• Previous Article
    Global classical large solution to compressible viscous micropolar and heat-conducting fluids with vacuum
  • DCDS Home
  • This Issue
  • Next Article
    Sufficiently strong dispersion removes ill-posedness in truncated series models of water waves
June  2019, 39(6): 3099-3122. doi: 10.3934/dcds.2019128

Classification of traveling waves for a quadratic Szegő equation

Département de mathématiques et applications, école normale supérieure, CNRS, PSL Research University, 75005 Paris, France

Received  February 2018 Revised  October 2018 Published  February 2019

We give a complete classification of the traveling waves of the following quadratic Szegő equation :
$i{\partial _t}u = 2J\Pi \left( {{{\left| u \right|}^2}} \right) + \bar J{u^2},\;\;\;\;u\left( {0, \cdot } \right) = {u_0},$
and we show that they are given by two families of rational functions, one of which is generated by a stable ground state. We prove that the other branch is orbitally unstable.
Citation: Joseph Thirouin. Classification of traveling waves for a quadratic Szegő equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3099-3122. doi: 10.3934/dcds.2019128
References:
[1]

C. J. Amick and J. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear neumann problem in the plane, Acta Mathematica, 167 (1991), 107-126. doi: 10.1007/BF02392447. Google Scholar

[2]

C. J. Amick and J. Toland, Uniqueness of Benjamin's solitary wave solution of the Benjamin-Ono equation, IMA J. Appl. Math., 46 (1991), 21-28. doi: 10.1093/imamat/46.1-2.21. Google Scholar

[3]

P. BizońB. CrapsO. EvninD. HunikV. Luyten and M. Maliborski, Conformal flow on $\mathbb S^3$ and weak field integrability in AdS4, Comm. Math. Phys., 353 (2017), 1179-1199. doi: 10.1007/s00220-017-2896-8. Google Scholar

[4]

P. Bizoń, D. Hunik-Kostyra and D. Pelinovsky, Ground state of the conformal flow on $\mathbb S^3$, arXiv: 1706.07726, 2017.Google Scholar

[5]

C. Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc., 77 (1971), 587-588. doi: 10.1090/S0002-9904-1971-12763-5. Google Scholar

[6]

P. Gérard, P. Germain and L. Thomann, On the cubic lowest Landau level equation, arXiv: 1709.04276, 2017.Google Scholar

[7]

P. Gérard and S. Grellier, Ann. Sci. Éc. Norm. Supér. (4), Acta Mathematica, 43 (2010), 761-810. doi: 10.24033/asens.2133. Google Scholar

[8]

P. Gérard and S. Grellier, Invariant tori for the cubic Szegő equation, Invent. Math., 187 (2012), 707-754. doi: 10.1007/s00222-011-0342-7. Google Scholar

[9]

P. Gérard and S. Grellier, An explicit formula for the cubic Szegő equation, Transactions of the American Mathematical Society, 367 (2015), 2979-2995. doi: 10.1090/S0002-9947-2014-06310-1. Google Scholar

[10]

P. Gérard and S. Grellier, The cubic Szegő equation and Hankel operators, Astérisque, 389 (2017), vi+112pp. Google Scholar

[11]

P. Gérard and H. Koch, The cubic Szegő flow at low regularity, In Séminaire Laurent Schwartz–Équations aux dérivées partielles et applications. Année 2016–2017, pages Exp No. XIV, 14 p. Ed. Éc. Polytech., Palaiseau, 2017. Google Scholar

[12]

F. John and L. Nirenberg, On functions of bounded mean oscillation, Communications on Pure and Applied Mathematics, 14 (1961), 415-426. doi: 10.1002/cpa.3160140317. Google Scholar

[13]

O. Pocovnicu, Traveling waves for the cubic Szegő equation on the real line, Anal. PDE, 4 (2011), 379-404. doi: 10.2140/apde.2011.4.379. Google Scholar

[14]

O. Pocovnicu, Soliton interaction with small Toeplitz potentials for the Szegö equation on $\mathbb R$, Dyn. Partial Differ. Equ., 9 (2012), 1-27. doi: 10.4310/DPDE.2012.v9.n1.a1. Google Scholar

