Exact solutions to the equation describing “cylindrical solitons”

Volume 71A, number 5,6

PHYSICS LETTERS

28 May 1979

EXACT SOLUTIONS TO THE EQUATION DESCRIBING “CYLINDRICAL SOLITONS” Ryogo HIROTA Department ofApplied Mathematics, Faculty of Engineering, Hiroshima University, Hiroshima, Japan Received 4 April 1979

A transformation which reduces the equation describing “cylindrical solitons” to the KdV equation is found. The transformation provides us with an explicit solution describing N solitons interacting with each other and moving in a timedependent and nonuniform background.

We consider the following equation:

solution to eq. (1):

u~+6uu~ +u~)~ +(l/2t)u0, (1) where the subscripts indicate partial differentiation with respect to the indicated variables. Eq. (1) is known to describe “cylindrical solitons” [l~~ First we observe that eq. (1) is transformed into

u(x,t)=x/12t+2(a2/ax2)logf(x, where (N)

f=

e ~=~j

(9)

t),

N

A

2

+

.

.

(10

XP [~

>•

11j,

Li1

=1

1~’~ Th

the equation 2/(~~+ p 2, 1)

6t~t~ + + (l/2t)xu~+ (1/t)r~= 0, through the transformation

(2)

expA11 = (p~ ~~)

u=x/(12t)+~.

(3)

n~=p~(t)x+ [2p~(t)]3t+i~,9,

+

Then, we find that eq. (2) is transformed into the KdV equation v +6uv

=0

+I~

through the transformations 2 (t) v(~,r), i~(x,t) = p

(4) (5) (6)

/

t

r=

—

(7)

‘~‘

pl=0,l,L12=0,1,...,andpN=0,l. As a special case, we have an expression for a soliton moving in the background x/(1 2t): u(x, t)~x/(l2t)+2p2(t 2r~, (13) 0ft)sech with [2p(t

2]3t+p(t

Let t ~0/t)h/ t 0 + t

with p(t)=p(t

2

.

(8)

0/t)L’ It is well known that the KdV equation exhibits an N-soliton solution [2]. Accordingly we have an explicit

(12)

with p 2for i = 1,2 N, where n~and1(t) t = p~(to/t)l/ 0 are constants, and the summation ~=o, indicates the sum over all possible combinations of

=

p3 (t’)dt’,

(11)

(ItoI

2x + const. 0/t)hI ~ Itj),x ~xo +x (!x01 ~ IxI)~

then eq. (13)2 reduces to the+usual sech2(—4p3t px), soliton solution (14) u(x, t)-~2p when t 0 ~x0. Finally, we note that the following “modified” KdV equation with nonuniform terms, 393

Volume 71A, number 5,6

u~.+ 6umu~+

+

‘y(t)xu~+ (2/m)’y(t)u

PHYSICS LETTERS =

0 (15)

where mis a constant and y(t)is an arbitrary function of time, is transformed into the “modified” KdV equation v +6vmv +v~ =0, (16) 7

~

=

r=

28 May 1979

p(t)x,

(18)

f

(19)

3(t’)dt,

p

/ p(t)pexp(~

f

t

y(t’)dt’~

.

(20)

through the transformations ~ U~X,t)=p(2/m)(t)u(~,r),

(17)

References ~ H. Ono considered the same transformations for the case ‘y(t)

394

=

const. (private communication).

Ifl 121

S. Maxon and J. Viecelli, Phys. Fluids 17 (1974) 1614. R. Hirofa, Phys. Rev. Lett. 27 (1971) 1192.