Infinite Number of Conservation Laws

✐ ✐ 2010/10/8 page 91 ✐ ✐ 3.6 Infinite Number of Conservation Laws 91 Notice that the only information used to construct the N-soliton solutions in the hierarchy (3.20) is the discrete scattering data {ζk , ζ¯k , vk0 , v¯ k0 , ≤ k ≤ N } and the dispersion relation of the hierarchy (3.20) as reflected in the exponents of Eqs (3.89)–(3.90) For a particular evolution equation in this hierarchy (3.20), the only caution the reader should heed is the symmetry reduction of this equation from the hierarchy This symmetry reduction corresponds to symmetry properties of the potential matrix Q in Eq (3.42), and it induces the corresponding involution properties in the discrete scattering data {ζk , ζ¯k , vk0 , v¯ k0 , ≤ k ≤ N} For instance, the symmetry reduction (3.35) leads to the symmetry (3.75) of the potential matrix Q and involution properties (3.80) and (3.82) of the discrete scattering data For some evolution equations (such as the Sasa–Satsuma equation (3.4)), the potential matrix Q admits more than one symmetry; thus the discrete scattering data also admits more than one involution These involution properties of the discrete scattering data must all be respected in order to obtain the correct N-soliton solutions To illustrate, explicit N-soliton solutions for the Manakov equations (3.1)–(3.2), the focusing-defocusing NLS system (3.29)–(3.30), and the Sasa–Satsuma equation (3.4) will be given later in this chapter (see Secs 3.10–3.12) 3.6 Infinite Number of Conservation Laws The hierarchy (3.20) also admits an infinite number of conservation laws These conservation laws can be derived analogously to what we did for the NLS equation in Sec 2.3 The only main difference is that conservation laws here will be generated by a coupled Riccati system rather than a single Riccati equation Let us consider a solution Y = (y1 , y2 , y3 )T to the third-order scattering operator (3.41) Defining µ(1) = y1 /y3 , µ(2) = y2 /y3 , (3.95) then it is easy to find from (3.41) that ˆ (1) + vµ ˆ (2) (ln y3 )x = iζ + uµ (3.96) Using the temporal equation (3.9) of this hierarchy, an analogous equation for (ln y3 )t can also be derived Cross-differentiating these two equations with respect to t and x, respectively, we see that if we expand µ(1) and µ(2) into the power series, ∞ µ(j ) (x, t, ζ ) = − n=1 (1) (j ) µn (x, t) , (−2iζ )n j = 1, 2, (3.97) (2) ˆ n would be the density of a local conservation law, and an infinite number then uµ ˆ n + vµ of conserved quantities are In = ∞ −∞ (1) (2) uµ ˆ n + vµ ˆ n dx, n = 1, 2, (3.98) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 92 ✐ ✐ 92 Chapter Integrable Equations with Higher-Order Scattering Operators (1,2) To determine the expansion coefficients µn , we use the scattering equation (3.41) Simple calculations show that µ(1,2) satisfy the following coupled Riccati system: (1) (3.99) (2) (3.100) µx = −µ(1) uµ ˆ (1) + vµ ˆ (2) − 2iζ µ(1) + u, ˆ (1) + vµ ˆ (2) − 2iζ µ(2) + v µx = −µ(2) uµ Then by inserting µ(1,2) ’s expansions (3.97) into this Riccati system and equating terms of (1,2) the same power in (−2iζ )−1 , we find that µn are given as (1) µ1 = u, (1) µ2 = ux , (2) (3.101) (2) (3.102) µ1 = v, µ2 = vx , and n−1 (1) (1) (1) (2) µn−k , (1) n ≥ 2, (3.103) (1) (2) µn−k , (2) n ≥ (3.104) uµ ˆ k + vµ ˆ k µn+1 = µn,x − k=1 n−1 (2) (2) uµ ˆ k + vµ ˆ k µn+1 = µn,x − k=1 Then the infinite number of conserved quantities (3.98) are obtained The first three conserved quantities are I1 = I2 = I3 = ∞ −∞ ∞ −∞ ∞ −∞ (uuˆ + v v) ˆ dx, (3.105) (ux uˆ + vx v) ˆ dx, (3.106) uxx uˆ + vxx vˆ − (uuˆ + v v) ˆ dx, (3.107) which are the mass (or power), momentum, and energy of the hierarchy (3.20) Higher conserved quantities can be similarly calculated If one also wishes to obtain the flux functions of local conservation laws, then one can first use the temporal equation (3.9) of the Lax pair to derive the (ln y3 )t equation, then insert the expansions (3.97) The coefficients of (ln y3 )t at various orders of ζ −n would then be the fluxes of local conservation laws For a particular evolution equation in the hierarchy (3.20), if the potential Q admits symmetry reductions, then these symmetries should also be inserted into the above general conservation laws For instance, the Manakov system (3.1)–(3.2) admits the symmetry reductions uˆ = −u∗ , vˆ = −v ∗ Utilizing these symmetries, the first three conserved quantities of the Manakov system are then I1 = ∞ −∞ (|u|2 + |v|2 ) dx, (3.108) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 93 ✐ ✐ 3.7 Closure of Eigenstates in the Higher-Order Scattering Operator I2 = I3 = ∞ −∞ (u∗ ux + v ∗ vx ) dx, ∞ −∞ 93 (3.109) u∗ uxx + v ∗ vxx + (|u|2 + |v|2 )2 dx (3.110) One may notice that these conserved quantities are not all the conserved quantities in the Manakov system For instance, the powers of each component I1u = ∞ −∞ |u|2 dx, I1v = ∞ −∞ |v|2 dx (3.111) are individually also the conserved quantities of the Manakov system, but these conserved quantities are not in the infinite number of conserved quantities (3.98) given above How all the conservation laws of the hierarchy (3.20) can be derived is still not clear 3.7 Closure of Eigenstates in the Higher-Order Scattering Operator In this section, we establish the closure of eigenstates in the third-order scattering operator (3.41) This proof is a simple extension of the one for the Zakharov–Shabat system in Sec 2.5 and will be only sketched below The proof of closure for the more general scattering operator (3.8) could be similarly given First, we define functions R + (x, y, ζ ) = χ + (x, ζ ) [θ (y − x)H1 − θ (x − y)H2 ] (χ + )−1 (y, ζ ), (3.112) R − (x, y, ζ ) = χ − (x, ζ ) [θ (x − y)H1 − θ (y − x)H2 ] (χ − )−1 (y, ζ ), (3.113) where χ + = (φ1 , φ2 , ψ3 ) , χ − = (ψ1 , ψ2 , φ3 ), (3.114) and θ(x) is the standard step function given in Eq (2.187) of the previous chapter Functions R ± are meromorphic for ζ ∈ C± , respectively, and are bounded as ζ → ∞ From similar relations as (2.188)–(2.189) but with H1,2 replaced by (3.54), we see that det χ + (x, ζ ) = e−iζ x sˆ33 (ζ ), det χ − (x, ζ ) = e−iζ x s33 (ζ ) (3.115) Thus R ± has pole singularities at the zeros of sˆ33 and s33 , respectively Then we define two complex contours, γ+ and γ− , with γ+ starting from ζ = −∞+i0+ , passing over all zeros of sˆ33 (ζ ) in C+ , and ending at ζ = ∞ + i0+ , and with γ− starting from ζ = −∞ + i0− , passing under all zeros of s33 (ζ ) in C− , and ending at ζ = ∞ + i0− Using the large-ζ asymptotics ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 94 ✐ ✐ 94 Chapter Integrable Equations with Higher-Order Scattering Operators (3.55)–(3.56) and (3.61)–(3.62) of Jost solutions and then bringing the contours to the real axis, we get R + (x, y, ζ )dζ + γ+ R − (x, y, ζ )dζ = 2π δ(x − y) (3.116) γ− Using the residue theorem, the left side of the above equation is found to be R + (x, y, ζ )dζ + γ+ R − (x, y, ζ )dζ = γ− ∞ −∞ R + (x, y, ξ ) + R − (x, y, ξ ) dξ Res R − (x, y, ζ ), ζ¯j − Res R + (x, y, ζ ), ζj + 2πi , (3.117) j where ζj ∈ C+ and ζ¯j ∈ C− are the zeros of sˆ33 (ζ ) and s33 (ζ ) Following similar calculations as in Sec 2.5, we also have ∞ −∞ R + (x, y, ξ ) + R − (x, y, ξ ) dξ = ∞ −∞ χ + (x, ξ ) H1 (χ + )−1 (y, ξ ) −χ − (x, ξ ) H2 (χ − )−1 (y, ξ ) dξ , (3.118) Res R + (x, y, ζ ), ζj = Res χ + (x, ζ ) H1 (χ + )−1 (y, ζ ), ζj , (3.119) Res R − (x, y, ζ ), ζ¯j = −Res χ − (x, ζ ) H2 (χ − )−1 (y, ζ ), ζ¯j (3.120) and These expressions can be further simplified Notice that (χ + )−1 = (φ1 , φ2 , ψ3 )−1 = ( H1 + H2 )−1  and   ψ1     +   ψ  = H1     + φ3 Using the relation H1 and thus = (3.121) −1 + H2 −1 (3.122) S, we get −1 + H2 −1 ( H1 + H2 ) = H1 SH1 + H2 S −1 S2 ,   s11   (χ + )−1 =  s21   −1  s12 s22 sˆ33       (3.123)   ψ1       ψ2      φ3 (3.124) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 95 ✐ ✐ 3.8 Squared Eigenfunctions of the Higher-Order Scattering Operator Similarly,   sˆ11   (χ − )−1 =  sˆ21   −1  sˆ12       sˆ22 s33 95   φ1       φ2      ψ3 (3.125) Substituting the above two equations into (3.116)–(3.120), the final closure relation is then obtained Assuming the zeros (ζj , ζ¯j ) are all simple, then the closure relation reads         ∞ 1  ψ1 (y, ξ )  (φ1 (x, ξ ), φ2 (x, ξ )) s−1 (ξ )  (x, ξ ) ψ (y, ξ ) dξ φ −  3  2π −∞  s33 (ξ )   ψ2 (y, ξ )          ψ1 (y, ζj )  −i φ3 (x, ζ¯j )ψ3 (y, ζ¯j ) φ1 (x, ζj ), φ2 (x, ζj ) sj−  +   s33 (ζ¯j )  j  ψ2 (y, ζj ) = δ(x − y) Here (3.126)    s11 (ζ ) s12 (ζ )  s(ζ ) ≡  , s21 (ζ ) s22 (ζ ) sj− ≡ lim (ζ − ζj )s−1 (ζ ) ζ →ζj (3.127) Note that det(s) = sˆ33 , thus s−1 (ζ ) has pole singularity at the zeros ζj of sˆ33 , and sj− in the above equation is the residue of s−1 (ζ ) at ζj The closure relation (3.126) shows that for the third-order scattering operator (3.41), its discrete and continuous eigenstates also form a complete set For multiple zeros of (ζj , ζ¯j ), the closure relation is similar, except that the residue terms in (3.126) need to be calculated differently 3.8 Squared Eigenfunctions of the Higher-Order Scattering Operator In this section, we derive the squared eigenfunctions for the third-order scattering operator (3.41) and its integrable hierarchy Most of the derivations are analogous to those for the Zakharov–Shabat system (2.2) in Sec 2.6, so our derivation will be brief in general Some new features arise though Such features will be elaborated in detail We will first assume the potential functions (u, v, u, ˆ v) ˆ in (3.42) to be independent, and hence derive the generic squared eigenfunctions for the third-order scattering operator (3.41) Afterwards, we will discuss the case when (u, v, u, ˆ v) ˆ are dependent on each other We will describe how to obtain the corresponding squared eigenfunctions from the generic ones through symmetry ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 96 ✐ ✐ 96 Chapter Integrable Equations with Higher-Order Scattering Operators reduction, and we explain how squared eigenfunctions can become sums of products of Jost functions due to the potential reduction 3.8.1 Squared Eigenfunctions for Generic Potentials We first consider the generic case when the potential functions (u, v, u, ˆ v) ˆ are independent from each other We also assume as before that s33 and sˆ33 have no zeros (the general case of s33 and sˆ33 having zeros will be added later) Following our procedure described in Sec 2.6, we first calculate variations of the scattering data in terms of variation of the potential Repeating the same calculation as in Sec 2.6, we also get the variation relations (2.221)–(2.224), i.e., ∞ δsij (ξ ) = −∞ and δ sˆij (ξ ) = − ψi (x, ξ ) δQ(x) φj (x, ξ ) dx (3.128) ∞ φi (x, ξ ) δQ(x) ψj (x, ξ ) dx, −∞ except that δQ is now  (3.129)     δQ =  0   δ uˆ δ vˆ δu    δv    (3.130) Defining new scattering coefficients ρ1 ≡ s31 , s33 ρ2 ≡ s32 , s33 ρˆ1 ≡ sˆ13 , sˆ33 ρˆ2 ≡ sˆ23 , sˆ33 (3.131) we can then find from the variation relations (3.128)–(3.129) that δρj (ξ ) = ∞ (ξ ) s33 −∞ and δ ρˆj (ξ ) = − sˆ33 (ξ ) ψ3 (x, ξ ) δQ(x) ϕj (x, ξ ) dx (3.132) ∞ −∞ ϕj (x, ξ ) δQ(x) ψ3 (x, ξ ) dx, (3.133) where ϕj = s33 φj − s3j φ3 , ϕj = sˆ33 φj − sˆj φ3 , j = 1, (3.134) Now we simplify these expressions of ϕj and ϕj and show that they are analytic in the same half planes of ψ3 and ψ3 , respectively For this purpose, we first rewrite ϕj and ϕj in terms ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 97 ✐ ✐ 3.8 Squared Eigenfunctions of the Higher-Order Scattering Operator  of functions: χ − = (ψ1 , ψ2 , φ3 ), 97   ψ1      χˆ + =  ψ2      φ3 (3.135) (S −1 H1 + H2 ) (3.136) Notice that χ− = H1 + H2 = and χˆ + = H1 −1 −1 + H2 = (H1 S + H2 ) −1 ; (3.137) thus = χ − B, −1 = B χˆ + , (3.138) where B = (S −1 H1 + H2 )−1 = (H1 + SH2 )−1 S (3.139) B = (H1 S + H2 )−1 = S −1 (H1 + H2 S −1 )−1 (3.140) and From the first formula of B in (3.139), we readily obtain the first two rows of B as the inverse of the × block in the upper left corner of S −1 , followed by a zero column From the second formula of B in (3.139), we readily obtain the third row of B as the third row of S divided by s33 The matrix B can be similarly determined from the two formulae in (3.140) Thus matrices B and B are found to be      sˆ22   B=  −ˆs21 s33   s31 −ˆs12 sˆ11 s32    ,    s33  s22   B=  −s21 sˆ33   −s12 s11 sˆ13    sˆ23    sˆ33 (3.141) Inserting Eqs (3.138) and (3.141) into (3.134), the expressions for ϕj and ϕj then reduce to    sˆ22 (ϕ1 , ϕ2 ) = (ψ1 , ψ2 )  −ˆs21 and     ϕ1   s22  = ϕ2 −s21 −ˆs12   sˆ11  (3.142)  −s12   ψ1    ψ2 s11 (3.143) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 98 ✐ ✐ 98 Chapter Integrable Equations with Higher-Order Scattering Operators Recalling the analytic properties of Jost solutions and scattering matrices summarized in (3.63)–(3.65), we see that (ϕ1 , ϕ2 ) are indeed analytic in C− as ψ3 , and (ϕ1 , ϕ2 ) are indeed analytic in C+ as ψ3 ; thus products of Jost solutions and adjoint Jost solutions in the variation formulae (3.132)–(3.133) for