Melnikov Analysis for a Singularly Perturbed DSII Equation

Transcription

1 Melnikov Analysis for a Singularly Perturbed DSII Equation By Y. Charles Li Rigorous Melnikov analysis is accomplished for Davey Stewartson II equation under singular perturbation. Unstable fiber theorem and center-stable manifold theorem are established. The fact that the unperturbed homoclinic orbit, obtained via a Darboux transformation, is a classical solution, leads to the conclusion that only local well posedness is necessary for such a Melnikov analysis. The main open issue regarding a proof of the existence of a homoclinic orbit to the perturbed Davey Stewartson II equation is discussed in the Appendix.. Introduction Under the assumption of () weakly nonlinear modulations, () slowly varying modulations, (3) propagation along nearly x-direction, and (4) a balance of these three effects; the time evolution of two-dimensional surfaces of water waves is described by the Davey Stewartson II (DSII) equations, which is a special case of the Benney Roskes equations. The DSII equations are an integrable system 3. Under decaying boundary condition, the inverse scattering transform was obtained by Fokas and Ablowitz 4. Under periodic boundary condition, finite-gap solutions of the DSII equations were obtained by Malanyuk 5. Address for correspondence: Y. C. Li, Department of Mathematics, University of Missouri, Columbia, MO 65; cli@math.missouri.edu STUDIES IN APPLIED MATHEMATICS 4: C 005 by the Massachusetts Institute of Technology Published by Blackwell Publishing, 350 Main Street, Malden, MA 048, USA, and 9600 Garsington Road, Oxford, OX4 DQ, UK.

2 86 Y. C. Li This article is a continuation of 6 on proving the existence of chaos in the perturbed DSII equations. In 6, an integrable foundation for such a study is built. Here, we shall study the perturbed system. In 6, families of homoclinic orbits asymptotic to periodic orbits are constructed via Darboux transformations. For the perturbed system, our aim is to locate a homoclinic orbit asymptotic to a saddle. Such a homoclinic orbit has two pieces. One piece is a perturbation of an unperturbed homoclinic orbit. The other is created through perturbation. This type of homoclinic orbit turns out to be responsible for the observed chaotic dynamics 7. To locate the first piece of such a homoclinic orbit, one needs Melnikov integral and in general Melnikov analysis (sometimes, called Melnikov measurement). The Melnikov analysis aims at a distance measurement between the unstable manifold of the saddle and a center-stable manifold containing the saddle. The current article completes this task. To locate the second piece of such a homoclinic orbit, one needs another distance measurement between the first piece and the stable manifold of the saddle. This will be a future work. The difficulty toward such a measurement is presented in the Appendix. Specifically, we study the Davey Stewartson II equation (DSII) under a singular perturbation { iqt = q ( q ω ) u y q iɛ q αq β, u = 4 y q (), q is a complex-valued function of the three variables (t, x, y), u is a real-valued function of the three variables (t, x, y), the external parameters ω, α, and β are all positive constants, and ɛ > 0isthe perturbation parameter, = xx yy, = xx yy, i =. Here, the first two terms in the perturbation are dissipative. The third term is a driving term. It can be a sinusoidal driver βe i t, then by a scaling q qe i t, e i t can be scaled away. Periodic boundary condition is imposed, q(t, x π/κ, y) = q(t, x, y) = q(t, x, y π/κ ), u(t, x π/κ, y) = u(t, x, y) = u(t, x, y π/κ ), κ and κ are positive constants. Even constraint is also imposed, q(t, x, y) = q(t, x, y) = q(t, x, y), u(t, x, y) = u(t, x, y) = u(t, x, y). Further constraints are placed upon ω, α, β, κ, and κ. The first one 0 < αω < β is the condition for the existence of a saddle, and the second one is the condition for the existence of only two unstable modes,

