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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 and Continuous Dynamical Systems, 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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

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.

[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.

[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.

[4]

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

[5]

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

[6]

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

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[19]

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

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