δρj and δ ρˆj are analytic, as we desired Furthermore, these products are only between Jost solutions and their inverse −1 whose boundary conditions are both set at x = +∞ The variation relations (3.132)–(3.133) can be rewritten into the following more convenient form: − δρj (ξ ) = (x, ξ ), (δu, δv, −δ u, ˆ −δ v) ˆ T (x) , (3.144) s33 (ξ ) j δ ρˆj (ξ ) = where j = 1, 2, (ξ ) sˆ33 + j (x, ξ ),    ψ31 ϕ3j    ψ32 ϕ3j −  = j   −ψ33 ϕ1j   −ψ33 ϕ2j (δu, δv, −δ u, ˆ −δ v) ˆ T (x) ,      ,      −ϕj ψ33    −ϕj ψ33 +  = j   ϕj ψ13   ϕj ψ23 (3.145)      ,     (3.146) ϕij , ϕj k are elements in vectors ϕj , ϕj as ϕj = (ϕ1j , ϕ2j , ϕ3j )T , ϕj = (ϕj , ϕj , ϕj ), (3.147) and the inner product is defined as f, g = ∞ −∞ fT(x) g(x) dx (3.148) ± These functions { ± , } are the adjoint squared eigenfunctions Notice that in the variation relations (3.144)–(3.145), the potential variation takes the form of (δu, δv, −δ u, ˆ −δ v) ˆT T rather than (δu, δv, δ u, ˆ δ v) ˆ The same goes to the expansion of the potential variation (3.156) below This form of the potential variation corresponds to the form of the integrable hierarchy (3.20) and is necessary so that the corresponding squared eigenfunctions and their adjoints are also eigenfunctions of the recursion operator LR and its adjoint operator LA R (see the next section) For the Zakharov–Shabat system (2.2), uˆ = −u∗ Thus the above form of the potential variation is consistent with that in variation relations (2.230)–(2.231) and (2.254) for the Zakharov–Shabat system Next, we calculate variation of the potential in terms of variations of the scattering data Defining new Jost solutions F + = P + diag 1, 1, , sˆ33 F − = (P − )−1 diag(1, 1, s33 ), (3.149) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 99 ✐ ✐ 3.8 Squared Eigenfunctions of the Higher-Order Scattering Operator 99 we find that the Riemann–Hilbert problem (3.66) becomes F + (ζ ) = F − (ζ ) G(ζ ), ζ ∈ R,   where (3.150)    G = E   ρ1 ρˆ1 ρˆ2 ρ2 + ρ1 ρˆ1 + ρ2 ρˆ2    −1 E   (3.151) Taking variation to this Riemann–Hilbert problem and repeating the same calculations as in Sec 2.6, we get (Yang and Kaup (2009)) δu(x) = − π δ u(x) ˆ = π ∞ −∞ 13 (x, ξ ) dξ , δv(x) = − 31 (x, ξ ) dξ , δ v(x) ˆ = ∞ −∞ = −∞ 23 (x, ξ ) dξ , (3.152) ∞ 32 (x, ξ ) dξ , −∞ (3.153)   where π ∞ π       δρ1 0 δρ2 δ ρˆ1    δ ρˆ2    −1 , (3.154) i.e., = φ3 φ1 δρ1 + φ3 φ2 δρ2 + φ1 φ3 δ ρˆ1 + φ2 φ3 δ ρˆ2 (3.155) Thus variation of the potential via variations of the scattering data is (δu, δv, −δ u, ˆ −δ v) ˆ T (x) = − where π ∞ −∞ j =1  φ φ  13 j      φ φ   23 j − ,  Zj =    φ33 φj      φ33 φj  Zj− (x, ξ )δρj (ξ ) + Zj+ (x, ξ )δ ρˆj (ξ ) dξ ,  φ φ  1j 33      φ φ   2j 33 + ,  Zj =    φ3j φ31      φ3j φ32 (3.156)  j = 1, 2, are the squared eigenfunctions, and φij , φij are the (i, j )th elements of matrices (3.157) , −1 ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 100 ✐ ✐ 100 Chapter Integrable Equations with Higher-Order Scattering Operators When the potential expansion (3.156) is inserted into the variation relations (3.144)– (3.145), we get the inner products between squared eigenfunctions and their adjoints as − − j (x, ξ ), Zj (x, ξ ) = −π s33 (ξ ) δ(ξ − ξ ), j = 1, 2, (3.158) + + j (x, ξ ), Zj (x, ξ ) = −π sˆ33 (ξ ) δ(ξ − ξ ), j = 1, 2, (3.159) and other inner products are zero Inserting the variation relations (3.144)–(3.145) into the potential expansion (3.156) and exchanging orders of integration, we get the closure relation − π ∞ −∞ k=1 (ξ ) s33 −T Z + (x, ξ ) k (y, ξ ) + sˆ33 (ξ ) k Zk− (x, ξ ) +T k (y, ξ ) = δ(x − y) I4 , dξ (3.160) where I4 is the unit matrix of rank In the general case where s33 and sˆ33 have zeros, the above closure relation needs to include the discrete-spectrum contribution As before, this discrete-spectrum contribution is nothing but the residues of integrand functions in the above closure relation Notice from − −T the above expressions of squared eigenfunctions that Z1− −T and Z2 are analytic in + +T the lower half plane, while Z1+ +T and Z2 are analytic in the upper half plane Also and 1/ˆ are meromorphic in the lower and upper half planes, respectively s33 notice that 1/s33 Thus the residues of integrand functions in (3.160) can be easily obtained These residue terms are similar to those in Eq (2.264) of Sec 2.6 Specifically, these terms are N − 2i s (ζ¯j ) j =1 33 N + 2i Zk− (x, ζ¯j ) −T ˙− ¯ ¯ k (y, ζj ) + Zk (x, ζj ) −T ¯ k (y, ζj ) k=1 sˆ (ζ ) j =1 33 j k=1 Zk+ (x, ζj ) +T ˙+ k (y, ζj ) + Zk (x, ζj ) where − ¯ k (x, ζj ) = ˙ − (x, ζ¯j ) − k +T k (y, ζj ) , (3.161) s33 (ζ¯j ) s (ζ¯j ) − ¯ k (x, ζj ), (3.162) sˆ33 (ζj ) sˆ33 (ζj ) + k (x, ζj ), (3.163) 33 + k (x, ζj ) = ˙ + (x, ζj ) − k and the dot above a variable represents its derivative with respect to ζ However, for each pair of zeros (ζj , ζ¯j ), Eq (3.161) contains eight terms, but we only expect six The reason for our expecting only six terms can be heuristically understood as follows Let us consider the reduction (3.27) which gives the Manakov equations In this case, due to the involution properties of zeros (ζj , ζ¯j ) and vectors {v¯j , vj }, the discrete spectral data in Eqs (3.73)–(3.74) for each pair of zeros contains only six real parameters, i.e., ζj and two ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 285 ✐ ✐ 6.2 One-Dimensional Gap Solitons Not Bifurcated from Bloch Bands 1.5 u (a) −1.5 Im(λ) (b) −1.5 x (a) −1.5 x (b) 0.7 −0.7 (c) 0.7 (c) −0.7 (d) −1.5 x x (d) −4 Re(λ) 1.5 0 −4 Re(λ) 1.5 0 −4 −0.7 1.5 285 −4 Re(λ) 0.7 −0.7 Re(λ) 0.7 Figure 6.8 Upper row: soliton profiles at the marked points in the semi-infinite gap of Fig 6.7 (σ = 1) Lower row: stability spectra of the solitons in the upper row in (−1, 1, 1) and (−1, 1, 1, −1) configurations, respectively (see Fig 6.9(f, g)) These three branches are also connected near the Bloch band, where the solitons are low and broad; see Fig 6.9(h) For this solution family, the lower and upper branches are symmetric, and the middle branch is asymmetric We have examined the stability properties of these dipole-soliton families and found that their lower branches are linearly stable, while their middle and upper branches are linearly unstable To demonstrate, the stability spectra for solitons in the upper rows of Figs 6.8 and 6.9 are displayed in the lower rows of those same figures For both families, the spectra of solitons on the lower branch not contain any unstable eigenvalue, and are thus stable The spectra of solitons on the middle and upper