Two-Dimensional Solitons
In Chap. 2 , we have shown that the KP hierarchy admits particular solutions, called the KP solitons, the main subject of this book, which are expressed by the Wronskian form. In this chapter, we show that this determinant structure is common for other t
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Two-Dimensional Solitons
Abstract In Chap. 2, we have shown that the KP hierarchy admits particular solutions, called the KP solitons, the main subject of this book, which are expressed by the Wronskian form. In this chapter, we show that this determinant structure is common for other two-dimensional integrable systems generated by several reductions of the modified bilinear identity proposed by Ueno-Takasaki [128] (see [123] for a further generalization of the bilinear identity). In addition to the KP hierarchy, these integrable systems also include the two-dimensional Toda lattice hierarchy and the Davey-Stewartson hierarchy. Here we construct their soliton solutions in the determinant form and show that their wave parameters for these solutions are chosen from conic curves, that is, the KP soliton from the parabola, the two-dimensional Toda soliton from the hyperbola, and the Davey-Stewartson soliton from the circle.
3.1 The τ -Functions for Two-Dimensional Soliton Equations In [128], Ueno and Takasaki proposed the modified bilinear identity for the twodimensional Toda lattice hierarchy whose τ -function depends on the variables (s, x, x¯ ), denoted by τ (s) (x, x¯ ) with s ∈ C, x = (xn : n ∈ Z>0 ) and x¯ = (xn : n ∈ Zk
where Λ I and θ I (x) for the ordered set I = {i 1 < · · · < i N } are defined by Λ(s) I
=
N
λisk
k=1
(λi j − λik ),
and
Θ I (x) =
N
j>k
θik (x).
k=1
Notice that the parameter s appears only in Λ(s) I and the τ -function has a simple form, Θ I (x) Δ I (A) Λ(s) . (3.9) τ (s) (x) = I e [M] I ∈( N ) We construct real and regular solutions of those two-dimensional soliton equations mentioned in the previous section by choosing appropriate parameters s and λ j ’s in the exponential functions in (3.7) and the coefficient matrix A in (3.8).
3.2.1 The KP Solitons As we discussed in the previous chapters, the bilinear equation (3.1) gives the KP equation through u(x, y, t) = 2∂x2 ln τ (s) (x, y, t) with the choice of the coordinates x = x1 ,
y = x2 ,
t = x3 .
For real solutions, we choose s = 0, λ j = κ j ∈ R and a real N × M matrix A. Then the τ -function is given by τ (0) (x, y, t) =
I ∈(
[M] N
Δ I (A) K I eΘ I (x,y,t) )
with
KI =
j>k
(κi j − κik ).
3.2 Soliton Solutions
45
If we choose the matrix A with the property that Δ I (A) ≥ 0 and K I > 0 for all , we have a positive definite τ -function which leads to a regular solution. I ∈ [M] N Such matrix A is called a totally nonnegative matrix, and we will discuss the details of those matrices in Chap. 5. The positivity of K I can be achieved by assuming the ordering, κ1 < κ2 < · · · < κ M . θj Let us write the phase function θ j (x, y, t) of the exponential function E (0) j = e in the form, θ j (x, y, t) = p j x + q j y + ω j t.
Then the wave parameters v j := ( p j , q j ) for the KP soliton is a point on the parabola, q = p 2 , i.e. p j = κ j and q j = κ 2j (see Fig. 3.1). Of particular interest is a duality that exists between the contour plot of a KP soliton and the triangulation of a polygon inscribed
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