On the cubic-quintic Schr odinger equation R emi Carles CNRS - PowerPoint PPT Presentation

On the cubic-quintic Schr¨ odinger equation R´ emi Carles CNRS & Univ Rennes Based on a joint work with Christof Sparber (Univ. Illinois) R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 1 / 23

Cubic Schr¨ odinger equation in 2D i ∂ u ∂ t + 1 2∆ u = λ | u | 2 u , x ∈ R d , with λ ∈ R . � Appears in various physical contexts: optics, superfluids, BEC, etc. � Often, cubic nonlinearity stems from Taylor expansion: f ( | u | 2 ) u . Conserved quantities: M = � u ( t ) � 2 Mass: L 2 ( R d ) , � Angular momentum: J = Im R d ¯ u ( t , x ) ∇ u ( t , x ) dx , E = �∇ u ( t ) � 2 L 2 ( R d ) + λ � u ( t ) � 4 Energy: L 4 ( R d ) . � The sign of λ plays a role at the level of the energy. . . but not only. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 2 / 23

Cubic Schr¨ odinger equation in 2D i ∂ u ∂ t + 1 2∆ u = λ | u | 2 u , x ∈ R d , with λ ∈ R . � Appears in various physical contexts: optics, superfluids, BEC, etc. � Often, cubic nonlinearity stems from Taylor expansion: f ( | u | 2 ) u . Conserved quantities: M = � u ( t ) � 2 Mass: L 2 ( R d ) , � Angular momentum: J = Im R d ¯ u ( t , x ) ∇ u ( t , x ) dx , E = �∇ u ( t ) � 2 L 2 ( R d ) + λ � u ( t ) � 4 Energy: L 4 ( R d ) . � The sign of λ plays a role at the level of the energy. . . but not only. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 2 / 23

Cubic Schr¨ odinger equation in 2D i ∂ u ∂ t + 1 2∆ u = λ | u | 2 u , x ∈ R d , with λ ∈ R . � Appears in various physical contexts: optics, superfluids, BEC, etc. � Often, cubic nonlinearity stems from Taylor expansion: f ( | u | 2 ) u . Conserved quantities: M = � u ( t ) � 2 Mass: L 2 ( R d ) , � Angular momentum: J = Im R d ¯ u ( t , x ) ∇ u ( t , x ) dx , E = �∇ u ( t ) � 2 L 2 ( R d ) + λ � u ( t ) � 4 Energy: L 4 ( R d ) . � The sign of λ plays a role at the level of the energy. . . but not only. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 2 / 23

Cubic Schr¨ odinger equation in 2D i ∂ u ∂ t + 1 2∆ u = λ | u | 2 u , x ∈ R d , with λ ∈ R . � Appears in various physical contexts: optics, superfluids, BEC, etc. � Often, cubic nonlinearity stems from Taylor expansion: f ( | u | 2 ) u . Conserved quantities: M = � u ( t ) � 2 Mass: L 2 ( R d ) , � Angular momentum: J = Im R d ¯ u ( t , x ) ∇ u ( t , x ) dx , E = �∇ u ( t ) � 2 L 2 ( R d ) + λ � u ( t ) � 4 Energy: L 4 ( R d ) . � The sign of λ plays a role at the level of the energy. . . but not only. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 2 / 23

Cubic Schr¨ odinger equation in 2D i ∂ u ∂ t + 1 2∆ u = λ | u | 2 u , x ∈ R d , with λ ∈ R . � Appears in various physical contexts: optics, superfluids, BEC, etc. � Often, cubic nonlinearity stems from Taylor expansion: f ( | u | 2 ) u . Conserved quantities: M = � u ( t ) � 2 Mass: L 2 ( R d ) , � Angular momentum: J = Im R d ¯ u ( t , x ) ∇ u ( t , x ) dx , E = �∇ u ( t ) � 2 L 2 ( R d ) + λ � u ( t ) � 4 Energy: L 4 ( R d ) . � The sign of λ plays a role at the level of the energy. . . but not only. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 2 / 23

