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Laboratory realization of KP-solitons
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2014 J. Phys.: Conf. Ser. 482 012046
(http://iopscience.iop.org/1742-6596/482/1/012046)
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Laboratory realization of KP-solitons
Harry Yeh1and Wenwen Li2
1Oregon State University, Corvallis, Oregon 97331, USA
2URS Corporation, Los Angeles, California 90017, USA
E-mail: harry@engr.orst.edu; wenwen.li@urs.com
Abstract. Kodama and his colleagues presented a classification theorem for exact
soliton solutions of the quasi-two-dimensional Kadomtsev-Petviashvili (KP) equation. The
classification theorem is related to non-negative Grassmann manifold, Gr(N , M) that is
parameterized by a unique chord diagram based on the derangement of the permutation group.
The cord diagram can infer the asymptotic behavior of the solution with arbitrary number
of line solitons. Here we present the realization of a variety of the KP soliton formations in
the laboratory environment. The experiments are performed in a water tank designed and
constructed for precision experiments for long waves. The tank is equipped with a directional-
wave maker, capable of generating arbitrary-shaped multi-dimensional waves. Temporal and
spatial variations of water-surface profiles are captured using the Laser Induces Fluorescent
method – a nonintrusive optical measurement technique with sub-millimeter precision. The
experiments yield accurate anatomy of the KP soliton formations and their evolution behaviors.
Physical interpretations are discussed for a variety of KP soliton formations predicted by the
classification theorem.
1. Introduction
Based on the assumption of weakly nonlinear, weakly dispersive water waves that have small
deviation in the transverse direction yfrom the main wave propagation direction x, we take the
following orders in the effects of wave nonlinearity, dispersion and directivity:
a0
h0
=O(),h0
λ02
=O(),tan2ψ0=O(),
respectiverly, where a0is the wave amplitude, h0is the constant water depth at the quiescent
state, λ0is the wavelength scale, and ψ0is the small oblique angle from the xdirection. Then,
the Euler formulation is approximated by the Kadomtsev-Petviashvili (KP) equation of the form
ηt+c0ηx+3c0
2h0
ηηx+c0h2
0
6ηxxxx
+c0
2ηyy = 0,(1)
in which ηis the water-surface elevation from h0, and c0=√gh0. The KP equation (1) can be
expressed in the following canonical non-dimensionalized form,
(4uT+ 6uuξ+uξξξ)ξ+ 3uY Y = 0,(2)
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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where the non-dimensional variables (ξ, Y, T ) and uare defined by
η=2h0
3u, x −c0t=h0ξ, y =h0Y, t =3h0
2c0
T.
With the form of the KP equation (2), we consider the solution in the form
u(ξ, Y , T) = 2∂2
ξ(ln τ(ξ, Y , T)) ,
where τis referred to as the τ-function defined in the Wronskian determinant. For a 2-soliton
case, τ-function can be written as
τ=Wronskian(f1, f2) =
f1∂ξf1
f2∂ξf2
.
In this paper, the term ’2-soliton’ is used to describe a wave condition with the presence of two
solitons each as y→ ±∞. The functions f1and f2satisfy the linear equations: ∂Yfi=∂2
ξfiand
∂Tfi=−∂3
ξfi(see e.g. Hirota [1]). Soliton solutions can be written by exponential functions of
the form:
Ej= exp(θj) = exp(kjξ+k2
jY−k3
jT).
Therefore,
f1
f2=a11 a12 a13 a14
a21 a22 a23 a24
E1
E2
E3
E4
=A
E1
E2
E3
E4
.
Here the coefficient matrix A= (aij ) is a constant 2 ×4 matrix. Thus each solution u(ξ, Y, T )
is completely determined by the A-matrix and the k-parameters. The foregoing is generalized
for arbitrary number N f-functions and M E-exponential functions. The τ-function leads to
the notion of Grassmannian variety: Gr(N , M). When constraints are applied for the regular
soliton solutions, the τ-function is identified as a point of the totally nonnegative Grassmannian
cell. The asymptotic soliton solutions for y→ ±∞ can be parameterized by the permutations,
which lead to the introduction of chord diagram to express the classification of soliton solutions
as a chord joining a pair of kj’s following its permutation representation. Detailed mathematical
descriptions are given by Chakravarty and Kodama [2] and Kodama [3]. Figure 1 shows seven
possible types of N= 2 solutions.