3 Melnikov Analysis 87 or { κ <κ < κ, κ < 4ω < min { κ κ, } 4κ, () { κ <κ < κ, κ < 4ω < min { κ κ, } 4κ. (3) DSII equation can be regarded as a generalization of the D cubic nonlinear Schrödinger equation (NLS) 8. In fact, it is a nontrivial generalization in the sense that the spatial part of the Lax pair of the DSII is a system of two first-order partial differential equations, for which there is no convenient Floquet discriminant to describe the isospectral property, in contrast to the case for NLS. It turns out that Melnikov vectors can still be obtained through quadratic products of Bloch eigenfunctions, instead of the gradient of the Floquet discriminant as in the NLS case. At the moment, there is no global well posedness for DSII in Sobolev spaces. In fact, DSII has finite-time blow-up solutions in H s (R ), (0 < s < ) 3, 9. Of course, DSII has local well-posedness in Sobolev spaces 0,. As mentioned before, the Melnikov measurement is built upon an unperturbed homoclinic orbit of the unperturbed DSII. Explicit expression of such a homoclinic orbit can be obtained through Darboux transformation 6. This homoclinic orbit is a classical solution. This enables us to iterate the local well-posedness result in time, and complete a Melnikov measurement. Unstable fiber theorem and center-stable manifold theorem are of course needed, and established along the same line as in Ref. 8. Novelties in regularity are introduced by the singular perturbation ɛ which generates the semigroup e ɛt. The article is organized as follows: Section deals with local theory, which includes unstable fiber theorem and center-stable manifold theorem, and we handle global theory in Section 3, which includes integrable theory and Melnikov analysis.. Local theory One can view the perturbed DSII () as an evolution equation in the q variable. First, one can define the spatial mean as π/κ π/κ q = κ κ 4π 0 Then, one may introduce the space Ḣ s as 0 qdxdy. Ḣ s ={q H s q =0}.

4 88 Y. C. Li The inverse Laplacian : Ḣ s Ḣ s is an isomorphism. The perturbed DSII () can be rewritten as iq t = q q q ω q iɛ( q αq β). (4) Denote by the D subspace.. Change of coordinates ={q x q = y q = 0}. (5) Dynamics in is the same as that given in Ref. 8. Denote by S ω the circle S ω ={q q =ω}. (6) When αω < β, there is a saddle Q ɛ near S ω in, which is located at q = Ie iθ { I = ω ɛ ω β α ω, cos θ = α I β, θ ( ) (7) 0, π. Its eigenvalues are µ, =± ɛ 4 I β sin θ ɛ ( β sin θ I ) ɛα, (8) I and θ are given in Equation (7). In the entire phase space, Q ɛ is still a saddle. Local theory will be built in a tubular neighborhood of S ω. Let Let q(t, x, y) = ρ(t) f (t, x, y)e iθ(t), f =0. I = q =ρ f, J = I ω. In terms of the new variables (J, θ, f ), Equation (4) can be rewritten as J = ɛ α(j ω ) β J ω cos θ ɛr J, (9) sin θ θ = J ɛβ J ω Rθ, (0) f t = L ɛ f V ɛ f in in 3, ()

5 Melnikov Analysis 89 L ɛ f = i f ɛ( α) f iω ( f f ), V ɛ f = ij ( f f ) iɛβ sin θ f, J ω R J = f f β cos θ J ω f J ω, R θ = ( f f ) ( f f ) ρ ( f f ) f ɛβ sin θ J ω f, J ω N = ρ f f ( f f ) f ( f f ), N 3 = ( f f ) ( f f ) f f ( f f ) f f f f ρ ( f f ) f f ɛβ sin θ J ω f f. J ω Since H s (s ) is a Banach algebra, we have R J O ( f s ), R θ O ( f s ), N s O ( f s), N3 s O ( f 3 s), (s )... Unstable fibers On (5), the saddle Q ɛ has an unstable and a stable curve, which lie in an annular neighborhood of S ω in. The width of this annular neighborhood is of order O( ɛ). DEFINITION. For any δ>0, we define the annular neighborhood of the circle S ω (6) in (5) as A(δ) ={(J,θ) J <δ}. () Unstable fibers with base points in A(δ) for some δ>0 persist, even under the singular perturbation. The spectrum of L ɛ consists of only point spectrum. The eigenvalues of L ɛ are: µ ± ξ = ɛ(α ξ ) ± ξ ξ ξ 4ω ξ, (3) ξ = (ξ, ξ ), ξ j = k j κ j,k j = 0,,,...,(j =, ), k k > 0, ξ = ξ ξ, and κ, κ, and ω satisfy the constraint () or (3).

