Math.,
27 (1974) 98.