Well-posedness i ∂ u ∂ t + 1 2∆ u = λ | u | 2 u , x ∈ R d , λ ∈ R . M = � u ( t ) � 2 E = �∇ u ( t ) � 2 L 2 ( R d ) + λ � u ( t ) � 4 L 2 ( R d ) , L 4 ( R d ) . Impose u | t =0 = u 0 . d = 1: u 0 ∈ L 2 � u ∈ C ( R ; L 2 ), higher regularity propagated (Tsutsumi 1987). d = 2: u 0 ∈ L 2 , λ > 0 � u ∈ C ( R ; L 2 ), higher regularity propagated (Dodson 2015). d = 3: u 0 ∈ H 1 , λ > 0 � u ∈ C ( R ; H 1 ), higher regularity propagated (Ginibre & Velo 1979). If λ < 0 and d � 2, finite time blow-up is possible. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 3 / 23

Well-posedness i ∂ u ∂ t + 1 2∆ u = λ | u | 2 u , x ∈ R d , λ ∈ R . M = � u ( t ) � 2 E = �∇ u ( t ) � 2 L 2 ( R d ) + λ � u ( t ) � 4 L 2 ( R d ) , L 4 ( R d ) . Impose u | t =0 = u 0 . d = 1: u 0 ∈ L 2 � u ∈ C ( R ; L 2 ), higher regularity propagated (Tsutsumi 1987). d = 2: u 0 ∈ L 2 , λ > 0 � u ∈ C ( R ; L 2 ), higher regularity propagated (Dodson 2015). d = 3: u 0 ∈ H 1 , λ > 0 � u ∈ C ( R ; H 1 ), higher regularity propagated (Ginibre & Velo 1979). If λ < 0 and d � 2, finite time blow-up is possible. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 3 / 23

Finite time blow-up i ∂ u ∂ t + 1 2∆ u = −| u | 2 u , x ∈ R d , u | t =0 = u 0 . E = �∇ u ( t ) � 2 L 2 ( R d ) − � u ( t ) � 4 L 4 ( R d ) . Theorem (Zhakharov 1972, Glassey 1977) Suppose d � 2 and u 0 ∈ H 1 ∩ F ( H 1 ) . If E < 0 , then ∃ T ± > 0 , �∇ u ( t ) � L 2 ( R d ) t →± T ± ∞ . − → Proof. � R d | x | 2 | u ( t , x ) | 2 dx is C 2 as long as u is H 1 , and The map t �→ � d 2 R d | x | 2 | u ( t , x ) | 2 dx � 2 E . dt 2 R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 4 / 23

Finite time blow up (continued) i ∂ u ∂ t + 1 2∆ u = −| u | 2 u , x ∈ R d , u | t =0 = u 0 . E = �∇ u ( t ) � 2 L 2 ( R d ) − � u ( t ) � 4 L 4 ( R d ) . Gagliardo-Nirenberg: � u � 4 L 4 ( R d ) � C � u � 4 − d L 2 ( R d ) �∇ u � d L 2 ( R d ) . � No blow-up if d = 1. � No blow-up if d = 2 and � u 0 � L 2 ≪ 1. � No blow-up if d = 3 and � u 0 � H 1 ≪ 1. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 5 / 23

Finite time blow up (continued) i ∂ u ∂ t + 1 2∆ u = −| u | 2 u , x ∈ R d , u | t =0 = u 0 . E = �∇ u ( t ) � 2 L 2 ( R d ) − � u ( t ) � 4 L 4 ( R d ) . Gagliardo-Nirenberg: � u � 4 L 4 ( R d ) � C � u � 4 − d L 2 ( R d ) �∇ u � d L 2 ( R d ) . � No blow-up if d = 1. � No blow-up if d = 2 and � u 0 � L 2 ≪ 1. � No blow-up if d = 3 and � u 0 � H 1 ≪ 1. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 5 / 23