6 90 Y. C. Li Denote µ ± (κ,0) by µ± x and µ± (0,κ ) by µ± y. The eigenfunctions corresponding to µ ± x and µ± y are u ± x = e±iϑ x cos κ x, e ±iϑ x = κ i 4ω κ, ω u ± y = e±iϑ y cos κ y, e ±iϑ y = κ ± i 4ω κ. ω Notice also that the singular perturbation ɛ ξ breaks the gap between the center spectrum and the stable spectrum. Nevertheless, the gap between the unstable spectrum and the center spectrum survives. This leads to the following unstable fiber theorem. THEOREM (Unstable Fiber Theorem). Forany s, there exists a δ>0 such that for any p A(δ), there is an unstable fiber Fp u, which is a D surface. Fp u has the following properties: () Fp u is a C smooth surface in s norm. () Fp u is also C smooth in ɛ, α, β, ω, and p in s norm, ɛ 0, ɛ 0 ) for some ɛ 0 > 0 depending on s. (3) p Fp u, F p u is tangent to span {u x, u y } at p when ɛ = 0. (4) Fp u has the exponential decay property: Let St be the evolution operator of (9) (), p Fp u, S t p S t p s Ce 3 µt p p s, t 0, µ = min{µ x, µ y }. (5) {F u p } p A(δ) forms an invariant family of unstable fibers, S t Fp u F S u t p, t T, 0, and T > 0(T can be ), such that S τ p A(δ), τ T, 0. The proof of this theorem follows from the same arguments as in Ref. 8. Notice, in particular, that Fp u H s for any s. It is this fact that leads to the C smoothness of Fp u in ɛ. Denote by W u (Q ɛ ) the unstable manifold of the saddle Q ɛ (7), which is three-dimensional. Denote by W u (Q ɛ) the unstable curve of Q ɛ in (5). W u (Q ɛ) = W u (Q ɛ ), and W u (Q ɛ) A(δ). W u (Q ɛ ) has the fiber representation W u (Q ɛ ) = Fp u. (4) Thus, W u (Q ɛ ) H s for any s. p W u (Q ɛ)

7 Melnikov Analysis 9.3. Center-stable manifold Also, due to the fact that the gap between unstable spectrum and center spectrum survives under the singular perturbation (3), a center-stable manifold persists. THEOREM (Center-Stable Manifold Theorem). There exists a C smooth codimension two locally invariant center-stable manifold Wn cs in H n for any n. () At points in the subset W cs n4 of Wcs n, Wcs n is C smooth in ɛ, in H n norm, for ɛ 0, ɛ 0 ), and some ɛ 0 > 0. () Wn cs is C smooth in (α, β, ω). (3) The annular neighborhood A(δ) in Theorem is included in W cs The proof of this theorem follows from the same arguments as in Ref. 8. Regularity of Wn cs in ɛ is crucial in Melnikov analysis. Melnikov integrals are the leading order terms in ɛ of the signed distances between W u (Q ɛ ) (4) and Wn cs. The signed distances are set up along an unperturbed homoclinic orbit, and the regularity of Wn cs in ɛ at ɛ = 0 determines the order of the signed distances in ɛ. Due to the singular perturbation, Wn cs is not C in ɛ at every point rather at points in the subset W cs n4. Here, one may be able to replace W cs cs n4 by W n. But we are not interested in sharper results, and the current result is sufficient for our purpose..4. Local well posedness Following a much easier argument than that in Refs. 3 and 4, one can prove the following local well-posedness theorem. THEOREM 3. For any q 0 H n (n ), there exists τ = τ( q 0 n ) > 0, such that the perturbed DSII (4) has a unique solution q(t) = S t (q 0 ; ɛ, α, β, ω) C 0 (0, τ, H n ), q(0) = q 0, S t denotes the evolution operator. S t ( ; ɛ, α, β, ω): H n H n is C in q 0 and (α, β, ω). S t ( ; ɛ, α, β, ω): H n4 H n is C in t and ɛ, ɛ 0, ɛ 0 ), ɛ 0 > 0. Here, C in q 0 and (α, β, ω) can be replaced by C in q 0 and (α, β, ω). H n4 can be replaced by H n. But we are not interested in sharper results. n. 3. Global theory Global theory refers to a theory that is global in phase space, which includes integrable theory and Melnikov analysis. Integrable theory provides two ingredients for a Melnikov analysis: () An explicit expression of the unperturbed homoclinic orbit, and () Melnikov vectors with explicit expressions.

8 9 Y. C. Li 3.. Integrable theory Calculations in this subsection are essentially the same as those in Ref. 6. The minor differences are introduced by the spatial periods π/κ and π/κ in contrast to π and π in Ref. 6. Proofs of theorems and lemmas can be found in Ref. 6. The DSII ɛ = 0in() is an integrable system with the Lax pair Lψ = λψ, (5) t ψ = Aψ, (6) ψ = (ψ, ψ ) T, and ( D ) q L = q D, ( ) ( A = i x q x r (D ) q) q x x (D, q) r D = α y x, D = α y x, α =, (7) r and r have the expressions, r = ( q ω ) u y iũ, r = ( q ω ) u y iũ, (8) ũ is also a real-valued function satisfying ũ = 4iα x y q. Notice that DSII is invariant under the transformation σ : σ (q, q, r, r ; α) = (q, q, r, r ; α). (9) Applying the transformation σ (9) to the Lax pair (5, 6), we have a congruent Lax pair for which the compatibility condition gives the same DSII. The congruent Lax pair is given as ˆL ˆψ = λ ˆψ, (0) t ˆψ = Â ˆψ, () ˆψ = ( ˆψ, ˆψ ), and ( D ) q ˆL = q D, ( ) ( ) x q x r (D q) Â = i. q x (D q) r x

