Hidden solitons in the Zabusky--Kruskal experiment ...christov.tmnt-lab.org/downloads/IMACS_Waves2009_talk.pdf · 2 Inverse Scattering Transform (IST): Construct the nonlinear Fourier

Introduction KdV Soliton Formation Nonlinear Fourier Analysis Uncovering the Hidden Solitons Closure

Hidden solitons in the Zabusky–Kruskal experiment:Analysis using the periodic, inverse scattering transform

Ivan Christov*

Department of Engineering Sciences and Applied Mathematics,Northwestern University, Evanston, IL 60208-3125, USA

[email protected]

Special Session on Nonlinear Wave Phenomena in Discrete and Continuous Models

6th IMACS International Conference on Nonlinear Evolution Equationsand Wave Phenomena: Computation and Theory

Athens, GeorgiaMarch 25, 2009

*A Travel Award from the organizers is kindly acknowledged.Ivan Christov (NU) Hidden Solitons in the ZK Experiment IMACS Waves 2009: SS #3 1 / 15

http://www.esam.northwestern.edu

http://people.esam.northwestern.edu/christov/

mailto:[email protected]


The Big Picture

Motivation:I Soliton formation is an important & fundamental phenomenon we wish

to understand.

I Nonlinear Fourier analysis provides a natural framework to answermany related questions.

Outline of the talk:1 The Korteweg–de Vries equation as a paradigm in solitonics.

2 The inverse scattering transform as a nonlinear Fourier analysis.

3 From Zabusky & Kruskal to Osborne & Bergamasco to Engelbrecht &Salupere.

4 Predicting the number and amplitudes of hidden solitons.

5 The notion of a soliton reference level for the periodic problem.

Ivan Christov (NU) Hidden Solitons in the ZK Experiment IMACS Waves 2009: SS #3 2 / 15



Background

ZK 1965: Discovery of solitons emerging from harmonic excitation ofKdV, results have become synonymous with “nonlinear science.”

GGKM 1967 ... AKNS 1973 ... Flaschka, McLaughlin / Dubrovin,Matveev, Novikov 1976: General method for solving nonlinearevolution equations termed the (inverse) scattering transform.

Osborne & Bergamasco 1986: Re-examination the ZK experimentusing the framework of the periodic, inverse scattering transform.

Osborne ca.1980–present: Application of the scattering transform forthe periodic KdV eq. to the analysis of oceanographic data — theso-called nonlinear Fourier analysis.

Salupere, Maugin, Engelbrecht & Kalda 1994, 1996 ... SEP/SPE2002-3 ... ES 2005: High-resolution direct numerical simulation ofthe periodic KdV eq. showing soliton formation, presence ofensembles, patterns in the trajectories and hidden solitons.




Statement of the Problem and Notation (I)

The paradigm in soliton theory is the Korteweg–de Vries equation:

PDE: ηt + c0ηx + αηηx + βηxxx = 0, (x , t) ∈ [0, L]× (0,∞);

IBC:

{η(x , t = 0) = η0(x), x ∈ [0, L],

η(x + L, t) = η(x , t), (x , t) ∈ [0, L]× [0,∞).

For surface water waves, for example, we have

c0 =√

gh,︸︷︷︸speed of linear waves

α =3c0

2h,︸︷︷︸

nonlinearity coeff.

β =c0h

2

6,︸︷︷︸

dispersion coeff.

λ =α

6β.︸︷︷︸

nl-to-disp. ratio for PIST

A particularly illustrative initial condition is

η0(x) = A sin(ωx), x ∈ [0, L], ω =2π

Ln, n ∈ Z.




Statement of the Problem and Notation (II)

Letting

η =1

αu, x = (ξ + c0t) mod L, t = τ, β = δ2, L = 2, n = 1,

transforms the previous initial-boundary-value problem into

PDE: uτ + uuξ + δ2uξξξ = 0, (ξ, τ) ∈ [0, 2]× (0,∞);

IBC:

{u(ξ, τ = 0) = αA sin(πξ), ξ ∈ [0, 2],

u(ξ + 2, τ) = u(ξ, τ), (ξ, τ) ∈ [0, 2]× [0,∞).

This is exactly the problem considered by ZK 1965 when A = 1α . And, to

be completely consistent, use δ2 = β =√

g6 h5/2 to deduce

h =

(6δ2√

g

)2/5

⇒ c0 =(6δ2g2

)1/5, α =

3

2

(6δ2

g3

)−1/5

.




Soliton Formation from a Harmonic Excitation [ZK 1965]

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“Fourier Analysis” of Integrable Nonlinear PDEs

Going back to the KdV equation in the moving frame:

ηt + c0ηx + αηηx + βηxxx = 0,

we can find its exact solution via the Scattering Transform:

1 Direct Scattering Transform (DST): Solve the associated Schrodingereigenvalue problem [

− d2

dx2− λη(x , 0)

]ψ = Eψ,

to get the discrete spectrum, aka “scattering data,” {Ej} ∪ {µ0j }.