[15]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1987. Google Scholar

[16]

E. M. Stein, Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals, volume 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ. Google Scholar

[17]

J. Thirouin, Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation, Transactions of the Am. Math. Soc., 371 (2019), 3673-3690. doi: 10.1090/tran/7535. Google Scholar

[18]

Y. Wu, Simplicity and finiteness of discrete spectrum of the Benjamin-Ono scattering operator, SIAM J. Math. Anal., 48 (2016), 1348-1367. doi: 10.1137/15M1030649. Google Scholar

[19]

H. Xu, The cubic Szegő equation with a linear perturbation, arXiv: 1508.01500, 2015.Google Scholar

show all references

References:
[1]

C. J. Amick and J. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear neumann problem in the plane, Acta Mathematica, 167 (1991), 107-126. doi: 10.1007/BF02392447. Google Scholar

[2]

C. J. Amick and J. Toland, Uniqueness of Benjamin's solitary wave solution of the Benjamin-Ono equation, IMA J. Appl. Math., 46 (1991), 21-28. doi: 10.1093/imamat/46.1-2.21. Google Scholar

[3]

P. BizońB. CrapsO. EvninD. HunikV. Luyten and M. Maliborski, Conformal flow on $\mathbb S^3$ and weak field integrability in AdS4, Comm. Math. Phys., 353 (2017), 1179-1199. doi: 10.1007/s00220-017-2896-8. Google Scholar

[4]

P. Bizoń, D. Hunik-Kostyra and D. Pelinovsky, Ground state of the conformal flow on $\mathbb S^3$, arXiv: 1706.07726, 2017.Google Scholar

[5]

C. Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc., 77 (1971), 587-588. doi: 10.1090/S0002-9904-1971-12763-5. Google Scholar

[6]

P. Gérard, P. Germain and L. Thomann, On the cubic lowest Landau level equation, arXiv: 1709.04276, 2017.Google Scholar

[7]

P. Gérard and S. Grellier, Ann. Sci. Éc. Norm. Supér. (4), Acta Mathematica, 43 (2010), 761-810. doi: 10.24033/asens.2133. Google Scholar

[8]

P. Gérard and S. Grellier, Invariant tori for the cubic Szegő equation, Invent. Math., 187 (2012), 707-754. doi: 10.1007/s00222-011-0342-7. Google Scholar

[9]

P. Gérard and S. Grellier, An explicit formula for the cubic Szegő equation, Transactions of the American Mathematical Society, 367 (2015), 2979-2995. doi: 10.1090/S0002-9947-2014-06310-1. Google Scholar

[10]

P. Gérard and S. Grellier, The cubic Szegő equation and Hankel operators, Astérisque, 389 (2017), vi+112pp. Google Scholar

[11]

P. Gérard and H. Koch, The cubic Szegő flow at low regularity, In Séminaire Laurent Schwartz–Équations aux dérivées partielles et applications. Année 2016–2017, pages Exp No. XIV, 14 p. Ed. Éc. Polytech., Palaiseau, 2017. Google Scholar

[12]

F. John and L. Nirenberg, On functions of bounded mean oscillation, Communications on Pure and Applied Mathematics, 14 (1961), 415-426. doi: 10.1002/cpa.3160140317. Google Scholar

[13]

O. Pocovnicu, Traveling waves for the cubic Szegő equation on the real line, Anal. PDE, 4 (2011), 379-404. doi: 10.2140/apde.2011.4.379. Google Scholar

[14]

O. Pocovnicu, Soliton interaction with small Toeplitz potentials for the Szegö equation on $\mathbb R$, Dyn. Partial Differ. Equ., 9 (2012), 1-27. doi: 10.4310/DPDE.2012.v9.n1.a1. Google Scholar

[15]

W. Rudin, Real and Complex Analysis, McGraw-Hill Book Co., New York, third edition, 1987. Google Scholar

[16]

E. M. Stein, Harmonic Analysis : Real-Variable Methods, Orthogonality, and Oscillatory Integrals, volume 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, Ⅲ. Google Scholar

[17]