9 Melnikov Analysis 93 The Bäcklund Darboux transformation can be formulated as follows. Let (q, u) beasolution to the DSII, and let λ be any value of λ. Let ψ = (ψ, ψ ) T be a solution to the Lax pair (5, 6) at (q, q, r, r ; λ ). Define the matrix operator: a b Ɣ =, c d =α y λ, and a, b, c, d are functions defined as: a = ψ ψ ψ ψ, b = ψ ψ ψ ψ, c = ψ ψ ψ ψ, d = ψ ψ ψ ψ, in which = α y λ, = α y λ, and = ψ ψ. Define a transformation as follows: { (q, r, r ) (Q, R, R ), φ ; Q = q b, R = r (D a), R = r (D d), = Ɣφ, () φ is any solution to the Lax pair (5, 6) at (q, q, r, r ; λ), D and D are defined in (7), we have the following theorem 6. THEOREM 4. The transformation () is a Bäcklund Darboux transformation. That is, the function Q defined through the transformation () is also a solution to the DSII. The function defined through the transformation () solves the Lax pair (5, 6) at (Q, Q, R, R ; λ). Consider the spatially independent solution, q c = η exp{ iη ω t iγ }, (3)

10 94 Y. C. Li η satisfies the constraint () and (3) with ω replaced by η. The dispersion relation for the linearized DSII at q c is ξ =± ξ 4η ( ξ ξ ξ ξ ), for δq qc exp{i(ξ x ξ y) t}, ξ = k κ, ξ = k κ, and k and k are integers. There are only two unstable modes (κ,0)and (0, κ ) under even constraint. The Bloch eigenfunction of the Lax pair (5) and (6) is given as, ψ = c(t) qc χ exp{i(ξ x ξ y)}, (4) c(t) = c 0 exp{ξ (iαξ λ) ir t}, r r = ( q c ω ), χ = (iαξ λ) iξ, (iαξ λ) ξ = η. For the iteration of the Bäcklund Darboux transformations, one needs two sets of eigenfunctions. First, we choose ξ =± κ,ξ = 0,λ 0 = η 4 κ (for a fixed branch), { ψ ± = c ± qc χ ± exp ±i } κ x, (5) c ± = c ± 0 exp{ κ λ 0 ir t}, χ ± = λ 0 i κ = ηe i( π ϑ ), i.e. ηe ±iϑ = κ ± iλ 0. We apply the Bäcklund Darboux transformations with ψ = ψ ψ, which generates the unstable foliation associated with the (κ,0)linearly unstable mode. Then, we choose ξ =± κ,λ= 0,ξ η 0 = 4 κ (for a fixed branch), { ( qc φ ± = c ± exp i ξ 0 χ x ± )} ± κ y, (6) c ± = c 0 ± exp { ±iακ ξ 0 ir t }, χ ± =±iα κ iξ 0 =±ηe iϑ, i.e., ηe ±iϑ = iα κ ± iξ 0.

11 Melnikov Analysis 95 We start from these eigenfunctions φ ± to generate Ɣφ ± through Bäcklund Darboux transformations, and then iterate the Bäcklund Darboux transformations with Ɣφ Ɣφ to generate the unstable foliation associated with all the linearly unstable modes (κ,0)and (0, κ ). It turns out that the following representations are convenient, ψ ± = c 0 c 0 eir t ( v ± ) v ±, (7) and φ ± = c 0 c 0 e iξ 0 xir t v ± = q ce τ ±i x, v ± = ηe τ ±i z, w ± = q ce ± ˆτ ±i ŷ, w ± =±ηe± ˆτ ±iẑ, ( w ± ) w ±, (8) c 0 /c 0 = eρiϑ, τ = κ λ 0 t ρ, x = κ x ϑ, z = x π ϑ, c 0 /c0 = e ˆρi ˆϑ, ˆτ = iακ ξ 0 t ˆρ, ŷ = κ y ˆϑ, ẑ = ŷ ϑ. The following representations are also very useful, ( ) ψ = ψ ψ = c v 0 c 0 eir t, (9) φ = φ φ = cc 0 e 0 iξ 0xir t v ( w w v = q c cosh τ cos x i sinh τ sin x, v = η cosh τ cos z i sinh τ sin z, w = q c cosh ˆτ cos ŷ i sinh ˆτ sin ŷ, w = η sinh ˆτ cos ẑ i cosh ˆτ sin ẑ. ), (30) Applying the Bäcklund Darboux transformations () with ψ given in (9), we have the representations,

