Rarefaction waves for the Toda equation via nonlinear steepest descent

Iryna Egorova; Johanna Michor; Gerald Teschl

doi:10.3934/dcds.2018081

Article Contents

2018, Volume 38, Issue 4: 2007-2028. Doi: 10.3934/dcds.2018081

This issue Previous Article Large deviations for stochastic heat equations with memory driven by Lévy-type noise Next Article Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems

Rarefaction waves for the Toda equation via nonlinear steepest descent

1.
B. Verkin Institute for Low Temperature Physics and Engineering, 47, Nauky ave, 61103 Kharkiv, Ukraine
2.
V.N. Karazin Kharkiv National University, 4, Svobody sq. 61022 Kharkiv, Ukraine
3.
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1,1090 Wien, Austria
4.
Erwin Schrödinger International Institute for Mathematics and Physics, Boltzmanngasse 9,1090 Wien, Austria

Received: January 2017

Revised: November 2017

Published: July 2018

Research supported by the Austrian Science Fund (FWF) under Grant No. V120

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Abstract

We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Toda lattice with steplike initial data corresponding to a rarefaction wave.

Keywords:

Toda equation.

Mathematics Subject Classification: Primary: 37K40, 35Q53; Secondary: 37K45, 35Q15.

Citation:

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Figure 1. Toda rarefaction problem with non-overlapping background spectra $\sigma(H_{\ell})=[1.2, 2.8]$, $\sigma(H_r)=[-1,1]$; $a=0.4$, $b=2$

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Figure 2. Signature table for $g(z)$

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Figure 3. Contour deformation of Step 2

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References

	K. Andreiev , I. Egorova , T. L. Lange and G. Teschl , Rarefaction waves of the Korteweg-de Vries equation via nonlinear steepest descent, J. Differential Equations, 261 (2016) , 5371-5410. doi: 10.1016/j.jde.2016.08.009.
	A. Boutet de Monvel , I. Egorova and E. Khruslov , Soliton asymptotics of the Cauchy problem solution for the Toda lattice, Inverse Problems, 13 (1997) , 223-237. doi: 10.1088/0266-5611/13/2/003.
	A. Boutet de Monvel and I. Egorova , The Toda lattice with step-like initial data. Soliton asymptotics, Inverse Problems, 16 (2000) , 955-977. doi: 10.1088/0266-5611/16/4/306.
	K. M. Case and M. Kac , A discrete version of the inverse scattering problem, J. Math. Phys., 14 (1973) , 594-603. doi: 10.1063/1.1666364.
	P. Deift , S. Kamvissis , T. Kriecherbauer and X. Zhou , The Toda rarefaction problem, Comm. Pure Appl. Math., 49 (1996) , 35-83. doi: 10.1002/(SICI)1097-0312(199601)49:1<35::AID-CPA2>3.0.CO;2-8.
	P. Deift , S. Venakides and X. Zhou , The collisionless shock region for the long time behavior of solutions of the KdV equation, Comm. Pure and Appl. Math., 47 (1994) , 199-206. doi: 10.1002/cpa.3160470204.
	I. Egorova , The scattering problem for step-like Jacobi operator, Mat. Fiz. Anal. Geom., 9 (2002) , 188-205.
	I. Egorova , J. Michor and G. Teschl , Scattering theory for Jacobi operators with general steplike quasi-periodic background, Zh. Mat. Fiz. Anal. Geom., 4 (2008) , 33-62.
	I. Egorova, J. Michor and G. Teschl, Inverse scattering transform for the Toda hierarchy with steplike finite-gap backgrounds, J. Math. Physics, 50 (2009), 103522, 9pp. doi: 10.1063/1.3239507.
	I. Egorova , J. Michor and G. Teschl , Scattering theory with finite-gap backgrounds: Transformation operators and characteristic properties of scattering data, Math. Phys. Anal. Geom., 16 (2013) , 111-136. doi: 10.1007/s11040-012-9121-y.
	I. Egorova, J. Michor and G. Teschl, Long-time Asymptotics for the Toda Shock Problem: Non-overlapping Spectra, arXiv: 1406.0720.
	I. Egorova and A. Pryimak, The Toda Rarefaction Problem: Construction of the Parametrix (in preparation).
	A. Its, Large N-asymptotics in random matrices, In: Random Matrices, Random Processes and Integrable Systems, CRM Series in Mathematical Physics, Springer, New York, (2011), 351-413.
	H. Krüger and G. Teschl , Long-time asymptotics for the Toda lattice in the soliton region, Math. Z., 262 (2009) , 585-602. doi: 10.1007/s00209-008-0391-9.
	H. Krüger and G. Teschl , Long-time asymptotics of the Toda lattice for decaying initial data revisited, Rev. Math. Phys., 21 (2009) , 61-109. doi: 10.1142/S0129055X0900358X.
	J. A. Leach and D. J. Needham , The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation: Ⅰ. Initial data has a discontinuous expansive step, Nonlinearity, 21 (2008) , 2391-2408. doi: 10.1088/0951-7715/21/10/010.
	J. A. Leach and D. J. Needham , The large-time development of the solution to an initial-value problem for the Korteweg-de Vries equation. Ⅱ. Initial data has a discontinuous compressive step, Mathematika, 60 (2014) , 391-414. doi: 10.1112/S0025579313000284.
	J. Michor , Wave phenomena of the Toda lattice with steplike initial data, Phys. Lett. A, 380 (2016) , 1110-1116. doi: 10.1016/j.physleta.2016.01.033.
	G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon., 72, Amer. Math. Soc., Rhode Island, 2000.
	M. Toda, Theory of Nonlinear Lattices, 2nd enl. ed., Springer, Berlin, 1989.
	S. Venakides , P. Deift and R. Oba , The Toda shock problem, Comm. Pure Appl. Math., 44 (1991) , 1171-1242. doi: 10.1002/cpa.3160440823.