The Cauchy problem of the Ward equation with mixed scattering data

THE CA UCHY PR OBLEM OF THE W ARD EQUA TION WITH MIXED SCA TTERING D A T A DERCHYI WU INSTITUTE OF MA THE MA TICS A CADEMIA SINICA T AIPEI, T AIW AN, R. O. C. mawudc@math.sinica.edu.t w Abstract W e solve the Cauch y problem of the W a rd eq uation with bo th c ontin uous and discrete scattering data. Keywor ds: Self-dual Y ang-Mills equation, La x pair, inv erse scattering problem, soliton solutions, Backlund transformation 1. Introduction The W ar d equation (or the modified 2 + 1 chiral model) (1.1) ∂ t  J − 1 ∂ t J  − ∂ x  J − 1 ∂ x J  − ∂ y  J − 1 ∂ y J  −  J − 1 ∂ t J, J − 1 ∂ y J  = 0 , for J : R 2 , 1 → S U ( n ), ∂ w = ∂ /∂ w , is obtained fro m a dimension reduction and a gauge fixing of the self-dual Y ang -Mills equatio n on R 2 , 2 [6], [11]. It is an int egra ble system which p os sesses the La x pair [15] (1.2)  λ∂ x − ∂ ξ − J − 1 ∂ ξ J, λ∂ η − ∂ x − J − 1 ∂ x J  = 0 with ξ = t + y 2 , η = t − y 2 . Note (1.2 ) implies that J − 1 ∂ ξ J = − ∂ x Q , J − 1 ∂ x J = − ∂ η Q . Then b y a ch ange of v ariables ( η , x, ξ ) → ( x, y , t ), (1.2) is equiv ale nt to ( ∂ y − λ∂ x )Ψ( x, y , t, λ ) = ( ∂ x Q ( x, y , t )) Ψ( x, y , t, λ ) , (1.3) ( ∂ t − λ∂ y )Ψ( x, y , t, λ ) = ( ∂ y Q ( x, y , t )) Ψ( x, y , t, λ ) (1.4) [7], and the W a rd equatio n (1.1) turns into: (1.5) ∂ x ∂ t Q = ∂ 2 y Q + [ ∂ y Q, ∂ x Q ] . The cons tr uction of solitons, and the study of the scattering prop erties of solitons of t he W a rd equation have bee n studied intensiv e ly by solving the degenerated Riemann-Hilb ert problem and studying the limiting method [15], [16], [8], [1], [2], [9]. In particular, Dai and T erng ga ve an explicit construction of all solitons of the W ard equa tion b y establishing a theory of B a cklund transfor mation [5]. On the other hand, Villarro el [14], F o k as and Ioannidou [7], Dai, T er ng and Uhlen b eck [6] inv estigate the scattering and inv erse scattering problem and solve the Cauch y problem of the W ard equation if the initial p otential is sufficiently 1 2 DERC HYI WU small. Under the sma ll data condition, the e ig enfunctions Ψ p oss esses c o nt inuous scattering data only and therefore the so lutions for the W ard equa tion do no t include the so litons in previous study . Observing the similarity b etw een Lax pairs of the W ar d equa tion a nd the AK NS system [3], [4] and transfor ming the sp ectral equation (1.3) into a ¯ ∂ -equation, the author uses the Bea ls -Coifman scheme, the L 2 -theory of the Cauch y integral o p- erator and estimation of the singular integral to solv e the scattering and inv erse scattering pro blem a nd the Cauch y problem of the W ard e q uation if the initial po tent ial p osses s es purely contin uous scattering data [18]. The purp ose of the present pap er is to apply the ab ov e existence theory and the Backlund transformation theory to solve the the Cauch y problem o f the W a rd equation with mixed scattering data. The scheme w e will follow was dev elop ed by [12], [6] to construc t AKNS flows and W ar d equations. Mor e precisely , we will generalize the re sults of [14], [7], [6], [18] by the following theorem: Theorem 1.1. If Q 0 ∈ P ∞ ,k, 1 , k ≥ 7 (se e Definition 1), and the p oles of Ψ 0 (in- cluding the multiplicity) ar e fin ite and c ontaine d in C \ R , then t he Cauchy pr oblem of t he War d e quation (1.5) with initial c ondition Q ( x, y , 0) = Q 0 ( x, y ) admits a smo oth glob al solution satisfying: for i + j + h ≤ k − 4 , i 2 + j 2 > 0 , ∂ x Q ( x, y , t ) ∈ su ( n ) , ∂ i x ∂ j y ∂ h t Q, ∂ t Q, Q ∈ L ∞ , ∂ i x ∂ j y ∂ h t Q, ∂ t Q, Q → 0 , a s x, y , t → ∞ . 