Integrability of Nonlinear Systems

Nonlinear Waves, Solitons, and IST 23
derivative of (4.18), i.e. operating on (4.18) by
∂
∂k = 1 ( ∂
+ i ∂ )
, and
2 ∂k R ∂k I
using, as we sometimes do, the notation G(x, y, k) to denote G(x, y, k R ,k I ) etc.,
we find that
∂G
∂k (x, y, k) =sgn (−k R)
e i(p0x+iq0y)
2π
(4.22a)
G(x, y; −¯k) =G(x, y, k) exp(−i(p 0 x + q 0 y)) (4.22b)
p 0 = −2k R , q 0 =4k R k I . (4.22c)
We find that m satisfies the following ¯∂ equation,
∂m
∂k (x, y, k) =R(k R,k I )e ip0x+iq0y m(x, y, −¯k), (4.23)
where R(k R ,k I ) plays the role of the scattering data, and is related to the potential
via
R(k R ,k I )= sgn (−k ∫∫
R)
u(x, y)m(x, y, k)e −ip0x−iq0y dx dy. (4.24)
2π
Thus, given an appropriate potential u(x, y) vanishing sufficiently fast at infinity
(cf. [12]), the direct problem establishes relations (4.23)–(4.24). The inverse
problem is fixed by giving R(k R ,k I ) (there are no discrete state solutions known
which lead to real, nonsingular, decaying states for KPII) in order to determine
m(x, y, k) and then u(x, y). The inverse problem is developed by using the
generalized Cauchy integral formula (cf. [13]),
m(x, y, k) = 1 ∫ ∫
∂m ∧ d¯z
(x, y, z)dz
2πi R ∞
∂¯z z − k + 1 ∫
m(x, y, z)
dz,
2πi C ∞
z − k
(4.25)
where R ∞ is the entire complex plane, C ∞ is a circular contour at infinity,
z = z R + iz I , and dz ∧ d¯z =2idz R dz I .Ask →∞, we can establish that m ∼ 1,
hence the second term on the right-hand side of (4.26) is unity. Using (4.23) we
find that
m(x, y, k) =1+ 1 ∫∫
i(p0x+q0y) m(x, y, −¯z)
R(z R ,z I )e dz ∧ d¯z. (4.26)
2πi
z − k
Once m(x, y, k) is found, the potential is reconstructed from
u(x, y) =− ∂ ( ∫∫ 2i
∂x π
)
R(z R ,z I )e i(p0x+q0y) m(x, y, −¯z)dz R dz I . (4.27)
The latter formula is obtained by comparing the limit as k →∞in (4.26) and
(4.18).
Finally, the time-dependence of the scattering data is shown from (4.1b) to
be
∂R
∂t = −4i(k3 + ¯k 3 )R. (4.28)
Thus, the IST framework for KPII is complete; namely at t = 0, u(x, y, 0)
determines R(k R ,k I , 0); (4.28) gives R(k R ,k I ,t) and from (4.26)–(4.27) the eigenfunction
m(x, y, t, k) and solution u(x, y, t) are obtained.