Presentation on Multichannel Nonlinear Scattering for Nonintegrable - PowerPoint PPT Presentation

Presentation on Multichannel Nonlinear Scattering for Nonintegrable equations by A. Soffer and M. I. Weinstein Presenter : Chulkwang Kwak Facultad de Matem´ aticas Pontificia Universidad Cat´ olica de Chile 2017 Participating School, KAIST August 21–25, 2017 C. Kwak August 21–25, 2017 1 / 19

Part I C. Kwak August 21–25, 2017 2 / 19

Nonlinear Schr¨ odinger equation Consider the nonlinear Schr¨ odinger equation with a potential V � i Φ t = [ − ∆ + V ( x ) + | Φ | 2 ]Φ , ( t, x ) ∈ R × R 3 (1) Φ(0) = Φ 0 ∈ H 1 ( R 3 ) . Aim : Asymptotic stability Given initial conditions which lie in a neighborhood of a solitary wave e iγ 0 ψ E 0 , the solution � � t � Φ( t ) = e − i 0 E ( s ) ds − γ ( t ) ( ψ E ( t ) + φ ( t )) converges asymptotically to a solitary wave of nearby energy E ± and phase γ ± in L 4 , as t → ±∞ , i.e., � t Φ( t ) ∼ e − i 0 E ( s ) ds e iγ ± ψ E ± , t → ±∞ . C. Kwak August 21–25, 2017 3 / 19

Hypotheses for a potential V Hypotheses Let V : R 3 → R be a smooth function satisfying (V1) V ∈ S ( R 3 ). (V2) − ∆ + V has exactly one negative eigenvalue E ∗ on L 2 ( R 3 ) with corresponding L 2 normalized eigenfunction ψ ∗ . (V3) V ( x ) = V ( | x | ). Nonresonance Condition (NR) V satisfies (NR) condition if 0 is neither an eigenvalue nor a resonance of − ∆ + V . C. Kwak August 21–25, 2017 4 / 19

Nonlinear bound state Consider a time periodic and spatially localized solution to (1) of the form Φ( t, x ) = e − iEt ψ E ( x ) . ψ E satisfies H ( E ) ψ E ≡ [ − ∆ + V ( x ) + | ψ E | 2 ] ψ E = Eψ E (2) ψ E ∈ H 2 , ψ E > 0 An H 2 -solution ψ E is called a nonlinear bound state or solitary wave profile . Note that the solution e − iEt ψ E does not converge to e − iE 0 t ψ E 0 , since there is a family of solitary waves. C. Kwak August 21–25, 2017 5 / 19

Nonlinear bound state Theorem-Existence of ψ E Let E ∈ ( E ∗ , 0). Then, there exists a solution ψ E > 0 to (2) such that (a) ψ E ∈ H 2 . (b) The function E �→ � ψ E � H 2 is smooth for E � = E ∗ , and E → E ∗ � ψ E � H 2 = 0 , lim i.e. ( E, ψ E ) bifurcates from the zero solution at ( E ∗ , 0) in H 2 (and therefore in L p , 2 ≤ p ≤ ∞ thanks to Sobolev embedding). (c) For all ε > 0, | ψ E ( x ) | ≤ C E,ε e − ( | E |− ε ) | x | . (d) As E → E ∗ , � E − E ∗ � 1 2 � ψ E = [ ψ ∗ + O ( E − E ∗ )] ψ 4 ∗ in H 2 . Here ψ ∗ is the normalized ground state of − ∆ + V with corresponding eigenvalue E ∗ . C. Kwak August 21–25, 2017 6 / 19

Nonlinear bound state Corollary For all E ∈ Ω, any compact subinterval of ( E ∗ , 0), we have � ψ E � H 2 ≤ C Ω � ψ E � L 2 . Theorem-Weighted estimates Let E ∈ ( E ∗ , 0). Also, E lie in a sufficiently small neighborhood of E ∗ . Then, for k ∈ Z + and s ≥ 0, �� x � k ψ E � H s ≤ C k,s � ψ E � H s and �� x � k ∂ E ψ E � H s ≤ C ′ k,s | E − E ∗ | − 1 � ψ E � H s Remark : By above theorems and corollary, we can regard any weighted L p norm of ψ E and ∂ E ψ E as a constant, which tends to 0 as E → E ∗ , in various estimates appearing in the analysis. C. Kwak August 21–25, 2017 7 / 19