branches, however, contain one and two positive eigenvalues, respectively, and are thus unstable What happens here is that, as the soliton moves from the lower branch to the branching point (where the three branches meet), when it crosses the power minimum point, a positive (unstable) eigenvalue bifurcates out, and hence the soliton becomes linearly unstable This change of stability at the power minimum point should occur in view of the general result in Sec 5.4 This positive eigenvalue, after its creation, persists as the soliton moves onto the middle and upper branches At the branching point (see Figs 6.8(d) and 6.9(h)), another eigenvalue bifurcates out from the origin Along the upper branch, this eigenvalue bifurcates along the real axis, hence creating a second positive (unstable) eigenvalue Along the middle and lower branches, however, this eigenvalue bifurcates along the imaginary axis, and hence no second unstable eigenvalue appears These eigenvalue bifurcations explain why solitons on ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 286 ✐ ✐ 286 Chapter Nonlinear Wave Phenomena in Periodic Media 1.6 u (e) −1.6 Im(λ) (f) −1.6 x (e) −1.6 x (f) 0.8 −0.8 (g) −1.6 x (g) 0.8 −0.8 (h) x (h) −4 Re(λ) 1.6 −4 Re(λ) 1.6 0 −4 −0.8 1.6 −4 Re(λ) 0.8 −0.8 Re(λ) 0.8 Figure 6.9 Upper row: soliton profiles at the marked points in the first gap of Fig 6.7 (σ = −1) Lower row: stability spectra of the solitons in the upper row the middle branch contain only one positive eigenvalue, while solitons on the upper branch contain two positive eigenvalues Stable Truncated-Bloch-Wave Solitons and Their Solution Families At every edge of a Bloch band, an infinitesimal spatially periodic Bloch wave exists (see Figs 6.1 and 6.2) When ω moves away from the edge (toward the left under self-focusing nonlinearity and toward the right under self-defocusing nonlinearity), this infinitesimal (linear) Bloch wave bifurcates out into a finite-amplitude (nonlinear) Bloch wave which is also spatially periodic (Alexander et al (2006), Wang et al (2009)) This nonlinear Bloch wave exists not only when ω lies in a bandgap, but also when ω lies in a Bloch band When ω lies inside a bandgap, we can truncate this nonlinear periodic Bloch wave to a finite number of intensity peaks and obtain a corresponding solitary wave This resulting solitary wave is called a truncated-Bloch-wave soliton (Wang et al (2009)) Such a soliton was first reported experimentally by Anker et al (2005) and theoretically investigated by Alexander et al (2006) and Wang et al (2009) To illustrate this type of solitons, we consider selfdefocusing nonlinearity and take V0 = A truncated-Bloch-wave soliton in the first gap is displayed in Fig 6.10(b) This soliton is truncated from the nonlinear Bloch wave which bifurcates out from the lower edge of the first Bloch band ω1 = 2.063182, and hence its seven intensity peaks are all in phase, similar to the linear Bloch wave at edge ω1 ; see Fig 6.2(a) This soliton, once in existence, will generate its own solution family The power curve of ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 287 ✐ ✐ 6.2 One-Dimensional Gap Solitons Not Bifurcated from Bloch Bands 60 287 (b) (a) (c) c 40 u P 0 b 20 first gap µ −2 x −2 x Figure 6.10 (a) Power curve for a family of truncated-Bloch-wave solitons in the first gap under self-defocusing nonlinearity (σ = −1); the dashed parts are unstable branches (b, c) Soliton profiles at the marked points on the lower and upper branches in (a) (After Wang et al (2009).) this family is shown in Fig 6.10(a) This curve also has triple branches While solutions on the lower branch have seven in-phase intensity peaks in (1, 1, 1, 1, 1, 1, 1) configuration (see Fig 6.10(b)), solutions on the middle branch develop one more intensity peak at one edge of the soliton and turn into (−1, 1, 1, 1, 1, 1, 1, 1) configuration, and solutions on the upper branch develop two more intensity peaks at both edges and turn into (−1, 1, 1, 1, 1, 1, 1, 1, −1) configuration; see Fig 6.10(c) Regarding the stability of this solution family, the lower branch is found to be linearly stable, while the middle and upper branches are linearly unstable (Wang et al (2009)) These behaviors are analogous to those of the in-phase dipole family under self-defocusing nonlinearity as shown in Figs 6.7 and 6.9 In fact, the in-phase dipole soliton in Fig 6.9(e) is also a truncated-Bloch-wave soliton, where the nonlinear Bloch wave is truncated to two intensity peaks instead of seven Thus these two soliton families are closely related and hence share similar properties One can carry this connection even further That is, we can also view the on-site gap soliton in Fig 6.4(d) as the simplest truncated-Bloch-wave soliton, where the nonlinear Bloch wave is truncated to a single intensity peak Under this viewpoint, truncated-Bloch-wave solitons then provide a link between infinitely extended nonlinear Bloch waves and the simplest on-site gap solitons (Alexander et al (2006)) If the nonlinear Bloch wave in the first gap is truncated to other numbers of intensity peaks, the corresponding solitons and their solution families can be found as well (Wang et al (2009)) The results above and in the previous section enable us to develop some intuition on the existence and stability of gap solitons in the lattice equation (6.3) On existence, these results suggest that for both self-focusing and self-defocusing nonlinearities, one can find gap solitons with an arbitrary arrangement of intensity peaks and phase structures For instance, one should be able to find solitons in (1, 1, 1), (1, 1, 1, −1), or (1, −1, 1, −1) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 288 ✐ ✐ 288 Chapter Nonlinear Wave Phenomena in Periodic Media configurations These intuitions are indeed correct, if the lattice potential is sufficiently deep For instance, with V0 = 6, we have found all these soliton structures for both selffocusing and self-defocusing nonlinearities Each of these solitons induces its own solution families, and thus infinite families of gap solitons exist in the lattice equation (6.3) Except for the on-site and off-site soliton families considered in the previous section, none of the other soliton families bifurcates from band edges, and thus their power curves always have multiple branches For each family, while the middle and upper branches of the power curve are always linearly unstable, the lowest branch can be stable The