1. Numerical Studies of the KP Line Solitons
Michelle Osborne & Tommy McDowell
University of Colorado Colorado Springs
April 17, 2015
2. Introduction
The Kadomtsev-Petviashvili (KP) equation models nonlinear wave
patterns in shallow water
Such patterns occur in nature as beach waves
KP equation has other applications (plasma physics, optics,
ferromagnetics, etc.)
Figure: Beach wave pattern (Venice beach, CA). Photo by D. Baldwin.
3. Introduction (Cont.)
Figure: Y-shape wave (Nuevo Vallarta, Mexico). Photo by M. Ablowitz.
The KP equation
(4ut + 6uux + uxxx )x + 3uyy = 0
is a 2-d, weakly dispersive, nonlinear wave equation for the wave
amplitude u(x, y, t). It admits a class of exact solitary wave
solutions called line-solitons.
4. Exact Solution of KP
A one-soliton solution of KP is given by
u(x, y, t) = 1
2(k2 − k1)2
sech2 1
2(k2 − k1) [x + (k1 + k2)y − ct − x0] .
Amplitude = 1
2(k2 − k1)2, Direction = tan Ψ[1,2] = k1 + k2,
Speed = c = k2
1 + k1k2 + k2
2
Solution depends on two distinct parameters k1, k2 with
k1 < k2 and is called a [1, 2]-soliton.
Solution is localized along the line (for a fixed value of t):
x + (k1 + k2)y − ct − x0 = 0
and decays exponentially in the xy-plane away from that line.
[1,2]
Ψ[1,2]
K[1,2]
x
y
E(1) E(2)
Figure: One-soliton solition of KP
5. Exact Solution of KP (Cont.)
2-soliton solutions of KP
-100 -50 0 50 100
-100
-50
0
50
100
(1,3)
(2,4)
(1,4)
(2,3)
[1,2]
[3,4]
[1,2]
[3,4]
-50 0 50 100
-100
-50
0
50
(1,2)
(3,4)
(2,3)
(1,4)
[1,3]
[1,3]
[2,4]
[2,4]
-100 -50 0 50 100
-100
-50
0
50
100
[1,4]
[1,4]
[2,3]
[2,3]
(3,4)
(1,2)
(2,4)
(1,3)
There are 7 distinct type of 2-soliton solutions of KP.
Each two-soliton solution depends on 4 distinct k-parameters
k1 < k2 < k3 < k4.
6. Solving KP Numerically
Goal: Numerically study the convergence of arbitrary initial data
to exact solutions of KP.
We set up initial condition by “gluing” together pieces of
one-soliton solution and solve the KP equation numerically using a
pseudospectral scheme with the given initial data as follows:
Assume u is periodic in the numerical domain such that
u(x, y, t) =
∞
l=−∞
∞
m=−∞
ˆu(l, m, t)ei(lx+my)
where ˆu(l, m, t) is the Fourier Transform of u(x, y, t).
Then the KP PDE reduces to an ODE for ˆu(l, m, t), which is
solved numerically using Runge-Kutta method.
The solution u(x, y, t) is then reconstructed by taking the
inverse Fourier Transform of ˆu(l, m, t)
7. Example: (3142)-soliton
Left panel shows a V-shape initial condition formed by gluing
together two one-soliton solutions.
Middle and right panels show the time-evolution of the initial
condition at t = 10 and t = 30, respectively. The initial data
converges to a 2-soliton solution of KP called the (3142)-soliton
which has a large amplitude stem shown inside the box.
8. 3142-soliton (Cont.)
The exact (3142)-solution for KP is given as follows:
uexact = 2(ln τ)xx
where τ(x, y, t) is a determinant of a 2 × 2 matrix
τ =
1 a 0 −c
0 0 1 b
eθ1 0 0 0
0 eθ2 0 0
0 0 eθ3 0
0 0 0 eθ4
1 k1
1 k2
1 k3
1 k4
with θi = ki x + k2
i y − k3
i t , i = 1, . . . , 4.
We choose {k1, k2, k3, k4} = {−0.9, −0.1, 0.1, 0.9} to obtain
the V-shape initial condition.
The parameters a, b, c are optimized so that the error
E(t) ≡ B |unum(x,y,t)−uexact (x,y,t)|2dxdy
B u2
exact (x,y,t)dxdy
1
2
is minimum over a
certain box B at a given time t and decreases for later times.
9. Error (3142 case)
Figure: E(t) vs t. Optimized at t=20. Optimal parameter values
{a, b, c} = {2.07, 0.23, 0.6}
10. Stem of (3142) Soliton
Left panel shows the stem amplitude at y = 0 growing with time.
It approaches the asymptotic value of 1
2(k4 − k1)2 = 1.62.
Right panel shows the position of the point on the stem at y = 0
as it evolves in time. Its slope yields the estimated stem speed
(≈ 0.812), which agrees with the theoretical value of
k2
1 + k1k4 + k2
4 = 0.81.
11. Stem of (3142) Soliton (Cont.)
Figure: Growth of the stem length. The theoretical value is computed as
L(t) = 2k3t + Lo, where Lo is a constant that depends on the parameters
a, b, c, and ki . The theoretical line is slightly higher than the best fit line;
however, their slopes are almost the same (slope=0.2).
12. Example: Y-soliton
Left panel shows an initial condition obtained by gluing two
one-soliton solutions of different amplitudes.
Middle (t = 10) and right (t = 21) panels show the time evolution
of the initial data into a Y-shape line-soliton solution of KP.
13. Y-soliton (Cont.)
The exact Y-soliton solution is given by
uexact(x, y, t) = 2(ln τ)xx
where the τ-function is given by
τ = (k2 − k1)eθ1+θ2
+ a(k3 − k1)eθ1+θ3
+ b(k3 − k2)eθ2+θ3
(k3 − k2) ,
with θi (x, y, t) = ki x + k2
i y − k3
i t , i = 1, 2, 3. In this case, the
k-parameters are chosen to be {k1, k2, k3} = {−1, 0, 1}.
As in the (3142)-case, the parameters a, b in the exact solution are
optimized such that the Error E(t) is minimized at a given t and
decreases monotonically for later times.
15. Example: O-Type soliton
Left panel shows a bow-shape initial condition obtained by gluing
two one-soliton solutions to a short, higher amplitude vertical stem.
Middle (t = 10) and right (t = 31) panels show the time evolution
of the initial data into an O-type exact solution of KP.
16. O-Type soliton (Cont.)
The exact O-type solution is given by
uexact(x, y, t) = 2(ln τ)xx
where the τ-function is given by
τ = (k3 − k1)eθ1+θ3
+ b(k4 − k1)eθ1+θ4
+ a(k3 − k2)eθ2+θ3
+ ab(k4 − k2)eθ2+θ4
,
with θi (x, y, t) = ki x + k2
i y − k3
i t , i = 1, 2, 3, 4. In this case, the
k-parameters are chosen to be
{k1, k2, k3, k4} = {−1.1, −0.1, 0.1, 1.1} .
The parameters a, b in the exact solution are optimized such that
the Error E(t) is minimized at a given t and decreases
monotonically for later times.
18. Amplitude of Intersection
Figure: Evolution of the peak amplitude.
As the stem shrinks, the peak amplitude decreases and converges
to the theoretical value umax = A1 + A2 + 2C
√
A1A2, where
A1 = 1
2(k2 − k1)2, A2 = 1
2(k4 − k3)2 are the amplitudes of the [1, 2]
and [3, 4] solitons, respectively, and C depends on the ki . For the
chosen k-values, umax = 1.288.
