# American Institute of Mathematical Sciences

October  2020, 40(10): 5973-5990. doi: 10.3934/dcds.2020255

## Filtering the $L^2-$critical focusing Schrödinger equation

 Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud Ⅺ, CNRS, Université Paris-Saclay, F-91405 Orsay, France

Received  February 2020 Published  June 2020

Fund Project: The author is partially supported by the grant "ANAÉ" ANR-13-BS01-0010-03 of the 'Agence Nationale de la Recherche'. This research is carried out during the author's PhD studies, financed by the PhD fellowship of École Doctorale de Mathématique Hadamard

We study the influence of Szegő projector on the
 $L^2-$
critical non linear focusing Schrödinger equation, leading to the quintic focusing NLS–Szegő equation on the line
 $\begin{equation*} i\partial_t u + \partial_x^2 u + \Pi(|u|^4 u) = 0, \quad (t, x)\in \mathbb{R}\times \mathbb{R}, \qquad u(0, \cdot) = u_0. \end{equation*}$
It has no Galilean invariance but the momentum
 $P(u) = \langle -i\partial_x u, u\rangle_{L^2}$
becomes the
 $\dot{H}^{\frac{1}{2}}-$
norm. Thus this equation is globally well-posed in
 $H^1_+ = \Pi(H^1(\mathbb{R}))$
, for every initial datum
 $u_0$
. The solution
 $L^2-$
scatters both forward and backward in time if
 $u_0$
has sufficiently small mass. By using the concentration–compactness principle, we prove the orbital stability of some weak type of the traveling wave :
 $u_{\omega, c}(t, x) = e^{i\omega t}Q(x+ct)$
, for some
 $\omega, c>0$
, where
 $Q$
is a ground state associated to Gagliardo–Nirenberg type functional
 $\begin{equation*} I^{(\gamma)}(f) = \frac{\|\partial_x f\|_{L^2}^2\|f\|_{L^2}^{4}+\gamma \langle -i \partial_x f , f\rangle_{L^2}^2 \|f\|_{L^2}^2}{\|f\|_{L^{6}}^{6}}, \qquad \forall f\in H^1_+ \backslash \{0\}, \end{equation*}$
for some
 $\gamma\geq 0$
. Its Euler–Lagrange equation is a non local elliptic equation. The ground states are completely classified in the case
 $\gamma = 2$
, leading to the actual orbital stability for appropriate traveling waves. As a consequence, the scattering mass threshold of the focusing quintic NLS–Szegő equation is strictly below the mass of ground state associated to the functional
 $I^{(0)}$
, unlike the recent result by Dodson [8] on the usual quintic focusing non linear Schrödinger equation.
Citation: Ruoci Sun. Filtering the $L^2-$critical focusing Schrödinger equation. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5973-5990. doi: 10.3934/dcds.2020255
##### References:
 [1] C. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation – a nonlinear Neumann problem in the plane, Acta Math., 167 (1991), 107-126.  doi: 10.1007/BF02392447. [2] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  doi: 10.1353/ajm.1999.0001. [3] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, Vol. 26, Instituto de Matemática UFRJ, 1996. [4] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, New York University, Courant Institute of Mathematical Sciences, AMS, Providence, RI, 2003. doi: 10.1090/cln/010. [5] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504. [6] T. Cazenave and F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta. Math., 61 (1988), 477-494.  doi: 10.1007/BF01258601. [7] T. Cazenave and F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A. [8] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.  