2. The Ca uchy pr oblem with purel y continuous sca ttering da t a W e summarize the results o f [1 8] w hich are nece s sary in proving Theor em 1.1. First of all, define the functional s paces Definition 1. P ∞ ,k 1 ,k 2 = { q x ( x, y ) : R × R → su ( n ) | | ξ i y s b q | L 1 ( dξ dy ) , || ξ h b q ( ξ , y ) | L 1 ( y ) | L 2 ( dξ ) , | ∂ j x ∂ l y q | L ∞ , sup y | ∂ j x ∂ l y q | L 1 ( dx ) , | ∂ j x ∂ l y q | L 1 ( dxdy ) < ∞ for 1 ≤ i ≤ max { 5 , k 1 } , 0 ≤ j, l ≤ max { 5 , k 1 } , 1 ≤ h ≤ k 1 , 0 ≤ s ≤ k 2 } . DH k = { f | ∂ i x f ( x, y ) ar e uniformly b ounde d in L 2 ( R , dx ) , 0 ≤ i ≤ k . } Theorem 2.1. L et Q ∈ P ∞ ,k, 0 , k ≥ 2 . Then t her e is a b oun de d s et Z ⊂ C \ R such that • Z ∩ ( C \ R ) is discr ete in C \ R ; • F or λ ∈ C \ ( R ∪ Z ) , the pr oblem (1.3) has a unique solution Ψ and Ψ − 1 ∈ DH k ; • F or ( x, y ) ∈ R × R , t he eigenfunction Ψ( x, y , · ) is mer omorphic in λ ∈ C \ R with p oles pr e cisely at the p oints of Z ; • Ψ( x, y , λ ) satisfies: lim | x |→∞ Ψ( · , y , λ ) = 1 , lim | y |→∞ Ψ( x, · , λ ) = 1 , for λ ∈ C \ ( R ∪ Z ) , (2.1) Ψ( x, y , · ) tends to 1 uniformly as | λ | → ∞ . (2.2) THE CA UCHY PROBLEM OF THE W ARD EQUA TION WITH MIXED SCA TTERING DA T A 3 • Ψ( x, 0 , λ ) satisfies: Ψ − 1 ar e uniformly b ounde d in D H k for λ ∈ C \  R ∪ λ j ∈ Z D ǫ ( λ j )  . (2.3) F or any z j ∈ C \ R , fi xing ǫ k for ∀ k 6 = j and letting ǫ j → 0 , t hese L 2 ( dx ) -norms incr e ase as C j ǫ − h j j with uniform c onstants C j , h j > 0 ; Ψ − 1 → 0 in D H k as λ → ∞ . (2.4) Wher e ǫ > 0 is any given c onstant, D ǫ ( λ j ) is t he disk of r adius ǫ c enter e d at λ j . Final ly, for λ / ∈ Z , det Ψ( x, y , λ ) ≡ 1 , (2.5) Ψ( x, y , λ )Ψ( x, y , ¯ λ ) ∗ = I . (2.6) Pr o of. See Theorem 1.1, Co r ollary 4.1 and Lemma 4.14 in [18].  Theorem 2. 2. F or Q ∈ P ∞ ,k, 1 , k ≥ 7 , if Ψ ± exist, then ther e exists uniquely a function v ( x, y , λ ) s uch that Ψ + ( x, y , λ ) = Ψ − ( x, y , λ ) v ( x, y , λ ) , λ ∈ R. Mor e over, v satisfies the algebr aic c onstr aints: det ( v ) ≡ 1 , (2.7) v = v ∗ > 0 , (2.8) and the analytic c onstra ints: for i + j ≤ k − 4 , L λ v = 0 , v ( x, y , λ ) = v ( x + λy , λ ) for ∀ x, y ∈ R , (2.9) ∂ i x ∂ j y ( v − 1) ar e uniformly b ounde d in L ∞ ∩ L 2 ( R , dλ ) ∩ L 1 ( R , dλ ); (2.10) ∂ i x ∂ j y ( v − 1) → 0 uniformly in L ∞ ∩ L 2 ( R , dλ ) ∩ L 1 ( R , dλ ) (2.11) as | x | or | y | → ∞ ; ∂ λ v are in L 2 ( R , dλ ) and the norms dep end co n tinuously o n x , y . (2.12) Wher e L λ = ∂ y − λ∂ x . Pr o of. Note it is of no harm to r e pla ce the condition Z = Z (Ψ) = φ by the ex is tence of Ψ ± in the pro o f of Theorem 1 .2 in [18].  Theorem 2. 