Finite time blow up (continued) i ∂ u ∂ t + 1 2∆ u = −| u | 2 u , x ∈ R d , u | t =0 = u 0 . E = �∇ u ( t ) � 2 L 2 ( R d ) − � u ( t ) � 4 L 4 ( R d ) . Gagliardo-Nirenberg: � u � 4 L 4 ( R d ) � C � u � 4 − d L 2 ( R d ) �∇ u � d L 2 ( R d ) . � No blow-up if d = 1. � No blow-up if d = 2 and � u 0 � L 2 ≪ 1. � No blow-up if d = 3 and � u 0 � H 1 ≪ 1. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 5 / 23

The 2D case E = �∇ u ( t ) � 2 L 2 ( R 2 ) − � u ( t ) � 4 L 4 ( R 2 ) . � u � 4 L 4 ( R 2 ) � C � u � 2 L 2 ( R 2 ) �∇ u � 2 L 2 ( R 2 ) . Best constant? M. Weinstein 1983, � � 2 � u � L 2 ( R 2 ) � u � 4 �∇ u � 2 L 4 ( R 2 ) � L 2 ( R 2 ) , � Q � L 2 ( R 2 ) where Q is the unique positive, radial solution to − 1 2∆ Q + Q = Q 3 , x ∈ R 2 . � If � u 0 � L 2 < � Q � L 2 , GWP. � If � u 0 � L 2 � � Q � L 2 , blow-up may happen. (M. Weinstein, Merle, Merle-Rapha¨ el, etc.) R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 6 / 23

The 2D case E = �∇ u ( t ) � 2 L 2 ( R 2 ) − � u ( t ) � 4 L 4 ( R 2 ) . � u � 4 L 4 ( R 2 ) � C � u � 2 L 2 ( R 2 ) �∇ u � 2 L 2 ( R 2 ) . Best constant? M. Weinstein 1983, � � 2 � u � L 2 ( R 2 ) � u � 4 �∇ u � 2 L 4 ( R 2 ) � L 2 ( R 2 ) , � Q � L 2 ( R 2 ) where Q is the unique positive, radial solution to − 1 2∆ Q + Q = Q 3 , x ∈ R 2 . � If � u 0 � L 2 < � Q � L 2 , GWP. � If � u 0 � L 2 � � Q � L 2 , blow-up may happen. (M. Weinstein, Merle, Merle-Rapha¨ el, etc.) R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 6 / 23

The 2D case E = �∇ u ( t ) � 2 L 2 ( R 2 ) − � u ( t ) � 4 L 4 ( R 2 ) . � u � 4 L 4 ( R 2 ) � C � u � 2 L 2 ( R 2 ) �∇ u � 2 L 2 ( R 2 ) . Best constant? M. Weinstein 1983, � � 2 � u � L 2 ( R 2 ) � u � 4 �∇ u � 2 L 4 ( R 2 ) � L 2 ( R 2 ) , � Q � L 2 ( R 2 ) where Q is the unique positive, radial solution to − 1 2∆ Q + Q = Q 3 , x ∈ R 2 . � If � u 0 � L 2 < � Q � L 2 , GWP. � If � u 0 � L 2 � � Q � L 2 , blow-up may happen. (M. Weinstein, Merle, Merle-Rapha¨ el, etc.) R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 6 / 23

Solitary waves i ∂ u ∂ t + 1 2∆ u = −| u | 2 u , x ∈ R 2 . Special solution u ( t , x ) = e i ω t φ ( x ): − 1 2∆ φ + ωφ = | φ | 2 φ. A priori estimates (Pohozaev identities): 1 (multiplier ¯ 2 �∇ φ � 2 L 2 + ω � φ � 2 L 2 − � φ � 4 L 4 = 0 φ ) , L 2 = 1 (multiplier x · ∇ ¯ ω � φ � 2 2 � φ � 4 φ ) . L 4 � Nec. ω > 0. Conversely, ∃ H 1 solution if ω > 0, with exponential decay. � Any solution satisfies E ( φ ) = 0: instability by blow-up. R´ emi Carles (CNRS & Univ Rennes) Cubic-quintic Schr¨ odinger equation 7 / 23