Integrability of Nonlinear Systems

Nonlinear Waves, Solitons, and IST 19
where δ is a constant and Tu is the singular integral operator,
(Tu)(x) = 1 ∫ ∞ ( π
)
2δ − cosh
2δ (y − x) u(y)dy, (3.49)
−∞
∫ ∞
and − denotes the principal value integral. (3.48) is referred to as the Intermediate
Long Wave (ILW) equation. Indeed, it has two well known limits:
−∞
a) δ → 0 ILW reduces to the KdV equation,
u t +2uu x + δ 3 u xxx =0, (3.50)
b) δ →∞ ILW reduces to the Benjamin-Ono (BO) equation.
u t +2uu x +Hu xx =0, (3.51)
where Hu is the Hilbert transform,
(Hu)(x) = 1 ∫ ∞
π − u(y)
dy. (3.52)
−∞ y − x
The BO equation was derived in the context of long internal gravity waves in
a stratified fluid [14–16], whereas the ILW equation was derived in a similar
context in [17,18]. In [7] the IST analysis associated with the ILW equation and
BO equation is reviewed. The unusual aspect of the IST scheme is the fact that
the scattering operator is a differential RHBVP. Related generalizations are also
discussed in [7].
4 IST for 2+1 Equations
In Sect. 2 we discussed the relevance of the KP equation in two-dimensional water
waves. The normalized KP equation is given by (1.14). In this section, a broad
outline of the main results of IST for the KP equation will be outlined. Just as
the KdV equation was the first 1+1 equation linearized by IST methods, the KP
equation was the first nontrivial equation linearized by 2+1 IST methods. After
the methods were established for the KP equation, they were quickly generalized
to other equations such as the Davey-Stewartson equation and the 2+1 N wave
equation (cf. [7]).
The compatible linear system for KP is given by
σv y + v xx + uv =0
(4.1a)
v t +4v xxx +6uv x +3u x v − 3σ(∂x −1 u y )v + γv =0, (4.1b)
∫ x
where ∂x −1 = dx ′ , and γ is an arbitrary constant. In the case when σ 2 = −1,
−∞
i.e., KPI, then (4.1a) is the nonstationary Schrödinger equation. When σ 2 = +1,