Figure 1. The chord diagrams for seven different types of 2-soliton solutions. The number in
the parenthesis is the permutation.
As for the simplest case, consider the solution of single line-soliton that can be expressed as
τ=E1+aE2
= 2√aexp 1
2(θ1+θ2) cosh 1
2(θ1−θ2−ln a).
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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Therefore,
u= 2∂2
ξ(ln τ) = 1
2(k1−k2)2sech21
2(θ1−θ2−ln a),
=A[i,j]sech2rA[i,j]
2(ξ+Ytan Ψ[i,j]−C[i,j ]T−x0
[i,j]),
where A[i,j]=1
2(kj−ki)2; tan Ψ[i,j]=ki+kj;C[i,j]=k2
i+kikj+k2
j=1
2A[i,j]+3
4tan2Ψ[i,j], and
Ψ[i,j]is the direction of wave propagation. The soliton formation and the corresponding code
diagram is shown in Fig. 2.
Figure 2. A line-soliton contour plot and the corresponding chord diagram.
2. Laboratory Experiments
Laboratory experiments are conducted in the wave tank, 7.3 m long, 3.6 m wide, and 0.30 m
deep: see Fig. 3. The tank is elevated 1.2 m above the laboratory floor to enable us to make
measurements with optical instruments. The bottom and sidewalls are made of 12.7 mm thick
glass plates. The wave basin is equipped with a 16-axis directional-wavemaker system along the
3.6-m long headwall, capable of generating arbitrary-shaped, multi-directional waves. Each wave
paddle is pushed through hinge connections by two adjacent linear-servo-motor motion devices.
The paddles are made of PVDF (Polyvinylidene fluoride) plates that are driven horizontally in
piston-like motions; they are optimally flexible to form a smooth curve between the driving axes.
Each paddle has a maximum horizontal stroke of 55 cm – more than adequate to generate very
long and multiple waves in a water depth. This wavemaker system, together with the precise
wave tank, is ideal for investigation of nonlinear dynamics of long-wave motion. It is emphasized
that the generated waves are precisely replicable: for example, the maximum error is less than
0.06 mm (or 0.1% of the depth) for a solitary wave with amplitude a= 1.73 cm in the water
depth of h0= 6.0 cm. Note that, the horizontal dimensions of the tank scaled by the depth are
120h0×60h0, which is considered extremely large. Incidentally, the wave Reynolds number is
on the order of 12,000 that we consider large enough to circumvent serious viscous scale effects.
The Laser Induced Fluorescent (LIF) method is used to examine temporal and spatial
variations of water-surface profiles. Figure 4 (left) shows a setup for the LIF method. A laser
beam (from a 5W diode-pumped solid-state laser) is converted to a thin laser sheet using a
cylindrical lens. Two front-surface mirrors direct the laser sheet to illuminate the vertical plane.
With the aid of fluorescein dye dissolved in water, the vertical laser-sheet illumination induces
the dyed water to fluoresce and identifies the water-surface profile directly and non-intrusively.
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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Figure 3. Schematic drawings of the laboratory apparatus: (upper) a plan view showing the
waveguide that creates an oblique wave reflection and the tank frame that can be seen through
the bed that is made of glass plates; (lower) an elevation view.
The illuminated profiles, as the wave passes through the light sheet, are recorded by a high-
speed high-resolution video camera. The captured images are rectified and processed with the
calibrated image so that the resulting images can be analyzed quantitatively. The transparent
glass bed of the tank minimizes the reflection of the laser illumination that could have caused
contamination in the image analysis for the wave profiles. The temporal and spatial variations
of the water-surface profiles captured with the LIF technique are shown in Fig. 4 (right).
Figure 4. (Left) Setup for the Laser-Induced Fluorescent (LIF) method. (Right) Temporal
variation of the water surface profile along the vertical wall at y= 0. The profiles were
constructed by making a montage of three LIF images. The time interval of each profile is
1/100 sec.