Decay estimates Decay estimate Let K = − ∆ + V acting on L 2 ( R 3 ), and assume Hypotheses on V . Also, V satisfies (NR). Let P c ( K ) denote the projection onto the continuous spectral part of K . If 1 /p + 1 /q = 1, 2 ≤ q ≤ ∞ , then � e itK P c ( K ) ψ � L q ≤ C q | t | − (3 / 2 − 3 /q ) � ψ � L p . If ψ is more regular ( ψ ∈ H 1 ), then � e itK P c ( K ) ψ � L q ≤ C q � t � − (3 / 2 − 3 /q ) ( � ψ � L p + � ψ � H 1 ) . A simple consequence is the following local decay estimate Local decay estimate Under the same assumption as in the above theorem, let σ > 3 / 2 − 3 /q . Then �� x � − σ e itK P c ( K ) ψ � L 2 ≤ C q | t | − (3 / 2 − 3 /q ) � ψ � L p . C. Kwak August 21–25, 2017 8 / 19

Decomposition of the solution Φ We decompose the solution to (1) as Φ( t ) = e − i Θ ( ψ E ( t ) + φ ( t )) where Φ(0) = Φ 0 = e iγ 0 ( ψ E 0 + φ 0 ) � t Θ = E ( s ) ds − γ ( t ) 0 E (0) = E 0 , γ (0) = γ 0 Orthogonality Condition d � ψ E 0 , φ 0 � = 0 and dt � ψ E 0 , φ ( t ) � = 0 The orthogonality condition ensures that φ ( t ) lies in the Range of P c ( H ( E 0 )). C. Kwak August 21–25, 2017 9 / 19

Decomposition of the solution Φ � iφ t = [ H ( E 0 ) − E 0 ] φ + [ E 0 − E ( t ) + ˙ γ ( t )] φ + F , (3) φ (0) = φ 0 where F = F 1 + F 2 , γψ E − i ˙ F 1 = ˙ E∂ E ψ E , F 2 = F 2 , lin + F 2 , nl . Here F 2 , lin is a linear term in φ of the form F 2 , lin = (2 ψ 2 E − ψ 2 E 0 ) φ + ψ 2 E φ and F 2 , nl is a nonlinear term in φ of the form F 2 , nl = 2 ψ E | φ | 2 + ψ E φ 2 + | φ | 2 φ. C. Kwak August 21–25, 2017 10 / 19

Decomposition of the solution Φ The Orthogonality condition says φ (0) = φ 0 = P c ( H ( E 0 )) φ 0 , which implies F = P c ( H ( E 0 )) F . Moreover, we know ˙ E ( t ) = � ∂ E ψ E , ψ E 0 � − 1 Im � F 2 , ψ E 0 � and γ ( t ) = −� ψ E , ψ E 0 � − 1 Re � F 2 , ψ E 0 � . ˙ C. Kwak August 21–25, 2017 11 / 19

Linear propagator of dispersive part φ Consider the homogeneous linear equation � iu t = ( H ( E 0 ) − E 0 ) u + ( E 0 − E ( t ) + ˙ γ ( t )) u, (4) u ( s ) = f. Let U ( t, s ) be the propagator associated to (4), i.e. u ( t ) = U ( t, s ) f, U ( s, s ) = Id. Using the gauge transform � t u ( t ) = e − i s [ E 0 − E ( τ )] dτ − i ( γ ( t ) − γ ( s )) v ( t ) , (4) is equivalent to the equation iv t = ( H ( E 0 ) − E 0 ) v with the initial data v ( s ) = f . The solution v is of the form v ( t ) = e − i ( H ( E 0 ) − E 0 )( t − s ) f. Hence � t U ( t, s ) = e − i s [ E 0 − E ( τ )] dτ − i ( γ ( t ) − γ ( s )) e − i ( H ( E 0 ) − E 0 )( t − s ) . (5) C. Kwak August 21–25, 2017 12 / 19

Linear propagator of dispersive part φ Now (3) can be rewritten as the integral equation, in addition to the Orthogonality condition, � t φ ( t ) = U ( t, 0) P c ( H ( E 0 )) φ 0 − i U ( t, s ) P c ( H ( E 0 )) F ( s ) ds. 0 We remark that the gauge transform (5) preserves L p or weighted L 2 norms, i.e., � U ( t, s ) g � X = � e − i ( H ( E 0 ) − E 0 )( t − s ) g � X where X = L p or a weighted L 2 . C. Kwak August 21–25, 2017 13 / 19