12 96 Y. C. Li a = λ 0 sech τ sin( x z) sin( x z) sech τ cos( x z) cos( x z), (3) b = q c b = λ 0q c cos( x z) i tanh τ sin( x z) η sech τ cos( x z) sech τ cos( x z) cos( x z), (3) c = b, d = ā = a. (33) The evenness of b in x is enforced by the requirement that ϑ ϑ =± π, and a ± = λ 0 sech τ cos ϑ sin(κ x) sech τ sin ϑ cos(κ x), (34) b ± = q c b ± = λ 0q c η sin ϑ i tanh τ cos ϑ ± sech τ cos(κ x) sech τ sin ϑ cos(κ x), (35) c = b, d = ā = a. (36) Notice also that a ± is an odd function in x. Under the above Bäcklund Darboux transformations, the eigenfunctions φ ± (6) and φ are transformed into ϕ ± = Ɣφ ±, ϕ = Ɣφ = Ɣφ Ɣφ, (37) a b Ɣ =, b a and = α y λ with λ evaluated at 0. Then ϕ ± = cc 0 e 0 iξ 0xir qc W ± t ηw ± W ± = ±i ακ a ± η be iϑ W ± =±e iϑ ϕ = cc 0 e 0 iξ 0xir t, e ± ˆτ ±i ŷ, ±i ακ a ± η be ±iϑ e ± ˆτ ±i ŷ ; qc W ηw,

13 Melnikov Analysis 97 W = cosh ˆτ a cos ŷ ακ sin ŷ iη b sin ẑ sinh ˆτ iακ cos ŷ iasin ŷ η b cos ẑ, iasin ẑ iακ cos ẑ η b cos ŷ W = cosh ˆτ sinh ˆτ ακ sin ẑ a cos ẑ iη b sin ŷ. We generate the coefficients in the Bäcklund Darboux transformations () with ϕ (the iteration of the Bäcklund Darboux transformations), a (I ) = W (α y W ) W (α y W ) W W, (38) b (I ) = q c η W (α y W ) W (α y W ) W W, (39) c (I ) = b (I ), d (I ) = a (I ), (40) W (α y W ) W (α y W ) = { ακ cosh ˆτ ακ a iaη( b b) cos ϑ 4 κ a η b W W cos(ŷ ẑ) sin ϑ sinh ˆτaη( b b) } sin ϑ, = cosh ˆτ a 4 κ η b iακ η ( b b) cos ϑ 4 κ a η b sin(ŷ ẑ) sin ϑ sinh ˆτ ακ η ( b b) sin ϑ, W (α y W ) W (α y W ) = { ακ cosh ˆτ ακ η b i ( a 4 ) κ η b cos ϑ sinh ˆτ a 4 } κ η b sin ϑ. The new solution to the DSII is given by Q = q c b b (I ). (4)

14 98 Y. C. Li The evenness of b (I ) in y is enforced by the requirement that ˆϑ ϑ =± π.in fact, we have LEMMA. Choosing the Bäcklund parameters ϑ and ˆϑ as follows: ϑ = ϑ ± π, and ˆϑ = ϑ ± π, b( x) = b(x), b (I ) ( x, y) = b (I ) (x, y) = b (I ) (x, y), (4) and Q = q c b b (I ) is even in both x and y. The asymptotic behavior of Q can be computed directly. In fact, we have the asymptotic phase shift lemma. LEMMA. (Asymptotic Phase Shift Lemma). For λ 0 > 0, ξ 0 > 0, and α = i; as t ±, Q = q c b b (I ) q c e iπ e i(ϑ ϑ ). (43) In comparison, the asymptotic phase shift of the first application of the Bäcklund Darboux transformations is given by q c b q c e iϑ. Next, we generate the Melnikov vectors. Starting from ψ ± and φ ± given in (5) and (6), we generate the following eigenfunctions corresponding to the solution Q given in (4) through the iterated Bäcklund Darboux transformations, ± = Ɣ (I ) Ɣψ ±, at λ = λ 0 = η 4 κ, (44) ± = Ɣ (I ) Ɣφ ±, at λ = 0, (45) a b a (I ) b (I ) Ɣ =, Ɣ b (I ) =, a b (I ) a (I ) = α y λ for general λ. LEMMA 3. The eigenfunctions ± and ± defined in (44) and (45) have the representations, ± =±iλ 0 κ η c 0 c 0 eir t v v qc (λ0 a (I ) )v η b (I ) v η, (46) η b (I ) v (λ 0 a (I ) )v