One of the difficulties associated with the LIF technique for measuring long waves is the
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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limitation in resolution. Unlike experiments for capillary waves or breaking waves, long waves
have an inherently small vertical-to-horizontal scale ratio. This causes insufficient resolution
in the vertical direction. Our laboratory experiments require measurements of small wave
amplitudes (a few centimeters) in a large horizontal span (more than 100 cm). To circumvent
this difficulty, LIF water-surface profiles are repeated on approximately 30 cm segments, creating
a montage of the segment profiles.
3. Results
We first demonstrate the relation between a wave pattern of two solitons and the corresponding
chord diagram. Let us consider, for example, two line solitons at y→ ±∞: namely the one has
the wave amplitude a[1,2] = 0.35 cm with the inclination angle ψ[1,2] =−20◦, and the other has
the amplitude a[3,4] = 0.70 cm with ψ[3,4] = 30◦in the water depth h0= 6.0 cm: see Fig. 5
(left). Using the four relations A[i,j]=1
2(kj−ki)2and tan Ψ[i,j]=ki+kj, we determine four k-
parameters: k1=−0.3785, k2= 0.01457, k3= 0.03250, and k4= 0.5448. Because the asymptotic
line-solitons are presented with [1,2] and [3,4] for y→ ±∞, the corresponding chord diagram is
π= (2143), which is also called the “O-type” 2-soliton because this solution was “originally”
found to express the two-soliton solution [2]. Because the chords are closed independently,
representing two individual line-solitons, the two solitons are not resonantly interacting and the
waveform simply translates in time, maintaining the small phase shift at the intersection of the
two solitons. It is noted that the chord diagram does not contain the initial phase information,
which is represented in A= (aij) matrix.
Figure 5. Theoretical water-surface profile of O-type 2-soliton solution of the KP equation,
and the corresponding chord diagram with π= (2143).
3.1. π= (2143) and π= (3142)
In the laboratory, we create 2-soliton waveforms that are symmetric about the x-axis by utilizing
the wave reflection at the vertical wall: in other words, only a half of the domain, 0 < y < ∞, is
realized in the laboratory taking the advantage of the symmetric wave pattern. Figure 6 shows
the measured 2-soliton patterns of π= (2143) (O-type) and π= (3142), respectively. The tank
sidewall is located along the top edge of the plot. The horizontal direction in the plot represents
time that increases to the right. Negligibly small capillary-wave noise emanating from the wall
can be detected in the figure. As shown in the figure, we observe non-resonant interaction in
π= (2143) and the stem-like formation along the wall remains unchanged, because this stem
wave is a consequence of the phase shift resulted from the wave intersection.
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
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Figure 6. (Upper) O-type wave-profile evolution ai/h0= 0.076 (κ= 1.395) and (lower) (3142)-
type ai/h0= 0.277 (κ= 0.731). ψi= 30◦,x/h0= 71.1, 96.1, and 121.1 from the left panel to
the right. Positive t-axis points to the right and positive y-axis points downward.
On the other hand, in the case of the (3142)-type, the stem-wave along the wall continually
grows as it propagates. According to Miles [4], π= (2143) type and π= (3142) type are
distinguished by the parameter ψi/√2A0; recently, Li et al. [5] found that the parameter should
be presented by κ=tan ψi
cos ψip3ai/h0
for real-world solitons, in which aiis the physical wave
amplitude. When κ > 1, the wave pattern is represented by π= (2143) type, and when κ < 1, π
= (3142). (Note that Kodama [6] formally derived the higher-order correction for the parameter
κ, which includes the correction in terms of ai.)
Figure 7 depicts the chord diagram of π= (2143) and π= (3142), respectively. The amplitude
of [1,4]-soliton (the stem wave) can be inferred from the chord diagram of π= (3142). According
to Miles [4], the stem-wave amplification is predicted by (1 + √αr)2where αris the reflected
wave amplification of the Mach reflection or equivalently, the ratio of the smaller wave amplitude
to the larger one: i.e. αr=A[3,4] /A[1,3] at y→+∞or A[1,2]/A[2,4] at y→ −∞. The theoretical
amplitude prediction of [1,4]-soliton can be found in the chord diagram as
A[1,4] =1
2(k4−k1)2=1
2[(k3−k1)+(k4−k3)]2=1
2"(k3−k1) 1 + s1
2(k4−k3)2
1
2(k3−k1)2!#2
,
which represents the amplification of in fact (1 + √αr)2. The transition between (2143)-type
and (3142)-type can be depicted as the condition (k3−k2)→0. At this critical state, the
Figure 7. Chord diagrams of π= (2143) and π= (3142) types and their critical state.