3. S u pp ose v ( x, y , λ ) satisfies (2.7), (2.8), and (2.10)-(2.12), k ≥ 7 . Then ther e exists a unique solution Ψ( x, y , · ) for the Ri emann-Hilb ert pr oblem ( λ ∈ R , v ( x, y , λ )) such that (2.13) Ψ − 1 , ∂ x Ψ , ∂ y Ψ ar e uniformly b ounde d in L 2 ( R , dλ ) . In additio n, for e ach fixe d λ / ∈ R , and i + j ≤ k − 4 , ∂ i x ∂ j y Ψ ∈ L ∞ ( dxdy ) , (2.14) ∂ i x ∂ j y (Ψ − 1) → 0 in L ∞ ( dxdy ) , as x or y → ∞ . (2.15) Mor e over, det Ψ( x, y , λ ) ≡ 1 , (2.16) Ψ( x, y , t, λ )Ψ( x, y , t, ¯ λ ) ∗ ≡ 1 . (2.17) Pr o of. See Theo r em 1 .3 a nd Lemma 6.6 in [18]. Note (2.9) is not used in the pr o of of Theorem 1.3 in [18]. It is only us ed in justifying the existence of Q ( x, y ) (see the pro of of Theorem 1.4 in [18]).  4 DERC HYI WU Theorem 2.4. If Q 0 ∈ P ∞ ,k, 1 , k ≥ 7 , and ther e ar e no p oles of the eigenfunction Ψ 0 of Q 0 , then the Cauchy pr oblem of t he War d e quation (1.5) with initial c ondition Q ( x, y , 0) = Q 0 ( x, y ) admits a smo oth glob al solution satisfying: for i + j + h ≤ k − 4 , i 2 + j 2 > 0 , ∂ x Q ( x, y , t ) ∈ su ( n ) , ∂ i x ∂ j y ∂ h t Q, ∂ t Q, Q ∈ L ∞ , ∂ i x ∂ j y ∂ h t Q, ∂ t Q, Q → 0 , a s x, y , t → ∞ . 3. The Ca uchy problem with purel y discrete sca ttering da t a W e r e view results of [5] which ar e necessary for the pro of of Theor em 1.1. Definition 2. D = { f : C → M n ( C ) satisfies the fol lowing c onditions: • f ( ¯ λ ) ∗ f ( λ ) = 1 , lim λ →∞ f = I . • f is mer omorphic in C \ R with p ossible fi nitely many p oles. • f ± exist. } ; D c = { f ∈ D such that f is holomorphic in C \ R and f ± exist. } ; D r = { f ∈ D such that f is r ational. } . Z ( f ) = the set of po les of f , for f ∈ D . Theorem 3. 1. F or ψ ∈ D r and Z ( f ) ⊂ C + , we c an factorize ψ as a pr o duct of f 1 · · · f p . Wher e f i ∈ D r , Z ( f i ) = { z i } ⊂ C + and z i 6 = z j if i 6 = j . Pr o of. See Theorem 5.4 in [13] and Coro llary 4.5 in [5].  Definition 3. L et A 0 ( x, y ) , A ( x, y , t ) , and B ( x, y , t ) ∈ M n ( C ) . • η 0 ( x, y ) is c al le d a fundamental solution of the syst em ( ∂ y − z ∂ x ) f = A 0 ( x, y ) f , (3.1) if for ∀ V ( x, y ) satisfies (3.1), t her e exists H ( x + z y ) such that V ( x, y ) = η 0 ( x, y ) H ( x + z y ) . • η ( x, y , t ) is c al le d a fundamental solution of ( ∂ y − z ∂ x ) f = A ( x, y , t ) f , (3.2) ( ∂ t − z ∂ y ) f = B ( x, y , t ) f , (3.3) if for ∀ V ( x, y , t ) satisfies (3.2), (3.3), ther e exists H ( x + z y + z 2 t ) such that V ( x, y , t ) = η ( x, y , t ) H ( x + z y + z 2 t ) . The next g oal is to solve the following Cauc hy problem (3.4 )-(3.13). Theorem 3 .2. Supp ose ther e exist A 0 ( x, y ) , Ψ 0 ( x, y , λ ) su ch that (3.4) ( ∂ y − λ∂ x )Ψ 0 ( x, y , λ ) = A 0 ( x, y )Ψ 0 ( x, y , λ ) , THE CA UCHY PROBLEM OF THE W ARD EQUA TION WITH MIXED SCA TTERING DA T A 5 and Ψ 0 ( x, y , · ) ∈ D r , Z (Ψ 0 ) = { z } with mu ltiplicity k , (3.5) ∂ i x ∂ j y Ψ 0 ∈ L ∞ for i + j ≤ N , and λ ∈ C \ { z } fix e d, (3.6) lim x 2 + y 2 →∞ ∂ i x ∂ j y (Ψ 0 − C 1 ) = 0 for i + j ≤ N , and λ ∈ C \{ z } fix e d. (3.7) Wher e C 1 is indep endent of x , y . Then ther e exist Ψ ( x, y , t, λ ) , A ( x, y , t ) , B ( x, y , t ) such t hat ( ∂ y − λ∂ x )Ψ( x, y , t, λ ) = A ( x, y , t )Ψ( x, y , t, λ ) , (3.8) ( ∂ t − λ∂ y )Ψ( x, y , t, λ ) = B ( x, y , t )Ψ( x, y , t, λ ) , (3.9) Ψ( x, y , 0 , λ ) = Ψ 0 ( x, y , λ ) , A ( x, y , 0) = A 0 ( x, y ) , (3.10) and Ψ( x, y , t, · ) ∈ D r , Z (Ψ) = { z } with multiplicity k , (3.1 1 ) ∂ i x ∂ j y ∂ h t Ψ ∈ L ∞ for i + j + h ≤ N , and λ ∈ C \{ z } fi xe d, (3.12) lim x 2 + y 2 + t 2 →∞ ∂ i x ∂ j y ∂ h t (Ψ 0 − C 1 ) = 0 for i + j + h ≤ N , and λ ∈ C \{ z } fixe d. (3.13) Pr o of. This theorem is indeed a r ephrase of a very decen t result (Theorem 8 .6) in [5]. F or c o nv enience, the pro o f is sk etched here. The theorem will be prov ed b y simult aneous ly es tablishing (3.8)-(3.1 3), and the following statements (3.14)-(3.18): There exis ts a fundament al solution, η 0 ( x, y ) , of (3.1); (3.14) There exis ts a fundament al solution, η ( x, y , t ) , of (3.2) a nd (3.3); (3.15) η ( x, y , 0) = η 0 ( x, y ); (3.16) ∂ i x ∂ j y ∂ h t η ∈ L ∞ for i + j + h ≤ N , and λ ∈ C \ { z } fixe d; (3.17) lim x 2 + y 2 + t 2 →∞ ∂ i x ∂ j y ∂ h t ( η − C 2 ) = 0 for i + j + h ≤ N , and λ ∈ C \{ z } fi xe d. (3.18) Where C 2 is indep endent of x , y . Suppo se k = 1. By the results of [13], [15], [1 7], one has Ψ 0 ( x, y , λ ) = 1 + z − ¯ z λ − z π ⊥ 0 ( x, y ) , (3.19) A 0 ( x, y ) = ( z − ¯ z ) ∂ x π 0 . (3.20) Where π 0 ( x, y ) : R × R → C n , π ∗ 0 = π 0 , π 2 0 = π 0 , Im π 0 is spanned by columns of (3.21) V ( x + z y ) , V : C → M n × r is holomorph ic exc ept at { p 1 , · · · , p s } , and M n × r denotes the space o f rank r co mplex n × r ma trices. Note by (3.6), one can find another ˜ V ( x + z y ), ˜ V : C → M n × r is holomor phic in the neighborho o d of p 1 , · · · , p s , a nd Im π 0 is spanned by columns of ˜ V . Hence (3.1 9)-(3.21) hold for ∀ x, y . Now let us define Ψ( x, y , t, λ ) = 1 + z − ¯ z λ − z π ⊥ ( x, y , t ) , π ( x, y , t ) : R × R × R → C n , π ∗ = π , π 2 = π , Im π is sp ann e d by c olumn s of V ( x + z y + z 2 t ) , 6 DERC HYI WU then (3.8)-(3.13) ar e satisfied with (3.22) A ( x, y , t ) = ( z − ¯ z ) ∂ x π , B ( x, y , t ) = ( z − ¯ z ) ∂ y π . T o prov e (3.14)-(3.18), b y a Dai and T erng ’s constructio n (Theorem 8.6 in [5]), one can find a fundamental solution η ( x, y , t ) of (3.2) and (3.3) by defining η ( x, y , t ) = [ ξ 1 · · · ξ n 1 u n 1 +1 · · · u n ] , ξ j = lim ǫ → 0 Ψ( x, y , t, z + ǫ )( a j ( x + ( z + ǫ ) y + ( z + ǫ ) 2 t )) , u j = lim ǫ → 0 Ψ( x, y , t, z + ǫ )( ǫb j ( x + ( z + ǫ ) y + ( z + ǫ ) 2 t )) , a 1 , · · · , a n 1 span Im π , π ⊥ ( b n 1 +1 ) , · · · , π ⊥ ( b n ) span Im π ⊥ . Defining η 0 ( x, y ) b y (3.16) and using (3.8)-(3.10), (3.1 2), (3 .1 3), the inv ertibility of η , one ha s (3.14), (3.17), and (3.1 8). Now suppos e (3.8)-(3.18) hold for k and as sume that Z (Ψ 0 ) = { z } with mul- tiplicit y k + 1 . Using the minimal factorization proper ty of elements in D r (see Section 4 and Theo rem 8.1 in [5]), o ne would o btain Ψ 0 ( x, y , λ ) = g z ,π k +1 , 0 Ψ k, 0 ( x, y , λ ) , (3.23) g z ,k +1 , 0 ( x, y , λ ) = 1 + z − ¯ z λ − z π ⊥ k +1 , 0 ( x, y ) , (3.24) ( ∂ y − λ∂ x )Ψ k, 0 = A k, 0 ( x, y )Ψ k, 0 , (3.2 5 ) and Ψ k, 0 ∈ D r , Z (Ψ k, 0 ) = { z } with mult iplicity k , (3.26) ∂ i x ∂ j y g z ,π k +1 , 0 , ∂ i x ∂ j y Ψ k, 0 ∈ L ∞ for i + j ≤ N , and λ ∈ C \ { z } fix e d; (3.27) ∂ i x ∂ j y  g z ,π k +1 , 0 − C 3  → 0 , ∂ i x ∂ j y (Ψ k, 0 − C 4 ) → 0 as | x | , or | y | → ∞ , (3.28) and λ ∈ C \{ z } fixe d. Where C 3 , C 4 are independent of x , y . Conditions (3.2 