Well-Posedness and Adiabatic Limit for Quantum Zakharov System - PowerPoint PPT Presentation

Well-Posedness and Adiabatic Limit for Quantum Zakharov System Yung-Fu Fang (joint work with Tsai-Jung Chen, Chi-Kun Lin, Jun-Ichi Segata, Hsi-Wei Shih, Kuan-Hsiang Wang, Tsung-fang Wu) Department of Mathematics National Cheng Kung University Tainan, 701 Taiwan Talk at NSYSU The 24th Annual Workshop on Differential Equations Feb. 22 ∼ 23, 2016

Abstract : For a Quantum Zakharov system, we study the LWP and GWP, adiabatic limit, and least energy solution. We obtain the adiabatic limit for ( QZ ) to a quantum modified NLS. We also prove the existence of homoclinic solutions with the least energy. We show an ill-posedness problem. Future study for ( QZ ) : LWP in 2D and 3D, semi-classical limit and subsonic limit, ground state, soliton waves, asymptotic behavior, and eventually the scattering problem.

1. Introduction : Zakharov System describes the propagation of Langmuir waves in an ionized plasma. ( rapid oscillations of the electron density)   iE t + ∂ 2 x E = nE, x ∈ R,   n tt − ∂ 2 x n = ∂ 2 x | E | 2 , (Z)    E ( x, 0) = E 0 ( x ) , n ( x, 0) = n 0 ( x ) , n t ( x, 0) = n 1 ( x ) . E = the rapidly oscillating electric field, n = the deviation of the ion density from its mean value. For the derivation of ( Z ) system, see [Z] and [OT].

Conservation of Mass: � | E ( x, t ) | 2 dx = constant Conservation of Hamiltonian: � | ∂ x E ( t ) | 2 + 1 2 n ( t ) 2 + n ( t ) | E ( t ) | 2 + 1 2 ν ( t ) 2 dx = constant � n + | E | 2 � where ∂ t n = ∂ x ν and ∂ t ν = ∂ x .

Global Well-Posedness: ( Z ) system with initial data ( E 0 , n 0 , n 1 ) ∈ H k ⊕ H l ⊕ H l − 1 ( R ) . Figure 1:

Taking quantum effects into account, we consider   iE t + ∂ 2 x E − ε 2 ∂ 4  x E = nE, x ∈ R ;  n tt − ∂ 2 x n + ε 2 ∂ 4 x n = ∂ 2 x | E | 2 (QZ)    E ( x, 0) = E 0 ( x ) , n ( x, 0) = n 0 ( x ) , n t ( x, 0) = n 1 ( x ) . (6 . 63 × 10 − 27 )(8 × 10 9 ) ε = � ω i Proton 2 π (1 . 38 × 10 − 16 )(10 5 ) ∼ 6 . 12 × 10 − 5 ∼ κ B T e � = Planck’s constant /2 π , ω i = ion plasma frequency κ B = Boltzmann constant, T e = electron fluid temperature

Conservation of Mass: � | E ( x, t ) | 2 dx = constant Conservation of Hamiltonian: � 2 ν 2 + ε 2 x E | 2 + 1 2 n 2 + n | E | 2 + 1 | ∂ x E | 2 + ε 2 | ∂ 2 2 | ∂ x n | 2 dx = constant � � n + | E | 2 − ε 2 ∂ 2 where ∂ t n = ∂ x ν and ∂ t ν = ∂ x . x n

Local Well-Posedness: ( QZ ) with initial data ( E 0 , n 0 , n 1 ) ∈ H k ⊕ H ℓ ε ⊕ H ℓ − 1 ε Figure 2:

Local Well-Posedness: ( Z ) with initial data ( E 0 , n 0 , n 1 ) ∈ H k ⊕ H ℓ ε ⊕ H ℓ − 1 ε Figure 3:

ILL-Posedness Problem: Figure 4:

2. Notations and Solution formulae � � 1 − ε 2 ∂ 2 1 + ε 2 ξ 2 . Denote D ε := and ξ ε := ξ x Decompose ( QZ ) we get  iE t + ∂ 2 x D 2  ε E = ( n + + n − ) E, x ∈ R (QZ ± ) ∂ t n ± ± ∂ x D ε n ± = ∓ 1 1 ∂ x | E | 2 + 1  2 n 1 L , 2 D ε The integral formulae for solutions: � � E ( t, x ) = U ε ( t ) E 0 ( x ) − iU ε ∗ R ( n + + n − ) E ( t, x ) (2 . 1) � ε ∂ x | E | 2 � D − 1 n ± ( t, x ) = W ε ± ( t )( n 0 , n 1 ) ∓ W ε ± ∗ R ( t, x ) (2 . 2)