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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amplification of the stem wave can be correctly interpreted from the chord diagram: namely the
4-fold amplification, A[1,4] =1
2(k4−k1)2= 4 ×1
2(k2−k1)2.
3.2. π= (3412)
According to Chakravarty and Kodama [2], π= (3412)-type solution is called the T-type
solution, and the T-type solution has four asymptotic solitons that form an exact “X” shape
near the origin. Nonetheless, the wave pattern deviates from the exact “X” shape near the
soliton intersection. Figure 8 compares the KP solution with our laboratory measurements for
the T-type solitons. At t= 0, the exact X-shaped wave pattern is created. The continual growth
of [1,4] and [2,3] waves (see Fig. 1) is observed in both theoretical prediction and the laboratory
observation, and the observed “box-shaped” wave pattern is consistent with the prediction.
Figure 8. Laboratory (upper) and theoretical (lower) wave patterns for π= (3412)-type (T-
type). Ai= 0.280; ψi= 20◦.
The exact X-shaped wave pattern is made in the laboratory wave tank as shown in Fig. 9.
Unlike (3412)-type, this is the O-type solution. Immediately after the formation of the exact X-
shaped waveform at t= 0, the waves adjacent to the intersection are bent so as to accommodate
the formation of the phase shift. Once it happens, the waveform remains steady.
3.3. V-Shaped Initial Waves
Initially V-shaped waves are generated along the sidewall in the laboratory with the aid of
directional wavemaker system (see Fig. 10). This initial waveform creates the incomplete chord
diagrams. Kao and Kodama [7] conducted the numerical experiments for the similar cases based
on the KP theory. Here we verify their numerical findings in the real-fluid environment.
Figure 11 shows the case with κ=tan ψ
cos ψq3a[2,3]/h0
= 1, and a[1,3]/a[2,3] = 4.0. This is the
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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Figure 9. Laboratory (upper) and theoretical (lower) wave patterns for π= (2143)-type (O-
type). Ai= 0.090; ψi= 30◦.
Figure 10. A sketch of the V-shaped wave generation in the laboratory.
critical condition of the Mach reflection [4]. The observation shows that the V-shaped wave is
quickly transformed to the Y-shaped wave by completing the chord diagram by linking k2to
k1(Fig. 11). Once the Y-shaped wave is formed by completing the chord diagram, the wave
becomes stable maintaining the constant crest length of the stem wave in the laboratory setup.
The actual wave amplitudes measured at x/h0= 50 are a[2,3]/h0= 0.133, a[1,2] /h0= 0.130, and
a[1,3]/h0= 0.459, consequently κ= 1.05. Note that a[2,3]/a[1,2] ≈1 as expected. The discrepancy
must be due to viscous dissipation that is unavoidable in the real-fluid experiments. Because of
the amplitude attenuation, the value of κcontinually but slowly increases in time.
Figure 12 shows the cases for κ > 1 and κ < 1, respectively. In the case of κ > 1, the crest
length of the stem wave continually decreases as it propagates, which means that the origin of k
in the chord diagram practically shifts to the right and ψ[1,3] is now slightly negative to make the
Y-shaped wave formation steady; in other words, the observation coordinates should be rotated
slightly in the clockwise direction. Also observed is a[2,3] < a[1,2]: the actual wave amplitudes
measured at x/h0= 50 are a[2,3]/h0= 0.090 and a[1,2] /h0= 0.124, consequently a[2,3]/a[1,2] <1.
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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Figure 11. Incomplete Y-shaped soliton generated in the laboratory for the case of κ= 1; the
generated wave amplitudes are a[2,3]/h0= 0.148 and a[1,3] /h0= 0.592 with the oblique angel
tan ψ[2,3] = 0.577.
