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

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[12]

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

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

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
[1]

Carlos Tomei. The Toda lattice, old and new. Journal of Geometric Mechanics, 2013, 5 (4) : 511-530. doi: 10.3934/jgm.2013.5.511

[2]

Gerald Teschl. On the spatial asymptotics of solutions of the Toda lattice. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 1233-1239. doi: 10.3934/dcds.2010.27.1233

[3]

Yong Liu. Even solutions of the Toda system with prescribed asymptotic behavior. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1779-1790. doi: 10.3934/cpaa.2011.10.1779

[4]

Manuel del Pino, Michal Kowalczyk, Juncheng Wei. The Jacobi-Toda system and foliated interfaces. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 975-1006. doi: 10.3934/dcds.2010.28.975

[5]

Isaac Alvarez-Romero, Gerald Teschl. On uniqueness properties of solutions of the Toda and Kac-van Moerbeke hierarchies. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2259-2264. doi: 10.3934/dcds.2017098

[6]

Weiwei Ao. Sharp estimates for fully bubbling solutions of $B_2$ Toda system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1759-1788. doi: 10.3934/dcds.2016.36.1759

[7]

Juncheng Wei, Jun Yang. Toda system and interior clustering line concentration for a singularly perturbed Neumann problem in two dimensional domain. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 465-508. doi: 10.3934/dcds.2008.22.465

[8]

Andreas Henrici. Symmetries of the periodic Toda lattice, with an application to normal forms and perturbations of the lattice with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2949-2977. doi: 10.3934/dcds.2015.35.2949

[9]

D. Blömker, S. Maier-Paape, G. Schneider. The stochastic Landau equation as an amplitude equation. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 527-541. doi: 10.3934/dcdsb.2001.1.527

[10]

Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145

[11]

Joachim Escher, Boris Kolev, Marcus Wunsch. The geometry of a vorticity model equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1407-1419. doi: 10.3934/cpaa.2012.11.1407

[12]

Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391

[13]

David Mumford, Peter W. Michor. On Euler's equation and 'EPDiff'. Journal of Geometric Mechanics, 2013, 5 (3) : 319-344. doi: 10.3934/jgm.2013.5.319

[14]

Saugata Bandyopadhyay, Bernard Dacorogna, Olivier Kneuss. The Pullback equation for degenerate forms. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 657-691. doi: 10.3934/dcds.2010.27.657

[15]

F. D. Araruna, F. O. Matias, M. P. Matos, S. M. S. Souza. Hidden regularity for the Kirchhoff equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1049-1056. doi: 10.3934/cpaa.2008.7.1049

[16]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731

[17]

Stéphane Mischler, Qilong Weng. On a linear runs and tumbles equation. Kinetic & Related Models, 2017, 10 (3) : 799-822. doi: 10.3934/krm.2017032

[18]

S. L. Ma'u, P. Ramankutty. An averaging method for the Helmholtz equation. Conference Publications, 2003, 2003 (Special) : 604-609. doi: 10.3934/proc.2003.2003.604

[19]

Luis Caffarelli, Serena Dipierro, Enrico Valdinoci. A logistic equation with nonlocal interactions. Kinetic & Related Models, 2017, 10 (1) : 141-170. doi: 10.3934/krm.2017006

[20]

Chunqing Lu. Asymptotic solutions of a nonlinear equation. Conference Publications, 2003, 2003 (Special) : 590-595. doi: 10.3934/proc.2003.2003.590

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (34)
  • HTML views (206)
  • Cited by (0)

Other articles
by authors

[Back to Top]