doi: 10.1016/j.aim.2015.04.030. [9] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS), 9 (2007), 739-774.  doi: 10.4171/JEMS/95. [10] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9. [11] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591. [12] P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  doi: 10.1051/cocv:1998107. [13] P. Gérard and S. Grellier, The cubic Szegő equation, Ann. Sci. l'Éc. Norm. Supér., 43 (2010), 761-810.  doi: 10.24033/asens.2133. [14] P. Gérard and S. Grellier, The cubic Szegő equation and Hankel operators, in Astérisque Vol. 389, 2017. [15] P. Gérard, E. Lenzmann, O. Pocovnicu and P. Raphaël, A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line, Ann. PDE, 4 (2018), 166 pp. doi: 10.1007/s40818-017-0043-7. [16] R. Glassey, On the blowing up of solutions to the Cauchy problem for non linear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.  doi: 10.1063/1.523491. [17] T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., (2005), No. 46, 2815–2828. doi: 10.1155/IMRN.2005.2815. [18] T. Hmidi and S. Keraani, Remarks on the blow-up for the $L^2-$critical nonlinear Schrödinger equations, SIAM J. Math. Anal., 38 (2006), 1035-1047.  doi: 10.1137/050624054. [19] C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4. [20] C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.  doi: 10.1007/s11511-008-0031-6. [21] R. Killip and M. Vişan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885.  doi: 10.2140/apde.2012.5.855. [22] R. Killip, M. Vişan and X. Zhang, Energy-critical NLS with quadratic potentials, Comm. PDE., 34 (2009), 1531-1565.  doi: 10.1080/03605300903328109. [23] J. Krieger and J. Lührmann, Concentration compactness for the critical Maxwell-Klein-Gordon equation, Ann. PDE, 1 (2015), no. 1, Art. 5,208 pp. doi: 10.1007/s40818-015-0004-y. [24] J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, Series of Lectures in Mathematics, European Mathematical Society, Zúrich, 2012. doi: 10.4171/106. [25] E. Lenzmann and J. Sok, A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, preprint, 2018, arXiv: 1805.06294. doi: 10.1093/imrn/rnz274. [26] P.-L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case. Part 1, Ann. Inst. Henri Poincaré, Analyse non linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0. [27] P.-L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case. Part 2, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 1 (1984), 223-283. [28] F. Merle and P. Raphaël, On universality of blow-up profile for $L^2-$critical non linear Schrödinger equation, Invent. Math., 156 (2004), 565-672.  doi: 10.1007/s00222-003-0346-z. [29] T. Ogawa and Y. Tsutsumi, Blow-up of $H^1-$solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B. [30] G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2 (2001), 605-673.  doi: 10.1007/PL00001048. [31] 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. [32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. [33] R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1. [34] R. Sun, Long time behavior of the NLS-Szegő equation, Dyn. Partial. Differ. Equ., 16 (2019), 325-357.  doi: 10.4310/DPDE.2019.v16.n4.a2. [35] R. Sun, Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds, preprint, arXiv: 2004.10007. [36] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106. [37] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.