15 Melnikov Analysis 99 ± =±i 4 ακ cc 0 e 0 iξ 0xir t W W qc, (47) η b (I ) = q c b(i ), and = W ( W W = W ( W W ) ( W W W ) W ( W W ) ( W W W ), ) ( W W W ). If we take r to be real in the Melnikov vectors, r appears in the form r r = ( q c ω ), then ± 0, ± 0, as t ±. (48) Next, we generate eigenfunctions solving the corresponding congruent Lax pair (0, ) with the potential Q, through the iterated Bäcklund Darboux transformations and the symmetry transformation (9) 6. LEMMA 4. Under the replacement α α (then ϑ π ϑ ), r r, r r, ˆϑ ˆϑ π ϑ, ˆρ ˆρ, the potentials are transformed as follows, Q Q, R R, (49) R R. The eigenfunctions ± and ± given in (46) and (47) depend on the variables in the replacement (49): ± = ± (α, r, r, ˆϑ, ˆρ), ± = ± (α, r, r, ˆϑ, ˆρ). Under replacement (49), ± and ± are transformed into ˆ ± = ± ( α, r, r, ˆϑ π ϑ, ˆρ), (50) ˆ ± = ± ( α, r, r, ˆϑ π ϑ, ˆρ). (5) LEMMA 5. ˆ ± and ˆ ± solve the congruent Lax pair (0, ) at (Q, Q, R, R ; λ 0 ) and (Q, Q, R, R ; 0), respectively. Notice that as a function of η, ξ 0 has two (plus and minus) branches. In order to construct Melnikov vectors, we need to study the effect of the replacement ξ 0 ξ 0.

16 300 Y. C. Li LEMMA 6. Under the replacement ξ 0 ξ 0 (then ϑ ϑ ), ˆϑ ˆϑ π ϑ, ˆρ ˆρ, (5) the potentials are invariant, Q Q, R R, R R. The eigenfunction ± replacement (5): given in (47) depends on the variables in the ± = ± ( ξ 0, ˆϑ, ˆρ ). Under the replacement (5), ± is transformed into ( ± = ± ξ 0, ˆϑ π ϑ, ˆρ ). (53) LEMMA 7. ± solves the Lax pair (5, 6) at (Q, Q, R, R ; 0). In the construction of the Melnikov vectors, we need to replace ± by ± to guarantee the periodicity in x of period π κ. The Melnikov vectors for the Davey Stewartson II equations are given by ( ˆ ) ( U ˆ ) = ˆ S ˆ, (54) U = () () () ˆ () ˆ S () () () ˆ () ˆ, (55) denotes complex conjugate, and S = ( 0 0 ). In fact, the even parts of U and U are the Melnikov vectors in our phase space. Nevertheless, the Melnikov integral formulas end up the same, as shown in Ref. 6. For simplicity, we just use U and U. 3.. Melnikov analysis The main difficulty in a rigorous Melnikov measurement is due to the lack of global well posedness. The main idea in resolving this difficulty is to iterate the small time interval in local well posedness by virtue of the fact that the unperturbed homoclinic orbit is a classical solution. Let p be any point on W u (Q ɛ), the unstable curve of Q ɛ in. Bythe Unstable Fiber Theorem, F u p is C in ɛ for ɛ 0, ɛ 0 ), ɛ 0 > 0; thus, there are two points q ɛ (0) and q 0 (0) on the unstable fibers F u p and F u p ɛ=0, such that

17 Melnikov Analysis 30 q ɛ (0) q 0 (0) n C n () ɛ, (n ). The key point here is that F u p H s for any fixed s. The expression of the unperturbed homoclinic orbit q 0 (t) has been given in (4), which represents a classical solution to the DSII. Let Ds = sup t (, ) { q 0 (t) s }, (s ). By the Local Well-Posedness Theorem 3, there exists τ = τ(d n ) > 0, such that q ɛ (t) q 0 (t) n C n () ɛ, t 0,τ, C () n = C () n (D n4 ). There is an integer N > 0 such that q 0 (Nτ) Wn cs ɛ=0, Wn cs is given by the Center-Stable Manifold Theorem. Iterating the Local Well-Posedness Theorem N times, one gets q ɛ (t) q 0 (t) n C n (3) ɛ, t 0, Nτ, C (3) n = C (3) n (D n4 ). Our goal is to determine when q ɛ(nτ) Wn cs through Melnikov measurement. The two Melnikov vectors U and U (54) and (55) are transversal to Wn cs. There is a unique point ˆq ɛ(nτ) Wn cs such that q ɛ (Nτ) ˆq ɛ (Nτ) span { U }, U ; thus, ˆq ɛ (Nτ) Wn4 cs.bythe Center-Stable Manifold Theorem, ˆq ɛ (Nτ) q 0 (Nτ) n C n (4) ɛ, C (4) n = C (4) n (D n4 ). Thus, q ɛ (Nτ) ˆq ɛ (Nτ) n C n ɛ, C n = C n (D n4 ). To determine when q ɛ(nτ) = ˆq ɛ (Nτ), one can define the signed distances d = U, q ɛ (Nτ) ˆq ɛ (Nτ), d = U, q ɛ (Nτ) ˆq ɛ (Nτ), q = (q, q) T, and A, B = π/κ π/κ 0 0 {A B A B } dx dy. The rest of the derivation for Melnikov integrals is completely standard. For details, see Refs. 5 and 6.