2 Inverse Scattering Transform (IST): Construct the nonlinear Fourierseries from the spectrum E ∈ {Ej} ∪ {µ0

j } using hyperellipticfunctions or the Riemann Θ-function.

? Together these two steps constitute the periodic, inverse scat-tering transform (PIST) for the KdV eq.




The Nonlinear Fourier Series

In terms of the hyperelliptic (aka Abelian) functions we have

η(x , t) =1

λ

{2

N∑j=1

µj(x , t)−2N+1∑j=1

Ej

}.

? All nonlinear waves and their interactions are obtained from thislinear superposition!

In the small amplitude limit, maxx ,t |µj(x , t)| � 1, we haveµj(x , t) ∼ cos(kjx − ωj t + φj) ⇒ we get the ordinary Fourier series!

If there are no interactions (N = 1, i.e., just one wave), we haveµ1(x , t) ∼ cn2(k1x − ω1t + φ1|m1), which is a Jacobian ellipticfunction with modulus m1 (a cnoidal wave).

The amplitudes of the nonlinear oscillations are given by

Aj =

{2λ(Eref − E2j), for solitons;12λ(E2j+1 − E2j), otherwise (radiation).




Of Moduli and Wave Numbers

In the hyperelliptic representation, we have kj = 2πj/L as in Fouriertheory (the wave numbers are commensurable).

We can compute the elliptic modulus mj of the hyperelliptic functions,which is called the soliton index, from the discrete spectrum as

mj =E2j+1 − E2j

E2j+1 − E2j−1, 1 ≤ j ≤ N.

Then, the oscillations fall into two generic categories:

I mj & 0.99 ⇒ solitons:

cn2(x |mj = 1) ≡ sech2(x),

I mj � 1.0 ⇒ radiation:

cn2(x |mj = 0) ≡ cos2(x).

m ! 1m ! 0.9m ! 0.75m ! 0




Back to Soliton Formation; δ = 0.022 [ZK 1965]

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(Left) The FFT spectrum gives us 30+ significant normal modes.

(Right) The DST gives 8 (well-defined) soliton modes (=ZK 1965),other nonlinear waves, and radiation.

N.B.: ∃ 4 non-soliton nonlinear waves with nontrivial amplitudes.

Amplitude curves DNS approach of SMEK 1994/1996 predicts12 waves for this case.




How Many Solitons in a Cosine Wave?

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A Note on the Large Dispersion Case: δ → O(1)

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+ λ−1(E − 2E3),

k1 =√E3 − E1, ω1 = 2Ek1,

E :=∑3

j=1Ej .




A Reference Level for the Periodic Problem

OB 1986 argue that a shift in the reference (or zero) level of thesolitons is equivalent to a shift in the energy level of the last soliton’sband gap edge with respect to E = 0, so:

Eref = E2j∗+1 − 0, j∗ is biggest j s.t. mj ≥ 0.99.

Whence, if we know Eref, then we can determine the solitons’reference level and reference wavenumber:

uref = −αEref/λ, kref = 2πj∗/L.

δ 1.0 0.318 0.142 0.0635 0.0253 0.022 0.0179

uref 1.29e-4 7.80e-4 0.337 -0.483 -0.555 -0.720 -0.844

In SMEK 1996, choice of uref affects the predicted # of solitons.

PIST gives a rigorous way to calculate uref, no guesswork.




Summary & Conclusions

∗ Applied the nonlinear Fourier analysis to the problem of solitonformation from a harmonic excitation of the periodic KdV equation.

∗ Addressed the issue of hidden solitons and corroborated Salupere etal.’s numerical results.

Found that all visible waves are indeed solitons, but this is not exactlytrue for the hidden ones. Hidden solitons → hidden modes?

I Fundamental difference between IST and PIST: periodic problem hasmuch richer solution space.

I 3 e-values determine 1 mode ⇒ ∞-line IST is only an analogy here.

Discovered a consistent feature: 4 nonlinear, non-soliton hiddenmodes exist across a wide range of δ values.

Within the framework of OB 1986, the exact number of nonlineroscillation modes, their amplitudes and the soliton referencelevel can be easily calculated for the ZK problem by the PIST.




Selected Bibliography

Osborne & Bergamasco, “The solitons of Zabusky and Kruskalrevisited: Perspective in terms of the periodic spectral transform.”Physica D 18 (1986) 26–46.

Salupere, Maugin, Engelbrecht & Kalda, “On the KdV solitonformation and discrete spectral analysis,”Wave Motion 23 (1998), 49–66.

Salupere, Engelbrecht & Peterson, “On the long-time behaviourof soliton ensembles.”Math. Comput. Simul. 62 (2003) 137–147.

Osborne, “Automatic algorithm for the numerical inversescattering transform of the Korteweg–de Vries equation.”Math. Comput. Simul. 36 (1994) 431–450.



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Hidden solitons in the Zabusky--Kruskal experiment ...christov.tmnt-lab.org/downloads/IMACS_Waves2009_talk.pdf · 2 Inverse Scattering Transform (IST): Construct the nonlinear Fourier