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April  2018, 38(4): 2007-2028. doi: 10.3934/dcds.2018081

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  January 2018

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

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.

Citation: Iryna Egorova, Johanna Michor, Gerald Teschl. Rarefaction waves for the Toda equation via nonlinear steepest descent. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2007-2028. doi: 10.3934/dcds.2018081
References:
[1]

K. AndreievI. EgorovaT. 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.  Google Scholar

[2]

A. Boutet de MonvelI. 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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

P. DeiftS. KamvissisT. 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.  Google Scholar

[6]

P. DeiftS. 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.  Google Scholar

[7]

I. Egorova, The scattering problem for step-like Jacobi operator, Mat. Fiz. Anal. Geom., 9 (2002), 188-205.   Google Scholar

[8]

I. EgorovaJ. Michor and G. Teschl, Scattering theory for Jacobi operators with general steplike quasi-periodic background, Zh. Mat. Fiz. Anal. Geom., 4 (2008), 33-62.   Google Scholar

[9]

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.  Google Scholar

[10]

I. EgorovaJ. 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.  Google Scholar

[11]

I. Egorova, J. Michor and G. Teschl, Long-time Asymptotics for the Toda Shock Problem: Non-overlapping Spectra, arXiv: 1406.0720. Google Scholar

[12]

I. Egorova and A. Pryimak, The Toda Rarefaction Problem: Construction of the Parametrix (in preparation). Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon., 72, Amer. Math. Soc., Rhode Island, 2000.  Google Scholar

[20]

M. Toda, Theory of Nonlinear Lattices, 2nd enl. ed., Springer, Berlin, 1989.  Google Scholar

[21]

S. VenakidesP. Deift and R. Oba, The Toda shock problem, Comm. Pure Appl. Math., 44 (1991), 1171-1242.  doi: 10.1002/cpa.3160440823.  Google Scholar

show all references

References:
[1]

K. AndreievI. EgorovaT. 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.  Google Scholar

[2]

A. Boutet de MonvelI. 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.  Google Scholar

[3]

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.  Google Scholar

[4]

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.  Google Scholar

[5]

P. DeiftS. KamvissisT. 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.  Google Scholar

[6]

P. DeiftS. 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.  Google Scholar

[7]

I. Egorova, The scattering problem for step-like Jacobi operator, Mat. Fiz. Anal. Geom., 9 (2002), 188-205.   Google Scholar

[8]

I. EgorovaJ. Michor and G. Teschl, Scattering theory for Jacobi operators with general steplike quasi-periodic background, Zh. Mat. Fiz. Anal. Geom., 4 (2008), 33-62.   Google Scholar

[9]

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.  Google Scholar

[10]

I. EgorovaJ. 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.  Google Scholar

[11]

I. Egorova, J. Michor and G. Teschl, Long-time Asymptotics for the Toda Shock Problem: Non-overlapping Spectra, arXiv: 1406.0720. Google Scholar

[12]

I. Egorova and A. Pryimak, The Toda Rarefaction Problem: Construction of the Parametrix (in preparation). Google Scholar

[13]

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.  Google Scholar

[14]

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.  Google Scholar

[15]

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.  Google Scholar

[16]

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.  Google Scholar

[17]

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.  Google Scholar

[18]

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.  Google Scholar

[19]

G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices, Math. Surv. and Mon., 72, Amer. Math. Soc., Rhode Island, 2000.  Google Scholar

[20]

M. Toda, Theory of Nonlinear Lattices, 2nd enl. ed., Springer, Berlin, 1989.  Google Scholar

[21]

S. VenakidesP. Deift and R. Oba, The Toda shock problem, Comm. Pure Appl. Math., 44 (1991), 1171-1242.  doi: 10.1002/cpa.3160440823.  Google Scholar

Figure 1.  Toda rarefaction problem with non-overlapping background spectra $\sigma(H_{\ell})=[1.2, 2.8]$, $\sigma(H_r)=[-1,1]$; $a=0.4$, $b=2$
Figure 2.  Signature table for $g(z)$
Figure 3.  Contour deformation of Step 2
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