18 30 Y. C. Li M = d k = ɛm k o(ɛ), k =,, U, G dt, M = G = ( f, f ) T, f = Q αq β. That is, M = M = π/κ π/κ 0 0 π/κ π/κ 0 0 Re {( ˆ Re {( () U, G dt, ) ( f ˆ ) } f dx dydt, ()) ( ˆ f () ()) ˆ } f dx dydt, η = ω, and we divide by the constant iλ 0 κ c 0 c 0 eiγ/, and by 4 iακ η cc 0 e 0 iγ/.ithas been verified numerically that multiplication of and byacomplex constant leads to equivalent results. It turns out that M j = M () j αm () j β cos γ M (3) j β sin γ M (4) j, ( j =, ), M (l) j = M (l) j (ω, ρ), ( j =, ; l 4), ρ = ˆρ iακ ξ 0κ λ 0 ρ, ˆτ = iακ ξ 0κ λ 0 τ ρ. M j = 0(j =, ) imply that α = α(ω, ρ,γ) = { M () (3) cos γ M sin γ M (4) M () (3) cos γ M sin γ M (4) } { M () (3) cos γ M sin γ M (4) M () β = β(ω, ρ,γ) = M () M() { M () M () cos γ M (3) M () (3) cos γ M cos γ M (3) sin γ M (4) M() sin γ M (4) sin γ M (4) }, (56) }. (57) THEOREM 5. There exists ɛ 0 > 0, such that for any ɛ (0, ɛ 0 ), there exists a domain D ɛ R R R, ω satisfies the constraint () or (3), and αω < β. For any (α, β, ω) D ɛ, there exists another orbit in W u (Q ɛ ) Wn cs other than the unstable curve W u (Q ɛ) of Q ɛ in, for the perturbed DSII ().

19 Melnikov Analysis 303 Proof: The zeros of M j ( j =, ) are given by (56, and (57). We need α>0and β > 0, which define a region in the external parameter space, parametrized by ρ and γ. Then the theorem follows from the implicit function theorem. For example, when κ = and κ =, α ( ) ( ) 0.,., π = 5.645, β 0.,., π =.336. Appendix The main obstacle toward proving the existence of a homoclinic orbit for the perturbed DSII () comes from a technical difficulty in the normal form transform 8. In this Appendix, we will present the difficulty. A.. The technical difficulty in the normal form transform To locate a homoclinic orbit to Q ɛ (7), we need to estimate the size of the local stable manifold of Q ɛ. The size of the variable J is of order O( ɛ). The size of the variable θ is of order O(). To be able to track a homoclinic orbit, we need the size of the variable f to be of order O(ɛ µ ),µ<. Such an estimate can be achieved, if the quadratic term N in () can be removed through a normal form transformation. In fact, it is enough to remove its leading order part: Ñ = ω f f ( f f ) f ( f f ). That is, our goal is to find a normal form transform g = f K ( f, f ), K is a bilinear form that transforms the equation f t = L ɛ f iñ, into an equation with a cubic nonlinearity g t = L ɛ g O ( g 3 s), (s ), L ɛ is given in (). In terms of Fourier transforms, f = k 0 ˆf (k)e ik ξ, f = k 0 ˆf ( k)e ik ξ,

20 304 Y. C. Li k = (k, k ) Z,ξ = (κ x,κ y). The terms in Ñ can be written as: f = a(k l) ˆf (k) ˆf ( l) ˆf (l) ˆf ( k)e i(kl) ξ, f f f f = f f f f = kl 0 a(k) a(l) ˆf (k) ˆf (l)e i(kl) ξ, kl 0 a(l) ˆf (k) ˆf ( l) a(k) ˆf (l) ˆf ( k)e i(kl) ξ, kl 0 a(k) = k κ k κ k κ. k κ We will search for a normal form transform of the general form, K ( f, f ) = kl 0 g = f K ( f, f ), ˆK (k,l) ˆf (k) ˆf (l) ˆK (k,l) ˆf (k) ˆf ( l) ˆK (l, k) ˆf ( k) ˆf (l) ˆK 3 (k,l) ˆf ( k) ˆf ( l) e i(kl)x, ˆK j (k,l), ( j =,, 3) are the unknown coefficients to be determined, and ˆK j (k,l) = ˆK j (l, k), ( j =, 3). To eliminate the quadratic terms, we first need to set which takes the explicit form: il ɛ K ( f, f ) ik(l ɛ f, f ) ik( f, L ɛ f ) = Ñ, (σ iσ ) ˆK (k,l) B(l) ˆK (k,l) B(k) ˆK (l, k) B(k l) ˆK 3 (k,l) = B(k) B(l), ω (A.) B(l) ˆK (k,l) (σ iσ ) ˆK (k,l) B(k l) ˆK (l, k) B(k) ˆK 3 (k,l) = B(k l) B(l), ω (A.) B(k) ˆK (k,l) B(k l) ˆK (k,l) (σ 3 iσ ) ˆK (l, k) B(l) ˆK 3 (k,l) = B(k l) B(k), ω (A.3) B(k l) ˆK (k,l) B(k) ˆK (k,l) B(l) ˆK (l, k) (σ 4 iσ ) ˆK 3 (k,l) = 0, (A.4)