Figure 12. Incomplete Y-shaped soliton generated in the laboratory for the case of (top)
κ= 1.22 >1: the generated wave amplitudes are a[2,3]/h0= 0.10 and a[1,3] /h0= 0.491 with
the oblique angel tan ψ[2,3] = 0.577; (bottom) κ= 0.92 <1: the generated wave amplitudes are
a[2,3]/h0= 0.175 and a[1,3] /h0= 0.645 with the oblique angel tan ψ[2,3] = 0.577.
In the case of κ < 1, the crest length of the stem wave continually increases as it propagates.
To make the Y-shaped wave pattern steady, the origin k= 0 in the chord diagram should be
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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shifted to the left and ψ[1,3] is now slightly positive, or the observation coordinate system is
rotated slightly in the counterclockwise direction. Opposite to the case of κ > 1, this case yields
a[2,3] > a[1,2]: the actual wave amplitudes measured at x/h0= 50 are a[2,3]/h0= 0.155 and
a[1,2]/h0= 0.129, consequently a[2,3]/a[1,2] >1. Those observations are in accord with the KP
theory and well described in the chord diagram.
4. Summary
The classification theorem for 2-soliton solutions of the KP theory is validated by performing
the experiments in the precisely controlled laboratory apparatus. The chord diagrams resulted
from the classification theorem do not only represent the asymptotic solitons at |y| → ∞, but
also heuristically imply the interaction behaviors, even quantitatively in some cases.
We first demonstrated the difference between the (2143)-type (i.e. O-type) and the (3142)-
type. The stem-wave-like formation is a consequence of the phase-shift associated with the
(2143)-type and the intersecting wave formation remains constant and steady. On the other
hand, the (3142)-type shows the resonant interaction behavior; consequently, the interacting
stem wave continually grows its crest length. The chord diagram can provide the quantitative
wave amplitude of the stem wave. Furthermore, the critical four-fold amplification at κ= 1.0
can be illustrated by the chord diagram.
The (3412)-type (i.e. T-type) soliton is realized in the laboratory, which creates the box-
shaped wave formation at the intersection. The box-shaped wave formation grows in time in
accordance with the KP theory. On the other hand, the exact X-shaped initial wave formation
does not necessarily lead to the (3412)-type. When the soliton amplitude is sufficiently small,
the X-shaped wave formation becomes the (2143)-type (i.e. O-type). In this case, the wave
crests near the intersection are bent so as to accommodate the necessary phase shift associated
with the (2143)-type.
Lastly, we studied the V-shaped KP-soliton formation at t= 0. This V-shaped soliton
formation can be interpreted as an incomplete Y-shaped wave with missing one upper arm, or the
corresponding chord diagram is not completely closed. It is observed that immediately after the
generation, the V-shaped wave becomes the Y-shaped wave formation by completing the chord
diagram. Three distinct cases of the incomplete Y-shaped wave formation are demonstrated in
the laboratory, presenting the wave transformation behaviors in accordance with the three sets
of the chord diagrams.
In short, we demonstrated the KP soliton formations in the real-fluid environment, and
presented the physical implications of the chord diagram without solving the KP equation.
Acknowledgments
We are grateful to Yuji Kodama of Ohio State University for inspiring discussions. This work
was supported by Oregon Sea Grant Program, and the Edwards Endowment through Oregon
State University.
References
[1] Hirota R 2004 The Direct Method in Soliton Theory (New York: Cambridge University Press)
[2] Chakravarty S and Kodama Y 2009 Studies in Applied Mathematics.87 83–151
[3] Kodama Y 2010 Journal of Physics A: Mathematical and Theoretical.43 434004
[4] Miles J W 1977 Journal of Fluid Mechanics.79 157–69
Miles J W 1977 Journal of Fluid Mechanics.79 171–79
[5] Li W, Yeh H and Kodama Y 2011 Journal of Fluid Mechanics.672 326–57
[6] Kodama Y 2012 KP solitons and Mach reflection in shallow water Preprint 1210.0281
[7] Kao C and Kodama Y 2012 Mathematics and Computers in Simulation.82,7 1185–218
Physics and Mathematics of Nonlinear Phenomena 2013 (PMNP2013) IOP Publishing
Journal of Physics: Conference Series 482 (2014) 012046 doi:10.1088/1742-6596/482/1/012046
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