Well-posedness theory Contraction mapping principle ⇒ Local well-posedness The equation (1) admits the following mass and energy conservation laws: � R 3 | Φ( x ) | 2 dx = N [Φ 0 ] N [Φ( t )] ≡ � � � H [Φ( t )] ≡ 1 R 3 |∇ Φ( x ) | 2 dx + 1 R 3 V ( x ) | Φ( x ) | 2 dx + 1 R 3 | Φ( x ) | 4 dx 2 2 4 = H [Φ 0 ] For C 0 > 0 such that | V ( x ) | ≤ C 0 , � Φ( t ) � 2 H 1 ≤ 2 H [Φ 0 ] + ( C 0 + 1) N [Φ 0 ] ≤ C ( � Φ 0 � 2 H 1 + � Φ 0 � 4 H 1 ) Local well-posedness implies Global well-posedness C. Kwak August 21–25, 2017 14 / 19

Main Theorem Theorem-Asymptotic stability Let Ω η = ( E ∗ , E ∗ + η ), where η is positive and sufficiently small. Then for all E 0 ∈ Ω η and γ 0 ∈ [0 , 2 π ), there exists a positive number ǫ = ǫ ( E 0 , η ) such that if Φ(0) = e iγ 0 ( ψ E 0 + φ 0 ) where � φ 0 � L 1 ( R 3 x ) + � φ 0 � H 1 ( R 3 x ) < ǫ then � t Φ( t ) = e − i 0 E ( s ) ds + iγ ( t ) ( ψ E ( t )+ φ ( t ) ) with ˙ γ ( t ) ∈ L 1 ( R t ) E ( t ) , ˙ ( ⇒ ∃ lim t →±∞ ( E ( t ) , γ ( t )) = ( E ± , γ ± )) C. Kwak August 21–25, 2017 15 / 19

Main Theorem Theorem A. - Asymptotic stability and φ ( t ) is purely dispersive in the sense that �� x � − σ φ ( t ) � L 2 ( R 3 ) = O ( � t � − 3 2 ) for σ > 2, and � φ ( t ) � L 4 ( R 3 ) = O ( � t � − 3 4 ) as | t | → ∞ . C. Kwak August 21–25, 2017 16 / 19

Decomposition of initial data Let � E ∈ ( E ∗ , 0) and � γ ∈ [0 , 2 π ) be given. Consider the initial data Φ 0 , which is nearby a nonlinear bound state: Φ 0 = e i � γ ψ � E + δ Φ . In general, � ψ � E , δ Φ � � = 0, so we can find E 0 and γ 0 such that � e − iγ 0 Φ 0 − ψ E 0 , ψ E 0 � = 0 , i.e. Φ 0 := e iγ 0 ( ψ E 0 + φ 0 ) = e iγ 0 ψ E 0 + [ e i � γ ψ � E − e iγ 0 ψ E 0 + δ Φ] . Indeed, let F [ E, γ, δ Φ] := � ψ E , φ 0 � = � e iγ ψ E , e i � γ ψ � E − e iγ ψ E + δ Φ � . Then F [ � E, � γ, 0] = 0. C. Kwak August 21–25, 2017 17 / 19

Decomposition of initial data We write F [ E, γ, δ Φ] = F 1 [ E, γ, δ Φ] + iF 2 [ E, γ, δ Φ] . The Jacobian matrix of ( E, γ, δ Φ) �→ ( F 1 , F 2 ) is given by | ψ E | 2 �   � �  − 1 d 0 � 2 dE | ψ E | 2 � E = �  E � � 0 � E = � E at ( � γ, 0). Since the curve E �→ � ψ E � 2 E, � L 2 has no critical point for E ∈ ( E ∗ , 0), the determinant of the Jacobian matrix at ( � E, � γ, 0) is nonzero. By the implicit function theorem, for any δ Φ near 0, there uniquely exists ( E 0 , γ 0 ) near ( � E, � γ ) such that F [ E 0 , γ 0 , δ Φ] = 0, i.e. the decomposition Φ 0 = e iγ 0 ( ψ E 0 + φ 0 ) with � ψ E 0 , φ 0 � = 0 holds. C. Kwak August 21–25, 2017 18 / 19

Thank You for Your Attention!! C. Kwak August 21–25, 2017 19 / 19