show all references

##### References:
 [1] C. Amick and J. F. Toland, Uniqueness and related analytic properties for the Benjamin-Ono equation – a nonlinear Neumann problem in the plane, Acta Math., 167 (1991), 107-126.  doi: 10.1007/BF02392447. [2] H. Bahouri and P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., 121 (1999), 131-175.  doi: 10.1353/ajm.1999.0001. [3] T. Cazenave, An Introduction to Nonlinear Schrödinger Equations, Textos de Métodos Matemáticos, Vol. 26, Instituto de Matemática UFRJ, 1996. [4] T. Cazenave, Semilinear Schrödinger Equations, Courant Lecture Notes in Mathematics, Vol. 10, New York University, Courant Institute of Mathematical Sciences, AMS, Providence, RI, 2003. doi: 10.1090/cln/010. [5] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys., 85 (1982), 549-561.  doi: 10.1007/BF01403504. [6] T. Cazenave and F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^1$, Manuscripta. Math., 61 (1988), 477-494.  doi: 10.1007/BF01258601. [7] T. Cazenave and F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in $H^s$, Nonlinear Anal., 14 (1990), 807-836.  doi: 10.1016/0362-546X(90)90023-A. [8] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, Adv. Math., 285 (2015), 1589-1618.  doi: 10.1016/j.aim.2015.04.030. [9] D. Foschi, Maximizers for the Strichartz inequality, J. Eur. Math. Soc. (JEMS), 9 (2007), 739-774.  doi: 10.4171/JEMS/95. [10] R. L. Frank and E. Lenzmann, Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}$, Acta Math., 210 (2013), 261-318.  doi: 10.1007/s11511-013-0095-9. [11] R. L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math., 69 (2016), 1671-1726.  doi: 10.1002/cpa.21591. [12] P. Gérard, Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var., 3 (1998), 213-233.  doi: 10.1051/cocv:1998107. [13] P. Gérard and S. Grellier, The cubic Szegő equation, Ann. Sci. l'Éc. Norm. Supér., 43 (2010), 761-810.  doi: 10.24033/asens.2133. [14] P. Gérard and S. Grellier, The cubic Szegő equation and Hankel operators, in Astérisque Vol. 389, 2017. [15] P. Gérard, E. Lenzmann, O. Pocovnicu and P. Raphaël, A two-soliton with transient turbulent regime for the cubic half-wave equation on the real line, Ann. PDE, 4 (2018), 166 pp. doi: 10.1007/s40818-017-0043-7. [16] R. Glassey, On the blowing up of solutions to the Cauchy problem for non linear Schrödinger equations, J. Math. Phys., 18 (1977), 1794-1797.  doi: 10.1063/1.523491. [17] T. Hmidi and S. Keraani, Blowup theory for the critical nonlinear Schrödinger equations revisited, Int. Math. Res. Not., (2005), No. 46, 2815–2828. doi: 10.1155/IMRN.2005.2815. [18] T. Hmidi and S. Keraani, Remarks on the blow-up for the $L^2-$critical nonlinear Schrödinger equations, SIAM J. Math. Anal., 38 (2006), 1035-1047.  doi: 10.1137/050624054. [19] C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case, Invent. Math., 166 (2006), 645-675.  doi: 10.1007/s00222-006-0011-4. [20] C. Kenig and F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., 201 (2008), 147-212.  doi: 10.1007/s11511-008-0031-6. [21] R. Killip and M. Vişan, Global well-posedness and scattering for the defocusing quintic NLS in three dimensions, Anal. PDE, 5 (2012), 855-885.  doi: 10.2140/apde.2012.5.855. [22] R. Killip, M. Vişan and X. Zhang, Energy-critical NLS with quadratic potentials, Comm. PDE., 34 (2009), 1531-1565.  doi: 10.1080/03605300903328109. [23] J. Krieger and J. Lührmann, Concentration compactness for the critical Maxwell-Klein-Gordon equation, Ann. PDE, 1 (2015), no. 1, Art. 5,208 pp. doi: 10.1007/s40818-015-0004-y. [24] J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, Series of Lectures in Mathematics, European Mathematical Society, Zúrich, 2012. doi: 10.4171/106. [25] E. Lenzmann and J. Sok, A sharp rearrangement principle in Fourier space and symmetry results for PDEs with arbitrary order, preprint, 2018, arXiv: 1805.06294. doi: 10.1093/imrn/rnz274. [26] P.-L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case. Part 1, Ann. Inst. Henri Poincaré, Analyse non linéaire, 1 (1984), 109-145.  doi: 10.1016/S0294-1449(16)30428-0. [27] P.-L. Lions, The concentration-compactness principle in calculus of variations. The locally compact case. Part 2, Ann. Inst. Henri Poincaré, Analyse Non Linéaire, 1 (1984), 223-283. [28] F. Merle and P. Raphaël, On universality of blow-up profile for $L^2-$critical non linear Schrödinger equation, Invent. Math., 156 (2004), 565-672.  doi: 10.1007/s00222-003-0346-z. [29] T. Ogawa and Y. Tsutsumi, Blow-up of $H^1-$solution for the nonlinear Schrödinger equation, J. Differential Equations, 92 (1991), 317-330.  doi: 10.1016/0022-0396(91)90052-B. [30] G. Perelman, On the formation of singularities in solutions of the critical nonlinear Schrödinger equation, Ann. Henri Poincaré, 2 (2001), 605-673.  doi: 10.1007/PL00001048. [31] 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. [32] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970. [33] R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke. Math. J., 44 (1977), 705-714.  doi: 10.1215/S0012-7094-77-04430-1. [34] R. Sun, Long time behavior of the NLS-Szegő equation, Dyn. Partial. Differ. Equ., 16 (2019), 325-357.  doi: 10.4310/DPDE.2019.v16.n4.a2. [35] R. Sun, Complete integrability of the Benjamin–Ono equation on the multi-soliton manifolds, preprint, arXiv: 2004.10007. [36] T. Tao, Nonlinear Dispersive Equations. Local and Global Analysis, CBMS Regional Conference Series in Mathematics, Vol. 106, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/cbms/106. [37] M. I. Weinstein, Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys., 87 (1983), 567-576.
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