21 Melnikov Analysis 305 B(k) = ω a(k), and σ = ɛ α ( k l κ k l κ ), σ = ( k l κ k l κ ) B(k l) B(k) B(l), σ = (k l )l κ (k l )l κ B(k l) B(k) B(l), σ 3 = (k l )k κ (k l )k κ B(k l) B(k) B(l), σ 4 = ( k k l l ) κ ( k k l l ) κ B(k l) B(k) B(l). Since these coefficients are even in (k, l), we will search for even solutions, that is, ˆK j ( k, l) = ˆK j (k,l), j =,, 3. The technical difficulty in the normal form transform comes from not being able to answer the following two questions in solving the linear system (A.) (A.4): () Is it true that for all k,l Z /{0}, there is a solution? () What is the asymptotic behavior of the solution as k and/or l?in particular, is the asymptotic behavior like k m and/or l m (m 0)? A.. A formal calculation Formally conducting the calculation for the second measurement to locate a homoclinic orbit 8, one gets the formulas M j = 0 (j =, ), βcos γ = αω γ sin γ, γ = 4(ϑ ϑ ). Thus, we have α = /χ, χ = χ(ω, ρ) = ( M () M(4) M () ) M(4) M () M(4) M () M(4) β = β(ω, ρ) = ω γ sin γ ( (3) M M(4) M (3) ) M(4), (αω γ) sin γ ( M (4) ) ( ( M () α M () M (3) ω γ sin γ ) ) /.

22 306 Y. C. Li For example, when κ = and κ =, ( ) χ 0.,. = References. A. DAVEY and K. STEWARTSON, On three-dimensional packets of surface waves, Proc. R. Soc. Lond. A338:0 (974).. D. J. BENNEY and G. J. ROSKES, Wave instabilities, Stud. Appl. Math. 48:377 (969). 3. M. J. ABLOWITZ and H. SEGUR, On the evolution of packets of water waves, J. Fluid Mech. 9:69 (979). 4. A. S. FOKAS and M. J. ABLOWITZ, The inverse scattering transform for multidimensional () problems, in Lecture Notes in Physics, vol. 89, p. 37, T. M. MALANYUK, Finite-gap solutions of the Davey-Stewartson equations, J. Nonlinear Sci. 4: (994). 6. Y. LI, Bäcklund-Darboux transformations and Melnikov analysis for Davey-Stewartson II equations, J. Nonlinear Sci. 0:03 (000). 7. Y. LI, Smale horseshoes and symbolic dynamics in perturbed nonlinear Schrödinger equations, J. Nonlinear Sci. 9:363 (999). 8. Y. LI, Homoclinic orbits for singularly perturbed NLS, Dyn. PDE : (004). 9. T. OZAWA, Exact blow-up solutions to the Cauchy problem for the Davey-Stewartson systems, Proc. R. Soc. Lond. 436:345 (99). 0. J. M. GHIDAGLIA and J. C. SAUT, On the initial value problem for the Davey-Stewartson systems, Nonlinearity 3:475 (990).. L. Y. SUNG, An inverse scattering transform for the Davey-Stewartson II equations, part I, II, III, J. Math. Anal. Appl. 83:, 89, 477 (994).. R. ADAMS, Sobolev Space, Academic Press, New York, T. KATO, Nonstationary flows of viscous and ideal fluids in R 3, J. Funct. Anal. 9:96 (97). 4. T. KATO, Quasi-linear equations of evolution, with applications to partial differential equations, in Lecture Notes in Mathematics, vol. 448, p. 5, Springer, Y. LI and D. W. MCLAUGHLIN, Homoclinic orbits and chaos in discretized perturbed NLS system, part I. Homoclinic orbits, J. Nonlinear Sci. 7: (997). 6. Y. LI et al., Persistent homoclinic orbits for a perturbed nonlinear Schrödinger equation, Comm. Pure Appl. Math. XLIX:75 (996). UNIVERSITY OF MISSOURI (Received February, 004)