Denote the Sobolev norms � � � ξ � 2 ℓ | � � ξ ε � 2 ℓ | � � f � 2 f ( ξ ) | 2 dξ, � f � 2 f ( ξ ) | 2 dξ, H ℓ := ε := (2 . 3) H ℓ � � | � | ξ ε | 2 ℓ | � � f � 2 f ( ξ ) | 2 dξ + f ( ξ ) | 2 dξ . ε := A ℓ | ξ |≤ 1 1 ≤| ξ | Denote the Sobolev norm for acoustic wave � �� �� � � n ( t ) � W ε := n ( t ) , ∂ t n ( t ) W ε := � n ( t ) � A ℓ ε + � ∂ t n ( t ) � A ℓ − 1 ε

Bourgain norm for Schr¨ odinger part: � � � � 1 ε � 2 b 1 | � 2 � ξ ε � 2 k � τ + ξ 2 E ( τ, ξ ) | 2 dτdξ � E � X Sε k,b 1 := (2 . 5) Bourgain norm for Wave part: � � � � 1 2 � ξ ε � 2 ℓ � τ ± ξ ε � 2 b | � n ( τ, ξ ) | 2 dτdξ � n ± � X := (2 . 6) Wε ± ℓ,b Y norm for Schr¨ odinger part: � � � � � 2 � 1 ε � − 1 | � 2 � ξ ε � k � τ + ξ 2 � E � Y Sε k := E ( τ, ξ ) | dτ (2 . 7) dξ Y norm for Wave part: � � � � � 2 � 1 � ξ ε � ℓ � τ ± ξ ε � − 1 | � 2 � n ± � Y := n ± ( τ, ξ ) | dτ (2 . 8) dξ Wε ± ℓ

3. Estimates for Iteration Argument Lemma 1. (Homogeneous Estimates) Lemma 2. (Duhamel Estimates) Lemma 3. (Multilinear Estimates) Let 0 < ε ≤ 1 . � n ± E � X Sε k, − c 1 � C ( ε ) � n ± � X ℓ,b � E � X Sε (3 . 1) k,b 1 . Wε ± ε ∂ x ( E 1 ¯ � D − 1 E 2 ) � X ℓ, − c � C ( ε ) � E 1 � X Sε k,b 1 � E 2 � X Sε (3 . 2) k,b 1 . Wε ± � n ± E � Y Sε k � C ( ε ) � n ± � X ℓ,b � E � X Sε (3 . 3) k,b 1 . Wε ± � � E 1 ¯ � D − 1 � Y � C ( ε ) � E 1 � X Sε k,b 1 � E 2 � X Sε (3 . 4) ε ∂ x E 2 k,b 1 . Wε ± ℓ

4. Proof of Multilinear Estimates Proof. We only outline the proof for � D − 1 ε ∂ x ( E 1 E 2 ) � X ℓ, − c � C 2 ( ε ) � E 1 � X Sε k,b 1 � E 2 � X Sε k,b 1 . Wε ± First we set ξ = ξ 1 − ξ 2 and decompose the ξ 1 - ξ 2 plane into Ω 1 ∪ Ω 2 ∪ Ω 3 ∪ Ω 4 , where Ω 1 = { ( ξ 1 , ξ 2 ) : | ξ 2 | ≪ | ξ 1 | ∼ | ξ |} , Ω 2 = { ( ξ 1 , ξ 2 ) : | ξ 1 | ≪ | ξ 2 | ∼ | ξ |} , Ω 3 = { ( ξ 1 , ξ 2 ) : | ξ 1 | ∼ | ξ 2 | ∼ | ξ |} , Ω 4 = { ( ξ 1 , ξ 2 ) : | ξ | ≪ | ξ 1 | ∼ | ξ 2 |} .