3)-(3.28), (3.4), and Prop o- sition 7.1, 7.4 in [5] conclude that ther e exist τ 1 , 0 ( x, y ) , · · · , τ n k +1 , 0 ( x, y ) sp an Im π k +1 , 0 , (3.29) ( ∂ y − z ∂ x ) τ j, 0 = A k, 0 ( x, y ) τ j, 0 , (3.30) ∂ i x ∂ h y τ j, 0 ∈ L ∞ for i + h ≤ N , and λ ∈ C \ { z } fix e d, (3.31) ∂ i x ∂ h y ( τ j, 0 − c j ) → 0 as | x | , or | y | → ∞ , and λ ∈ C \{ z } fixe d. (3.32) Where c j are indep endent of x , y . Besides, by (3.25)-(3.28), and the induction hypothesis, there exis ts a fundamen ta l so lution η k, 0 ( x, y ) of (3.30). Hence there ar e H 1 , · · · , H n k +1 : C → M n × 1 such that τ j, 0 ( x, y ) = η k, 0 ( x, y ) H j ( x + z y ) , (3.33) ∂ i x ∂ h y H j ∈ L ∞ for i + h ≤ N , and λ ∈ C \{ z } fi xe d, (3.34) ∂ i x ∂ h y  H j − c ′ j  → 0 as | x | , or | y | → ∞ , and λ ∈ C \{ z } fixe d. (3 .3 5) Where c ′ j are indep endent of x , y . Moreov er, b y induction, one ca n find Ψ k ( x, y , t, λ ) solution o f (3.8)-(3.1 3), η k a so lution of (3.15)-(3.18) with Ψ 0 , A 0 , A , B , η 0 , η THE CA UCHY PROBLEM OF THE W ARD EQUA TION WITH MIXED SCA TTERING DA T A 7 replaced b y Ψ k, 0 , A k, 0 , A k , B k , η k, 0 , η k . So by (3.3 3), (3.29), and (3.23), one ca n extend τ j, 0 ( x, y ), π k +1 , 0 , Ψ by τ j ( x, y , t ) = η k ( x, y , t ) H ( x + z y + z 2 t ) , π k +1 ( x, y , t ) , a Hermitia n pro jection on the space spanned by τ j , Ψ( x, y , t, λ ) = g z ,π k +1 Ψ k ( x, y , t, λ ) . Therefore, we can apply Lemma 8.2, and Prop os ition 7.1 in [5] to prov e (3.8)-(3.1 3) for the cas e of k + 1. T o prove (3.1 4)-(3.18) for the case o f k + 1, by a Dai and T e r ng’s co nstruction, one can find a fundamental solution η ( x, y , t ) of (3.2) and (3.3) by defining η ( x, y , t ) =  ξ 1 · · · ξ n k +1 ζ 1 · · · ζ n − n k +1  , ξ j = lim ǫ → 0 g z ,π k +1 ( x, y , t, z + ǫ )  a j ( x + ( z + ǫ ) y + ( z + ǫ ) 2 t )  , a 1 , · · · , a n k +1 span Im π k +1 , ζ 1 , · · · , ζ n − n k +1 sp an π ⊥ k +1 η k . Defining η 0 ( x, y ) b y (3.16) and using (3.8)-(3.10), (3.1 2), (3 .1 3), the inv ertibility of η , one ha s (3.14), (3.17), and (3.1 8).  Theorem 3. 3. Supp ose b oth f ( x, y , t, λ ) and g ( x, y , t, λ ) satisfy (3.8), (3.9) with differ ent A , B . If Z ( g ) = { z } with multiplicity of k , and f is holomorphic and non-de gener ate at z , ¯ z . Then ther e exist un ique ˜ f and ˜ g such that • Z ( ˜ g ) = { z } with mu ltiplicity of k , ˜ f is holomorphic and n on-de gener ate at z , ¯ z . • Ψ = ˜ f g = ˜ g f satisfies (3.8), (3.9) with n ew A , B . Mor e over, if for i + j + h ≤ N , and λ ∈ C \ ( { z } ∪ Z ( f )) fixe d, ∂ i x ∂ j y ∂ h t f , ∂ i x ∂ j y ∂ h t g ∈ L ∞ (3.36) lim x 2 + y 2 + t 2 →∞ ∂ i x ∂ j y ∂ h t ( f − C 1 ) = lim x 2 + y 2 + t 2 →∞ ∂ i x ∂ j y ∂ h t ( g − C 2 ) = 0 (3.37) then for i + j + h ≤ N , and λ ∈ C \ ( { z } ∪ Z ( f )) fixe d, ∂ i x ∂ j y ∂ h t Ψ ∈ L ∞ (3.38) lim x 2 + y 2 + t 2 →∞ ∂ i x ∂ j y ∂ h t (Ψ − C 3 ) = 0 . (3.39) Wher e C 1 , C 2 , C 3 ar e indep endent of x , y , t . Pr o of. See Theorem 6.1 in [5]. Note the regular it y pro pe r ty (3 .3 8) and the asymp- totic prop erty (3.39) ar e prov ed by using the proper ty ”‘ eac h comp onent of the minimal factorization o f g satisfies (3.36), and (3.37)”’.  The same ar gument yields Theorem 3. 