Student Seminar: Classical and Quantum Integrable Systems

We thus obtain
u 2 x = k − 2eu + vu 2 − 2u 3 = −2(u − b 1 )(u − b 2 )(u − b 3 ) ,
where k is another integration constant. In the last equation we traded the integration
constants e, k for three parameters b 3 ≥ b 2 ≥ b 1 which satisfy the relation
Equation
v = 2(b 1 + b 2 + b 3 ) .
u 2 x = −2(u − b 1 )(u − b 2 )(u − b 3 ) ,
describes motion of a ”particle” with the coordinate u and the time x in the potential
V = 2(u − b 1 )(u − b 2 )(u − b 3 ). Since u 2 x ≥ 0 for b 2 ≤ u ≤ b 3 the particle oscillates
between the end points b 2 and b 3 with the period
l = 2
∫ b3
b 2
du
√
−2(u − b1 )(u − b 2 )(u − b 3 ) = 2 √ 2
K(m) ,
(b 3 − b 2 )
1/2
where m is an elliptic modulus 0 ≤ m = b 3−b 2
b 3 −b 1
≤ 1.
The equation
u 2 x = −2(u − b 1 )(u − b 2 )(u − b 3 ) ,
can be integrated in terms of Jacobi elliptic cosine function cn(x, m) to give
(√ )
u(x, t) = b 2 + (b 3 − b 2 ) cn 2 (b3 − b 1 )/2(x − vt − x 0 ), m ,
where x 0 is an initial phase. This solution is often called as cnoidal wave. When
m → 1, i.e. b 2 → b 1 the cnoidal wave turns into a solitary wave
u(x, t) = b 1 +
A
(√ ) .
cosh 2 A
(x − vt − x 2 0)
Here the velocity v = 2(b 1 + b 2 + b 3 ) = 2(2b 1 + b 3 ) = 2(3b 1 + b 3 − b 1 ) is connected to
the amplitude A = b 3 − b 1 by the relation
v = 6b 1 + 2A .
Here u(x, t) = b 1 is called a background flow because u(x, t) → b 1 as x → ±∞.
One can further note that the background flow can be eliminated by a passage to
a moving frame and using the invariance of the KdV equation w.r.t. the Galilean
transformation u → u + d, x → x − 6dt, where d is constant.
To sum up the cnoidal waves form a three-parameter family of the KdV solutions
while solitons are parametrized by two independent parameters (with an account of
the background flow).
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