J. Thirouin, Optimal bounds for the growth of Sobolev norms of solutions of a quadratic Szegő equation, Transactions of the Am. Math. Soc., 371 (2019), 3673-3690. doi: 10.1090/tran/7535. Google Scholar

[18]

Y. Wu, Simplicity and finiteness of discrete spectrum of the Benjamin-Ono scattering operator, SIAM J. Math. Anal., 48 (2016), 1348-1367. doi: 10.1137/15M1030649. Google Scholar

[19]

H. Xu, The cubic Szegő equation with a linear perturbation, arXiv: 1508.01500, 2015.Google Scholar

Figure 1.  The action of $ H_u $, $ K_u $ and $ S^* $ on $ F $
[1]

Masaharu Taniguchi. Instability of planar traveling waves in bistable reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2003, 3 (1) : 21-44. doi: 10.3934/dcdsb.2003.3.21

[2]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753

[3]

Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295

[4]

Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017

[5]

Reika Fukuizumi. Stability and instability of standing waves for the nonlinear Schrödinger equation with harmonic potential. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 525-544. doi: 10.3934/dcds.2001.7.525

[6]

Zhaosheng Feng. Traveling waves to a reaction-diffusion equation. Conference Publications, 2007, 2007 (Special) : 382-390. doi: 10.3934/proc.2007.2007.382

[7]

Xiao-Biao Lin, Stephen Schecter. Traveling waves and shock waves. Discrete & Continuous Dynamical Systems - A, 2004, 10 (4) : i-ii. doi: 10.3934/dcds.2004.10.4i

[8]

Xiang-Sheng Wang, Haiyan Wang, Jianhong Wu. Traveling waves of diffusive predator-prey systems: Disease outbreak propagation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3303-3324. doi: 10.3934/dcds.2012.32.3303

[9]

Yaping Wu, Niannian Yan. Stability of traveling waves for autocatalytic reaction systems with strong decay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1601-1633. doi: 10.3934/dcdsb.2017033

[10]

Grigori Chapiro, Lucas Furtado, Dan Marchesin, Stephen Schecter. Stability of interacting traveling waves in reaction-convection-diffusion systems. Conference Publications, 2015, 2015 (special) : 258-266. doi: 10.3934/proc.2015.0258

[11]

Jian Fang, Jianhong Wu. Monotone traveling waves for delayed Lotka-Volterra competition systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3043-3058. doi: 10.3934/dcds.2012.32.3043

[12]

Je-Chiang Tsai. Global exponential stability of traveling waves in monotone bistable systems. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 601-623. doi: 10.3934/dcds.2008.21.601

[13]

Judith R. Miller, Huihui Zeng. Stability of traveling waves for systems of nonlinear integral recursions in spatial population biology. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 895-925. doi: 10.3934/dcdsb.2011.16.895

[14]

Oana Pocovnicu. Explicit formula for the solution of the Szegö equation on the real line and applications. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 607-649. doi: 10.3934/dcds.2011.31.607

[15]

Yuqian Zhou, Qian Liu. Reduction and bifurcation of traveling waves of the KdV-Burgers-Kuramoto equation. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 2057-2071. doi: 10.3934/dcdsb.2016036

[16]

Rui Huang, Ming Mei, Yong Wang. Planar traveling waves for nonlocal dispersion equation with monostable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3621-3649. doi: 10.3934/dcds.2012.32.3621

[17]

Emile Franc Doungmo Goufo, Abdon Atangana. Dynamics of traveling waves of variable order hyperbolic Liouville equation: Regulation and control. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 645-662. doi: 10.3934/dcdss.2020035

[18]

Orlando Lopes. A linearized instability result for solitary waves. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 115-119. doi: 10.3934/dcds.2002.8.115

[19]

Reinhard Racke. Instability of coupled systems with delay. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1753-1773. doi: 10.3934/cpaa.2012.11.1753

[20]

Chiun-Chuan Chen, Li-Chang Hung. An N-barrier maximum principle for elliptic systems arising from the study of traveling waves in reaction-diffusion systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1503-1521. doi: 10.3934/dcdsb.2018054

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (36)
  • HTML views (78)
  • Cited by (0)

Other articles
by authors

[Back to Top]