4 . Su pp ose b oth f ( x, y , λ ) and g ( x, y , λ ) satisfy (3.4) with differ ent A 0 . If Z ( f ) = { z } wi th multiplicity of k , and g is holomorphic and non-de gener ate at z , ¯ z . Then t her e exist unique ˜ f and ˜ g such that • Z  ˜ f  = { z } with mult iplici ty of k , ˜ g is holomo rphic and n on-de gener ate at z , ¯ z . • Ψ = ˜ f g = ˜ g f satisfies (3.4) with a new A 0 . 8 DERC HYI WU 4. The Ca uchy problem: Mixed sca ttering d a t a Lemma 4.1. F or Q ∈ P ∞ ,k, 1 , k ≥ 7 , if Z = Z (Ψ ) is fi nite and Z ∩ R = φ , then ther e exist un iquely f , ˜ f ∈ D c , g , ˜ g ∈ D r , such that (4.1) Ψ = ˜ f g = ˜ g f . Mor e over, for i + j ≤ k − 4 , and λ ∈ C \ ( R ∪ Z ) fix e d, ∂ i x ∂ j y f , ∂ i x ∂ j y g , ∂ i x ∂ j y ˜ f , ∂ i x ∂ j y ˜ g ∈ L ∞ ; (4.2) ∂ i x ∂ j y ( f − 1) , ∂ i x ∂ j y ( g − 1) , ∂ i x ∂ j y ( ˜ f − 1) , ∂ i x ∂ j y ( ˜ g − 1) → 0 , as x, y → ∞ ; (4.3) det f = det g = det ˜ f = det ˜ g = 1 . (4.4) Pr o of. Since Z is finite. Theor em 2.1 implies the existence of Ψ ± . Hence Theorem 2.2 implies that: ∃ v c ( x, y , λ ), λ ∈ R such that Ψ + = Ψ − v c ; (4.5) v c satisfies the algebr aic and analytic co ns traints (2.7)-(2.12). By Theorem 2.3, we ca n find a function f ( x, y , λ ), such that f satisfies (4.2)-(4.4), f ( x, y , · ) ∈ D c , and (4.6) f + = f − v c . Hence if we define ˜ g = Ψ f − 1 , then Theor em 2.1, f ( x, y , · ) ∈ D c , (4.5), (4.6) and Z is finite imply ˜ g ( x, y , λ ) ∈ D r and ˜ g ( x, y , · ) satisfies (4.2)- (4.4). Hence we pro ve the existence of f a nd ˜ g . Besides, using a similar ar gument as that in the pro of of Theor em 2.2, ther e exists ˜ v c ( x, y , λ ), λ ∈ R such that Ψ + = ˜ v c Ψ − ; (4.7) ˜ v c satisfies the a lgebraic and analytic co ns traints (2.7)- (2 .12) except (2.9). By a similar arg ument as that in the pro of of Theor e m 2.3, there ex ists a function ˜ f ( x, y , λ ), such tha t ˜ f satisfies (4.2), (4.3 ), ˜ f ( x, y , · ) ∈ D c , and (4.8) ˜ f + = ˜ v c ˜ f − . Hence if we define g = ˜ f − 1 Ψ, then Theo rem 2.1, ˜ f ( x, y , · ) ∈ D c , (4.7), (4.8) and Z is finite imply g ( x, y , · ) ∈ D r and g s atisfies (4.2)-(4.4). Suppo se Ψ = ˜ g 1 f 1 = ˜ g f wher e f 1 , ˜ g 1 satisfy the statement of the theorem. Hence ˜ g − 1 ˜ g 1 = f f − 1 1 . The r ight hand side is holo morphic in C \ R , the le ft hand side is c o nt inuous a cross the r eal line and tends to 1 at infinit y . Thu s the Liouville theorem yields ˜ g − 1 ˜ g 1 = f f − 1 1 = 1 and we prove the uniqueness of f a nd ˜ g . The uniqueness of ˜ f and g can b e obtained by analogy .  Lemma 4. 2. Supp ose Ψ satisfies (3.4 ), Ψ( x, y , · ) ∈ D , and Z (Ψ ) ⊂ C + . If Ψ = ψ 1 g 1 = g 2 ψ 2 , (4.9) g i ∈ D r , Z ( g i ) = { z i } ⊂ Z (Ψ) , z i ∈ C + , (4.10) z i / ∈ Z ( ψ i ) , (4.11) for i = 1 , 2 . Then ( L λ g 1 ) g − 1 1 , ( L λ ψ 2 ) ψ − 1 2 ar e indep endent of λ . THE CA UCHY PROBLEM OF THE W ARD EQUA TION WITH MIXED SCA TTERING DA T A 9 Wher e L λ = ∂ y − λ∂ x . Pr o of. The λ -indep endence of ( L λ g 1 ) g − 1 1 has b een prov ed b y Theor em 6.5 in [5]. W e can us e the same metho d to s how the λ -indep endence o f ( L λ ψ 2 ) ψ − 1 2 . That is , by (4.9) a nd a direct co mputation, we ha ve (4.12) g − 1 2  ( L λ Ψ) Ψ − 1 − ( L λ g 2 ) g − 1 2  g 2 = ( L λ ψ 2 ) ψ − 1 2 . Note (4.1 0), (4.11) imply that ( L λ ψ 2 ) ψ − 1 2 is holomorphic at z , ¯ z . Using (4.1 0), (3.4) and the left side of (4.12), one has that ( L λ ψ 2 ) ψ − 1 2 is holomor phic outside { z , ¯ z } . Moreover, ( L λ ψ 2 ) ψ − 1 2 is bo unded at ∞ b y g 2 ∈ D r , (3.4), and the left side of (4.12). By the Liouville’s theor em, ( L λ ψ 2 ) ψ − 1 2 is indep endent of λ .  