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 6 th IMACS International Conference on Nonlinear Evolution Equations and Wave Phenomena: Computation and Theory Athens, Georgia March 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
15
Embed
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
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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.
Ivan Christov (NU) Hidden Solitons in the ZK Experiment IMACS Waves 2009: SS #3 3 / 15
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).
Ivan Christov (NU) Hidden Solitons in the ZK Experiment IMACS Waves 2009: SS #3 8 / 15
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:
∗ 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.
Ivan Christov (NU) Hidden Solitons in the ZK Experiment IMACS Waves 2009: SS #3 14 / 15
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.