By duality argument, the estimate � D − 1 ε ∂ x ( E 1 E 2 ) � X Wε + ℓ, − c � � E 1 � X Sε k,b 1 � E 2 � X Sε k,b 1 � �� � � � D − 1 ∼ ε ∂ x ( E 1 E 2 ) , g � � � E 1 � X Sε k,b 1 � E 2 � X Sε k,b 1 � g � X Wε + − ℓ,c . � We set − ℓ,c = �� ξ ε � − ℓ � τ + ξ ε � c � � g � X Wε + g � L 2 ≡ � � v � L 2 ε � b 1 � k,b 1 = �� ξ ε � k � τ + ξ 2 � E j � X Sε E j � L 2 ≡ � � v j � L 2 � � D − 1 We can rewrite ε ∂ x ( E 1 E 2 ) , g as � � ξ ε � ℓ � v ( τ, ξ ) v 2 ( τ 2 , ξ 2 ) � � v 1 ( τ 1 , ξ 1 ) iξ � dτ 2 dξ 2 dτ 1 dξ 1 , � ξ 1 ε � k � ξ 2 ε � k � σ � c � σ 2 � b 1 � σ 1 � b 1 1 + ε 2 ξ 2 τ = τ 1 − τ 2 , ξ = ξ 1 − ξ 2 , σ = τ + ξ ε , σ 2 = τ 2 + ξ 2 2 ε , σ 1 = τ 1 + ξ 2 1 ε . � � v � L 2 � v 1 � L 2 � v 2 � L 2 .

5. Existence of a Least Energy Solution Consider the stationary solution and static solution to ( QZ ) in 1D by setting E ( x, t ) = e iωt Q ( x ) and n ( x, t ) = n ( x ) . Then � − ωQ + Q ′′ − ε 2 Q (4) = nQ, ⇒ ( QZ ) − n + ε 2 n ′′ = Q 2 . Question 1. Does solution Q ε,ω ( x ) and n ε,ω ( x ) exist? Question 2. Does lim ε → 0 ( Q ε,ω , n ε,ω ) = ( Q 0 ,ω , − Q 2 0 ,ω )? Question 3. Numerical Results? Question 4. Stability of such solutions? Soliton?

Lemma 4. Let n = 1 . � 1 ε Q 2 ( y ) dy → − Q 2 ( x ) as ε → 0 . −| x − y | n ε,Q ( x ) = − 2 εe R Plug n ε,Q back into the system to get an ODE with a nonlocal term. � 1 − ωQ + Q ′′ − ε 2 Q (4) = − Q ε Q 2 ( y ) dy. −| x − y | (E ε ) 2 εe R For 2D and 3D, we also have the solution formula for − ωQ + ∆ Q − ε 2 ∆ 2 Q = − Q (1 − ε 2 ∆) − 1 Q 2 . (E ε ) Theorem 1. (Fang & Wu, 2015) Let n = 1 , 2 , 3 . For each 0 < ε ≤ 1 , Problem ( E ε ) has a least energy homoclinic solution Q ε .

Consider the Zakharov system in 1D,   iE t + ∂ 2  x E = nE, x ∈ R,  λ − 2 n tt − ∂ 2 x n = ∂ 2 x | E | 2 , (Z)    E ( x, 0) = E 0 ( x ) , n ( x, 0) = n 0 ( x ) , n t ( x, 0) = n 1 ( x ) . When wave speed λ tends to infinity, naturally acoustic wave passed and disappeared, only remained the Schr¨ odinger part.

Review the ground state for ( Z ), � − ωQ + Q ′′ = nQ, − n = Q 2 . Thus we get − ωQ + Q ′′ + Q 3 = 0 . The solution can be derived via Newton’s method starting with Gaus- sian function. Hence the exact solution is √ 2 βsech ( βx ) , where β = √ ω. Q ( x ) = Finally solitary solutions for ( Z ) can be given as E λ, 0 ( x, t ) = e i ( β 2 − α 2 ) t e iαx � 1 − 4 λ − 2 α 2 √ � � 2 βsech β ( x − 2 αt ) . n λ, 0 ( x, t ) = − 2 β 2 sech 2 � � β ( x − 2 αt ) .

Consider a system that is close related to ( QZ ),  iE t + ∂ 2 x E − ε 2 ∂ 4  x E = nE, x ∈ R ; n = −| E | 2 + ε 2 � 3 � 2 | E | 4 + 3( ∂ x E ) 2 ¯ E E + 2 | ∂ x E | 2 + E∂ 2  x ¯ E + 4 ¯ E∂ 2 x E . It possesses solitary waves E ,ε ( x, t ) = e i ( ω − αc ) t e iαx √ � � 2 βsech β ( x − ct ) , ω − αc = − ε 2 ( α 4 + β 4 − 6 α 2 β 2 ) − α 2 + β 2 where and c = ε 2 α (4 α 2 − 4 β 2 ) + 2 α . Question : can we estimate � E λ,ε − E ∞ ,ε � ? � n λ,ε − n ∞ ,ε � ?