Pro of o f Theorem 1.1 : Since (1.5) is the compatibility condition of (1.3), (1.4). T o prove Theor em 1.1, it is sufficient to show the following theore m. Theorem 4. 1. F or Q 0 ∈ P ∞ ,k, 1 , k ≥ 7 , if Z = Z (Ψ 0 ) is finite and Z ∩ R = φ , then ∃ Ψ ( x, y , t, λ ) , Q ( x, y , t ) such that ( ∂ y − λ∂ x )Ψ( x, y , t, λ ) = ( ∂ x Q ( x, y , t )) Ψ( x, y , t, λ ) , (4.13) ( ∂ t − λ∂ y )Ψ( x, y , t, λ ) = ( ∂ y Q ( x, y , t )) Ψ( x, y , t, λ ) , (4.14) Q ( x, y , 0) = Q 0 ( x, y ) . Mor e over, for i + j + h ≤ N , and λ ∈ C \ ( R ∪ Z ) fix e d, ∂ i x ∂ j y ∂ h t Ψ ∈ L ∞ , (4.15) ∂ i x ∂ j y ∂ h t (Ψ − 1) → 0 as x , or y , or t → ∞ . (4.16) Pr o of. By Lemma 4 .1, one has Ψ 0 = ˜ f 0 g 0 = ˜ g 0 f 0 with f 0 , ˜ f 0 ∈ D c , g 0 , ˜ g 0 ∈ D r and (4.2), (4.3) b eing satisfied. Since Z = Z (Ψ 0 ) is finite a nd Z ∩ R = φ . Succes sively m ultiplying Ψ 0 by facto rs as λ − z j λ − ¯ z j , z j ∈ Z (Ψ 0 ) ∩ C − , one obtains ( ∂ y − λ∂ x )Ψ ′ 0 ( x, y , λ ) = A ′ 0 ( x, y )Ψ ′ 0 ( x, y , λ ) , (4.17) Ψ ′ 0 = ˜ f 0 g ′ 0 = ˜ g ′ 0 f 0 , (4.18) f 0 , ˜ f 0 ∈ D c , g ′ 0 , ˜ g ′ 0 ∈ D r , (4.19) Z (Ψ ′ 0 ) = Z ( g ′ 0 ) = Z ( ˜ g ′ 0 ) ⊂ C + , (4.20) and for i + j ≤ k − 4, and λ ∈ C \ ( R ∪ Z ), ∂ i x ∂ j y f 0 , ∂ i x ∂ j y g ′ 0 ∈ L ∞ , (4.21) lim x 2 + y 2 →∞ ∂ i x ∂ j y ( f 0 − I ) = lim x 2 + y 2 →∞ ∂ i x ∂ j y ( g ′ 0 − C I ) = 0 (4.22) Where C is a scalar whic h is indep endent of x , y . Moreover, by (4.19), (4.20), and Theorem 3.1, one can factorize g ′ 0 = g ′ 0 , 1 · · · g ′ 0 ,k , ˜ g ′ 0 = ˜ g ′ 0 , 1 · · · ˜ g ′ 0 ,k , g ′ 0 ,j , ˜ g ′ 0 ,j ∈ D r , (4.23) Z ( g ′ 0 ,j ) = { α j } ⊂ C + , Z ( ˜ g ′ 0 ,j ) = { β j } ⊂ C + , (4.24) α i 6 = α j , β i 6 = β j , if i 6 = j . (4.25) 10 DERC HYI WU T o prov e the theorem, it is equiv a lent to showing the solv ability of the follo wing Cauch y problem ( ∂ y − λ∂ x )Ψ ′ ( x, y , t, λ ) = A ′ ( x, y , t )Ψ ′ ( x, y , t, λ ) , (4.26) ( ∂ t − λ∂ y )Ψ ′ ( x, y , t, λ ) = B ′ ( x, y , t )Ψ ′ ( x, y , t, λ ) , (4.27) Ψ ′ ( x, y , 0 , λ ) = Ψ ′ 0 ( x, y , λ ) , A ′ ( x, y , 0) = A ′ 0 ( x, y ) , (4.28) as well for i + j + h ≤ k − 4, and λ ∈ C \ ( R ∪ Z ) fixed, ∂ i x ∂ j y ∂ h t Ψ ′ ∈ L ∞ (4.29) ∂ i x ∂ j y ∂ h t (Ψ ′ − C I ) → 0 as | x | , or | y | , or | t | → ∞ . (4.30) Since (4.1 3), (4.14) follow by computing the compa tibilit y conditions of (4.26), (4.27). W e are going to justify the solv a bilit y b y inducing on the length k (the m ultiplicity) in (4.23). F or k = 1 , b y (4.17), (4.18), Lemma 4.2, there exist A ′ r, 0 ( x, y ), A c, 0 ( x, y ) such that ( ∂ y − λ∂ x ) g ′ 0 ( x, y , λ ) = A ′ r, 0 ( x, y ) g ′ 0 ( x, y , λ ) , (4.31 ) ( ∂ y − λ∂ x ) f 0 ( x, y , λ ) = A c, 0 ( x, y ) f 0 ( x, y , λ ) . ( 4.32 ) By (4.31), (4.19), (4 .21), (4.22) and Theo rem 3.2, one can find ( ∂ y − λ∂ x ) g ′ ( x, y , t, λ ) = A ′ r ( x, y , t ) g ′ ( x, y , t, λ ) , (4.33) ( ∂ t − λ∂ y ) g ′ ( x, y , t, λ ) = B ′ r ( x, y , t ) g ′ ( x, y , t, λ ) , (4 .3 4) g ′ ( x, y , 0 , λ ) = g ′ 0 ( x, y , λ ) , A ′ r ( x, y , 0) = A ′ r, 0 ( x, y ) , (4.35 ) g ′ ( x, y , t, · ) ∈ D r , Z ( g ′ ( x, y , t, · )) = { α 1 } , (4.36) and for i + j + h ≤ k − 4, λ ∈ C \{ z } fix e d, ∂ i x ∂ j y ∂ h t g ′ ∈ L ∞ (4.37) lim x 2 + y 2 + t 2 →∞ ∂ i x ∂ j y ∂ h t ( g ′ − C I ) = 0 . (4.38) By (4.32), (4.19), (4 .21), (4.22), Lemma 4 .1, and Theorem 2.4, one can find ( ∂ y − λ∂ x ) f ( x, y , t, λ ) = A c ( x, y , t ) f ( x, y , t, λ ) , (4.39) ( ∂ t − λ∂ y ) f ( x, y , t, λ ) = B c ( x, y , t ) f ( x, y , t, λ ) , (4.40) f ( x, y , 0 , λ ) = f 0 ( x, y , λ ) , A c ( x, y , 0) = A c, 0 ( x, y ) , (4.41) f ( x, y , t, · ) ∈ D c , (4.42) and for i + j + h ≤ k − 4, λ ∈ C \ R , ∂ i x ∂ j y ∂ h t f ∈ L ∞ , (4.43) lim x 2 + y 2 + t 2 →∞ ∂ i x ∂ j y ∂ h t ( f − I ) = 0 . (4.44) Therefore, applying Theor em 3.3, we can find a unique (4.45) Ψ ′ ( x, y , t, λ ) = ˜ f g ′ = ˜ g ′ f such that Ψ ′ satisfies (4.26), (4 .27), (4.29), (4.30), and Z ( ˜ g ′ ( x, y , t, · )) = { α 1 } . One the o ther hand, by (4.18), (4.19), (4.31), (4.32), and Theo rem 3.4, we ha ve Ψ 0 ( x, y , λ ), ˜ f 0 ( x, y , λ ), ˜ g ′ 0 ( x, y , λ ) are the unique functions which satisfiy (4.17). Since g ′ ( x, y , 0 , λ ) = g ′ 0 ( x, y , λ ), f ( x, y , 0 , λ ) = f 0 ( x, y , λ ), we derive Ψ ′ ( x, y , 0 , λ ) = THE CA UCHY PROBLEM OF THE W ARD EQUA TION WITH MIXED SCA TTERING DA T A 11 Ψ ′ 0 ( x, y , λ ). As a r e sult, A ′ ( x, y , 0) = A ′ 0 ( x, y ) by (4.17), (4.26) and we prove the theorem in cas e of k = 1. F or k > 1, we define ˜ f k − 1 , 0 = ˜ f 0 g ′ 0 , 1 · · · g ′ 0 ,k − 1 , (4.46) f k − 1 , 0 = ˜ g ′ 0 , 2 · · · ˜ g ′ 0 ,k f 0 . (4.47) Then (4.18), (4.23), (4.4 6), (4.47) imply Ψ ′ 0 = ˜ f k − 1 , 0 g ′ 0 ,k = ˜ g ′ 0 , 1 f k − 1 , 0 , (4.48) Z ( g ′ 0 ,k ) = { α } ⊂ C + , α / ∈ Z ( ˜ f k − 1 , 0 ) ⊂ C + , (4.49) Z ( ˜ g ′ 0 , 1 ) = { β } ⊂ C + , β / ∈ Z ( f k − 1 , 0 ) ⊂ C + . (4.50) So by (4.17), (4.48)-(4.50), a nd Lemma 4.2, there exist A ′′ L, 0 ( x, y ), A ′′ R, 0 ( x, y ) such that ( ∂ y − λ∂ x ) g ′ 0 ,k ( x, y , λ ) = A ′′ r, 0 ( x, y ) g ′ 0 ,k ( x, y , λ ) , ( ∂ y − λ∂ x ) f k − 1 , 0 ( x, y , λ ) = A ′′ c, 0 ( x, y ) f k − 1 , 0 ( x, y , λ ) . Hence one can pr ove the theorem in case of k > 1 by rep eating the previous a r- guments except using the induction hypo thesis to o btain the solutio n of the Cauch y problem ( ∂ y − λ∂ x ) f k − 1 ( x, y , t, λ ) = A ′′ c ( x, y , t ) f k − 1 ( x, y , t, λ ) , ( ∂ t − λ∂ y ) f k − 1 ( x, y , t, λ ) = B ′′ c ( x, y , t ) f k − 1 ( x, y , t, λ ) , f k − 1 ( x, y , 0 , λ ) = f k − 1 , 0 ( x, y , λ ) , A ′′ c ( x, y , 0) = A ′′ c, 0 ( x, y ) .  W e conclude this re p o r t by a brief re mark on examples of Q 0 ∈ P ∞ ,k, 1 , k ≥ 7, and Z (Ψ 0 ) < ∞ . First we let v ( x, y , λ ) = v ( x + λy , λ ) satisfy det( v ) = 1 , v = v ∗ > 0 , v − 1 ∈ S , and for ∀ i , j, h ≥ 0, ∂ i x ∂ j y ∂ h λ ( v − 1) ∈ L 2 (R , dλ ) ∩ L 1 (R , dλ ) uniformly , ∂ i x ∂ j y ∂ h λ ( v − 1) → 0 in L 2 (R , dλ ) uniformly, as | x | , | y | → ∞ . Here S is the spac e of Sch wartz functions. W e can solve the inv er se problem a nd obtain ψ 0 ∈ S by the a rgument in proving Theorem 2.1. 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