above stability results allow us to develop some intuition on the stability of gap solitons as well Notice that under self-focusing nonlinearity (σ = 1), out-phase dipoles (1, −1) and (−1, 1) (see Fig 6.8(a)) are stable, while in-phase dipoles (1, 1) and (−1, −1) (see Fig 6.4(e)) are unstable Under self-defocusing nonlinearity (σ = −1), the situation is just the opposite: in-phase dipoles are stable, while out-phase dipoles are unstable (see Figs 6.4(h) and 6.9(e)) These stability results on dipoles can help us predict the stability of gap solitons in more general configurations Specifically, if a soliton contains an in-phase dipole such as (1, 1) under self-focusing nonlinearity, or out-phase dipole such as (1, −1) under self-defocusing nonlinearity, then this soliton is expected to be linearly unstable For example, for the (1, 1, −1) soliton in Fig 6.8(b) under self-focusing nonlinearity, since it contains an in-phase dipole (1, 1), it is then expected to be unstable, in agreement with its spectrum in Fig 6.8 For another example, the (−1, 1, 1) soliton in Fig 6.9(f) contains an outphase dipole (−1, 1) under self-defocusing nonlinearity and thus is expected to be unstable, again in agreement with its spectrum in Fig 6.9 On the other hand, the (1, −1, 1, −1) soliton can be expected to be stable under self-focusing nonlinearity since it comprises only (stable) out-phase dipoles; and the (1, 1, 1, 1) soliton can be expected to be stable under self-defocusing nonlinearity since it comprises only (stable) in-phase dipoles These expectations have been verified for deep potentials (such as V0 = 6) We note in passing that the above intuition on gap-soliton stability in the lattice equation (6.3) under self-focusing nonlinearity is consistent with the analytical results by Pelinovsky et al (2005) on soliton stability in the discrete NLS equation under self-focusing nonlinearity in the weak intersite coupling limit (which corresponds to deep potentials in the lattice equation (6.3)) The above results and intuitions on the existence and stability of gap solitons hold only when the potential is sufficiently deep In shallow potentials, the lower branches of many solution families can become unstable as well due to oscillatory instabilities (Wang et al (2009)) What is more, many solution families can even totally disappear if the potential is too shallow (Wang et al (2009)) Lastly, we point out that the gap solitons considered in this section are relatively simple There are also many other gap solitons which are related to edges of higher Bloch bands and thus have more complicated profiles Such examples can be found in Efremidis and Christodoulides (2003), Pelinovsky et al (2004), and Zhang and Wu (2009) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 289 ✐ ✐ 6.3 Two-Dimensional Gap Solitons Bifurcated from Bloch Bands 289 6.3 Two-Dimensional Gap Solitons Bifurcated from Bloch Bands In this section, we study gap solitons in two spatial dimensions Many of the results for the 1D case above can be carried over to the 2D case In such cases, our description will be brief However, many new phenomena also arise in two dimensions, such as a much wider variety of gap-soliton structures, which often have no counterparts in one dimension Such results will be described in more detail The model equation (6.1) in two dimensions is iUt + Uxx + Uyy − V (x, y)U + σ |U |2 U = 0, (6.38) where σ = ±1 is the sign of the cubic nonlinearity, V (x, y) = V0 sin2 x + sin2 y (6.39) is the 2D periodic potential, and V0 is the depth of this potential Notice that the spatial periods of this potential along the x and y directions are the same This type of potential is sometimes called a square potential in the literature This potential is separable in x and y, which makes the theoretical analysis a little easier Similar analysis can be repeated for other types of periodic potentials and nonlinearities with minimal changes Solitary waves of Eq (6.38) are sought in the form U (x, y, t) = u(x, y)e−iωt , (6.40) where amplitude function u(x, y) is a solution of the following equation: uxx + uyy − [F (x) + F (y)]u + ωu + σ |u|2 u = 0, (6.41) F (x) = V0 sin2 x, (6.42) and ω is the wave’s frequency In the rest of this section, we will determine solitary waves in Eq (6.41) which are bifurcated from Bloch bands To so, we first need to understand Bloch bands and bandgaps of this 2D equation 6.3.1 2D Bloch Bands and Bandgaps When the solution u(x, y) is infinitesimal, Eq (6.41) becomes a linear equation, uxx + uyy − [F (x) + F (y)]u + ωu = (6.43) Since the periodic potential in this equation is separable, its Bloch solutions and Bloch bands can be constructed from solutions of the 1D equation Specifically, the 2D Bloch solution ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 290 ✐ ✐ 290 Chapter Nonlinear Wave Phenomena in Periodic Media u(x, y) of (6.43) and its frequency ω can be written in a separable form, ω = ωa + ωb , u(x, y; ω) = p (x; ωa ) p (y; ωb ), (6.44) where p (x; ω) is a Bloch solution of the linear 1D equation (6.7) at frequency ω Using the 1D results of Sec 6.1 and the above connection (6.44) between the 1D and 2D Bloch solutions, we can construct the dispersion surfaces and bandgap structures of the 2D problem (6.43) In particular, the 2D Bloch modes are of the form u(x, y; ω) = eikx x+iky y p [x; ω(kx )] p [y; ω(ky )], (6.45) where kx , ky are wavenumbers in the first Brillouin zone −1 ≤ kx , ky ≤ 1, p (x; ω) is a π-periodic function in x, ω(k) is the 1D dispersion relation, and ω = ω(kx ) + ω(ky ) (6.46) is the 2D dispersion relation This 2D dispersion relation at the potential depth V0 = is shown in Fig 6.11(a) This figure contains a number of dispersion surfaces whose ω values form the 2D Bloch bands Between these surfaces, two bandgaps can be seen At other potential depth V0 , the 2D bandgap structure is summarized in Fig 6.11(b) This figure reveals that, unlike the 1D case, there is only a finite number of bandgaps in the 2D problem at any given potential depth V0 In addition, bandgaps appear only when V0 is above a certain threshold (which is approximately 1.4 here) As the potential gets deeper, so does (b) (a) ω 14 12 A C E B D 10 V0 A: 1+1 B: 2+2 D: 2+4 C: 1+3 E: 1+5 k x −1 −1 ky 0 ω 10 Figure 6.11 (a) Dispersion surfaces of the linear 2D equation (6.43) at potential depth V0 = 6; (b) 2D bandgap structure for various potential depths V0 Letters A, B, C, D, E mark the edges of Bloch bands at V0 = Insets like “A:1 + 1” are explained in the text (After Shi and Yang (2007).) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 291 ✐ ✐ 6.3 Two-Dimensional Gap Solitons Bifurcated from Bloch Bands 291 the number of bandgaps At V0 = 6, two bandgaps exist The edges of these Bloch bands are marked in Fig 6.11(b) by letters A, B, C, D, E, respectively The locations of band edges in the first Brillouin zone are important, as they determine the symmetry properties of the corresponding Bloch modes In the first Brillouin zone, band edges A and B in Fig 6.11(b) are located at (kx , ky ) = (0, 0) and (1, 1), respectively These two locations are the center and corner of the Brillouin zone and are called and M points in the literature At these two points, only a single Bloch wave exists Band edges C and D, however, are located at (kx , ky ) = (0, 1) and (1, 0), which are on the edges of the Brillouin zone and are called X points in the literature At these X points, two linearly independent Bloch waves exist Bloch solutions at these band edges are u(x, y) = p (x; ω1 )p (y; ω1 ) for A, p (x; ω2 )p (y; ω2 ) for B, p (x; ω1 )p (y; ω3 ) and p (y; ω1 )p (x; ω3 ) for C, p (x; ω2 )p (y; ω4 ) and p (y; ω2 )p (x; ω4 ) for D, where ωk (1 ≤ k ≤ 4) are the lowest four 1D band edges (see Fig 6.1), and p (x; ωk ) are the corresponding 1D Bloch waves at these edges (see Fig 6.2) For convenience, we denote edge A as “1 + 1”, B as “2 + 2,” C as “1 + 3,” and D as “2 + 4.” The Bloch modes at A, B have the 90◦ rotational symmetry, i.e., u(x, y) = u(y, x) Adjacent peaks in the Bloch mode of A are in-phase, while those of B are out of phase The Bloch modes p (x; ω1 )p (y; ω3 ) and p (x; ω2 )p (y; ω4 ) at C, D not have this 90◦ rotational symmetry however Since the square lattice V (x, y) in (6.39) admits this rotational symmetry, this means that the companion modes p (y; ω1 )p (x; ω3 ) and p (y; ω2 )p (x; ω4 ), which are 90◦ rotations of p (x; ω1 )p (y; ω3 ) and p (x; ω2 )p (y; ω4 ), are Bloch modes at C, D as well The coexistence of several linearly independent Bloch modes at a band edge is a new feature in two spatial dimensions, and it has important implications for soliton bifurcations in the next subsection 6.3.2 Envelope Equations of 2D Bloch Waves In this subsection, we study bifurcations of small-amplitude Bloch-wave packets from edges of 2D Bloch bands and derive their envelope equations The main difference between 1D and 2D cases is that, because there can be two linearly independent Bloch modes at a 2D band edge, solitons can bifurcate from a linear combination of them This leads to coupled envelope equations for the two Bloch modes and a wider variety of soliton structures that often have no counterparts in 1D Our derivation will follow Shi and Yang (2007) In certain special cases, envelope equations similar to (6.54)–(6.55) but without the γ terms were also derived by Brazhnyi et al (2006, 2007) Let us first consider a generic 2D band edge ω0 , where two linearly independent Bloch modes coexist At such an edge one can write ω0 = ω0,1 + ω0,2 , where ω0,1 and ω0,2 are two 1D band edges with ω0,1 = ω0,2 The corresponding 2D Bloch modes are p1 (x)p2 (y) and p1 (y)p2 (x), where pn (x) = p (x; ω0,n ) is the 1D Bloch mode at the 1D edge ω0,n When ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 292 ✐ ✐ 292 Chapter Nonlinear Wave Phenomena in Periodic Media the solution u(x, y) of Eq (6.41) is infinitesimal, this solution (at the band edge) is a linear superposition of these two Bloch modes When u(x, y) is small but not infinitesimal, this solution then becomes a combination of these two Bloch-wave packets and can be expanded into a multiscale perturbation series, u = u0 + u1 + ω = ω0 + τ where u2 + · · · , (6.47) (6.48) , is the small amplitude of this solitary wave, u0 = A1 (X, Y )p1 (x)p2 (y) + A2 (X, Y )p2 (x)p1 (y) (6.49) is the leading-order two-Bloch-wave packets, X = x, Y = y are slow spatial variables of envelope functions A1 and A2 , and τ = ±1 The derivation of envelope equations for A1 and A2 below is analogous to that for the 1D case, and thus will only be sketched Substituting the above expansions into the 2D lattice equation (6.41), this equation at O( ) is automatically satisfied At O( ), the equation for u1 is ∂ u0 ∂ u0 ∂ u1 ∂ u1 [F + + − (x) + F (y)] u + ω u = −2 1 ∂x∂X ∂y∂Y ∂x ∂y (6.50) It is easy to verify that the solution to this equation is ∂A1 ∂A1 ∂A2 ∂A2 ν1 (x)p2 (y) + ν2 (y)p1 (x) + ν2 (x)p1 (y) + ν1 (y)p2 (x), ∂X ∂X ∂X ∂X where νn (x) is a periodic solution of the 1D equation u1 = νn,xx − F (x)νn + ω0,n νn = −2pn,x , n = 1, (6.51) (6.52) At O( ), the equation for u2 is ∂ u2 ∂ u2 + − [F (x) + F (y)] u2 + ω0 u2 ∂x ∂y ∂ u1 ∂ u1 ∂ u0 ∂ u0 =− + + τ u0 + |u0 |2 u0 +2 + ∂x∂X ∂y∂Y ∂X2 ∂Y (6.53) Substituting the formulae (6.49) and (6.51) for u0 and u1 into the right-hand side of this equation, utilizing the Fredholm condition, (which requires the inhomogeneous term in (6.53) to be orthogonal to the homogeneous solutions p1 (x)p2 (y) and p1 (y)p2 (x) over 2π period in x and y), and recalling the relation (6.19), the following coupled nonlinear equations for envelope functions A1 and A2 are obtained (Shi and Yang (2007)): D1 ∂ A1 ∂ A1 + D2 + τ A1 + σ α|A1 |2 A1 + β A∗1 A22 + 2A1 |A2 |2 ∂X ∂Y + γ |A2 |2 A2 + A∗2 A21 + 2A2 |A1 |2 = 0, (6.54) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 293 ✐ ✐ 6.3 Two-Dimensional Gap Solitons Bifurcated from Bloch Bands D2 293 ∂ A2 ∂ A2 + D + τ A2 + σ α|A2 |2 A2 + β A∗2 A21 + 2A2 |A1 |2 ∂X2 ∂Y + γ |A1 |2 A1 + A∗1 A22 + 2A1 |A2 |2 = (6.55) Here, the superscript “*” represents complex conjugation, Dn ≡ ω (k) (6.56) ω=ω0,n is the 1D second-order dispersion at 1D edge ω0,n , and α= 2π 2π 0 2π 2π 0 β= 2π 2π 2 2 0 p1 (x)p (x)p1 (y)p (y) dxdy , 2π 2π 2 0 p1 (x)p (y) dxdy (6.58) γ= 2π 2π 3 0 p1 (x)p (x)p (y)p1 (y) dxdy 2π 2π 2 0 p1 (x)p (y) dxdy (6.59) p14 (x)p 24 (y) dxdy p12 (x)p 22 (y) dxdy , (6.57) are nonlinear coefficients Notice that α and β are always positive, but γ may be positive, negative, or zero For instance, at edges C and D of Fig 6.11, γ = But at edge E, γ < At special 2D band edges where only a single Bloch mode exists (such as points A and B in Fig 6.11), the envelope equation for this single Bloch mode would be simpler In this case, the single Bloch mode has the form p1 (x)p1 (y) with frequency ω0 = 2ω0,1 , where p1 (x) = p (x; ω0,1 ) is the 1D Bloch wave at edge ω0,1 The leading-order solution now becomes u0 = A1 (X, Y )p1 (x)p1 (y), (6.60) and the envelope equation for A1 (X, Y ) can be readily found to be D1 ∂ A1 ∂ A1 + τ A1 + σ α0 |A1 |2 A1 = 0, + ∂X2 ∂Y (6.61) where D1 is as given in Eq (6.56), and the nonlinear coefficient α0 is α0 = 2π 2π 0 2π 2π 0 p14 (x)p 14 (y) dxdy p12 (x)p 12 (y) dxdy (6.62) For this single Bloch-wave packet, an envelope equation similar to (6.61) has also been derived by Baizakov et al (2002) Similarly to the 1D case, envelope solutions (A1 , A2 ) must be centered at certain special positions in the lattice owing to constraints that are the counterparts of (6.25) for the ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 294 ✐ ✐ 294 Chapter Nonlinear Wave Phenomena in Periodic Media 1D case Following similar calculations as in one dimension, we can show that the center (x0 , y0 ) of envelope solutions can only be located at four possible positions (Shi and Yang (2007)): (x0 , y0 ) = (0, 0), 0, π π π π , ,0 , , 2 2 (6.63) Solitary waves centered at (x0 , y0 ) = (0, 0) of minimum potential are called on-site solitons, while those centered at other three positions are called off-site solitons Envelope equations (6.54)–(6.55) and (6.61) admit localized solutions only when their coefficients satisfy certain conditions Thus, soliton bifurcations at 2D band edges are possible only under such conditions as well For instance, at edge A in Fig 6.11, D1 > Thus soliton bifurcation is possible only when σ = 1, i.e., the nonlinearity is self-focusing The situation at edge C is similar But situations at edges B and D are just the opposite, i.e., soliton bifurcations at these edges are possible only under self-defocusing nonlinearity 6.3.3 Families of 2D Gap Solitons Bifurcated from Band Edges Envelope equations (6.54)–(6.55) and (6.61) admit various types of solutions, and each envelope solution generates four families of gap solitons corresponding to the four envelope locations (6.63) Thus, many soliton families can bifurcate out from each edge of a 2D Bloch band Among these solution families, off-site soliton families are always linearly unstable, just like the 1D case Because of this, we will only consider on-site soliton families in the rest of this subsection At the band edge A in Fig 6.11, the envelope equation is the scalar 2D NLS equation (6.61) with D1 > Under self-focusing nonlinearity (σ = 1), this envelope equation admits a positive and radially symmetric ground-state solution A1 (X, Y ), whose corresponding gapsoliton solution u0 (x, y) in (6.60) is a positive wave packet (since the Bloch wave at edge A is positive everywhere as well) From this positive wave packet, a family of positive on-site solitons bifurcates out in the semi-infinite gap The power curve of this solution family is displayed on the left side of Fig 6.12 Here the power is defined as P = |u|2 dxdy as before A distinctive feature of this power curve is that it is nonmonotonic It has a nonzero minimum, below which solitons not exist This contrasts the 1D case where solitons exist at all power levels (see Fig 6.3) It should be noted that as ω approaches the edge A, the power P approaches a finite value, not infinity (see the inset in this power figure) In addition, the power P linearly approaches this finite value on the band edge Near edge A, the soliton is a slowly modulated positive Bloch-wave packet described by the analysis in the previous subsection This profile is displayed in Fig 6.12(b) It has one main peak at a lattice site, flanked by in-phase tails on all four sides Away from this edge, the soliton becomes more localized, and its tails gradually disappear; see Fig 6.12(a) This family ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 295 ✐ ✐ 6.3 Two-Dimensional Gap Solitons Bifurcated from Bloch Bands b b c d c a σ=1 σ=−1 c b semi−infinite gap a d P 295 first gap 4A ω B Figure 6.12 Left: power curves of on-site solitons bifurcated from edge points “A, B” of the first Bloch band under self-focusing and self-defocusing nonlinearities, respectively (V0 = 6) The insets show power curves near band edges, where asterisks are the limit power values on the edges Right: soliton profiles at the marked points in the left figure (After Shi et al (2008).) of solitons has been investigated both theoretically and experimentally by Fleischer et al (2003b), Yang and Musslimani (2003), and Efremidis et al (2003) At band edge B, the envelope equation is the scalar 2D NLS equation (6.61) with D1 < 0, which admits a positive ground-state solution under self-defocusing nonlinearity (σ = −1) The corresponding gap soliton is a sign-changing wave packet (since the Bloch wave at edge B is sign changing) From this wave packet, a family of on-site solitons bifurcates out in the first gap The power curve of this soliton family is also displayed at the left side of Fig 6.12 Similar to the soliton family bifurcated from edge A, this solution family also has a nonzero minimum power In addition, the power remains finite as ω approaches edge B Near the edge, the profile of the soliton is displayed in Fig 6.12(c) It has one main peak at a lattice site, flanked by out-of-phase tails on all sides Away from the edge, the soliton becomes more localized; see Fig 6.12(d) This solution family has been investigated theoretically and experimentally by Fleischer et al (2003b), Efremidis et al (2003), and Lou et al (2007) At edge C, there are two linearly independent Bloch waves, and their envelope equations are (6.54)–(6.55) with D1 , D2 > 0, D1 = D2 , and γ = This coupled system admits several types of envelope solutions under self-focusing nonlinearity One of them is A1 = 0, A2 = 0, i.e., the soliton is a wave packet of a single Bloch wave In this case, the A1 equation (6.54) is a single 2D NLS equation with different dispersion coefficients along the X and Y directions Thus, it admits a positive elliptical-shaped envelope soliton ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 296 ✐ ✐ 296 Chapter Nonlinear Wave Phenomena in Periodic Media 16 σ=1 a b c d c C 12 b P a C C first gap 4A B ω C Figure 6.13 Left: power curves of on-site solitons bifurcated from edge C of the second Bloch band under self-focusing nonlinearity (V0 = 6) Right: (a, b) soliton profiles at the marked points “a,b” in the left figure; (c, d) amplitude and phase distributions of the soliton at point “c” of the left figure The grids in (a, b, c) are the lattice sites (of minimal potentials) (After Shi et al (2008).) Bifurcating from the resulting Bloch-wave packet, a family of on-site gap solitons is then generated from edge C We denote this solution family as the C1 family, and its power curve is displayed at the left side of Fig 6.13 At ω = away from the edge C (see point “a” on the power curve), the soliton in this family is displayed in Fig 6.13(a) This soliton contains two equal-intensity peaks aligned along the vertical direction, and the two peaks are out of phase with each other However, different from the dipole solitons considered earlier (such as Fig 6.8(a)), the two peaks of the dipole here are located at a single lattice site, not at two different lattice sites Thus we can call this soliton a single-site dipole soliton Near edge C, the soliton in this C1 family becomes broad along the vertical direction, but remains relatively narrow along the horizontal direction Because of this, it was called a reduced-symmetry soliton by Fischer et al (2006), who first reported this type of solutions At edge C, envelope equations (6.54)–(6.55) also admit other solutions One of them is A1 > 0, A2 > These envelope solutions are both ellipse shaped but stretched along orthogonal directions (Shi and Yang (2007)) In this case, the soliton consists of wave packets of both Bloch waves From this solution, a family of on-site gap solitons bifurcates out from edge C We denote this solution family as the C2 family, and its power curve is also displayed at the left side of Fig 6.13 At ω = (see point “b” on the power curve), the soliton in this family is displayed in Fig 6.13(b) This soliton is also an out-phase dipole residing at a single lattice site, similar to the one in Fig 6.13(a) However, this single-site dipole is now aligned along the diagonal direction Near edge C, the soliton in this C2 ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 297 ✐ ✐ 6.4 Stability of 2D Gap Solitons Bifurcated from Bloch Bands 297 family becomes broad along both the horizontal and vertical directions This C2 family of solutions was first reported by Shi and Yang (2007) Another envelope solution admitted at edge C is where A1 > 0, and A2 is purely imaginary In this case, these envelopes of the two Bloch waves have a π/2 phase delay The family of on-site gap solitons bifurcating from these solutions is denoted as the C3 family, and its power curve is also displayed at the left side of Fig 6.13 Unlike the (realvalued) solitons in the C1 and C2 families, solitons in the C3 family are complex valued At ω = (see point “c” on the power curve), this complex soliton is displayed in Fig 6.13(c, d) Its intensity field consists of a ring at a lattice site, and weaker tails in the far distance The phase plot shows that when winding around this ring, the phase of the soliton increases by 2π Thus this soliton has a vortex structure with a unit topological charge These singlesite vortex solitons, for saturable nonlinearity, were first reported theoretically by Manela et al (2004), and then experimentally observed by Bartal et al (2005) At other band edges (such as D, E in Fig 6.11), various families of gap solitons bifurcate out under either self-focusing or self-defocusing nonlinearity as well See Shi and Yang (2007) for details 6.4 Stability of 2D Gap Solitons Bifurcated from Bloch Bands Stability of 2D gap solitons in the previous section is an important issue In 1D, on-site gap solitons can be linearly stable both near band edges and away from them (see Sec 6.1.5) The situation turns out to be different in two dimensions One may notice that the power curves of these 2D soliton families all have a nonzero power minimum (see Figs 6.12 and 6.13) Thus, in view of the results in Sec 5.4, solitons on one side of the power minimum must be linearly unstable But we not know yet which side of solitons are unstable The Vakhitov–Kolokolov stability criterion is not applicable either since these gap solitons are sign-indefinite in general In this section, we determine stability properties of 2D gap solitons bifurcated from Bloch bands As in one dimension, off-site 2D solitons are all linearly unstable Thus we only consider on-site solitons below First, we will show analytically that for solitons near band edges, if σ P (ω) > 0, then these solitons are linearly unstable due to the presence of a positive eigenvalue For self-focusing nonlinearity (σ = 1), solitons with power slope P (ω) > are linearly unstable This power-slope condition is the same as that in the Vakhitov–Kolokolov criterion of Sec 5.3 (notice the notational difference of ω = −µ, where µ is the propagation constant in Sec 5.3, thus P (ω) = −P (µ)) But for self-defocusing nonlinearity (σ = −1), solitons with power slope P (ω) < are linearly unstable This ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 298 ✐ ✐ 298 Chapter Nonlinear Wave Phenomena in Periodic Media power-slope condition becomes the opposite of that in the Vakhitov–Kolokolov criterion Thus our instability condition σ P (ω) > is an extension and modification of the Vakhitov– Kolokolov criterion to sign-indefinite solitons When combining this stability condition with the power curves in Figs 6.12 and 6.13, we see that these 2D on-site solitons near Bloch bands are all linearly unstable This means that solitons on the Bloch-band side of the power minimum point are linearly unstable Hence solitons on the other side of the power minimum point away from Bloch bands could be linearly stable (if no other instabilities arise) Numerically, we will show that on-site solitons bifurcating from edges A, B of the first band (see Fig 6.12) are indeed linearly stable on the side of the power minimum point away from the edges But solitons bifurcating from edge C of the second band (see Fig 6.13) remain unstable away from the edge due to additional instabilities (the same holds for solitons bifurcating from edges D and E as well) 6.4.1 Analytical Calculations of Eigenvalue Bifurcations near Band Edges In this subsection, we analytically calculate eigenvalues of on-site 2D solitons near band edges and show that positive (unstable) eigenvalues appear when σ P (ω) > Near a band edge, these solitons are low-amplitude Bloch-wave packets whose envelope equations are (6.54)–(6.55) or (6.61) These envelope equations are translation invariant (along both x and y directions), phase invariant, and their envelope solitons have a constant power independent of the frequency Due to these properties, the linearization spectrum of these envelope solitons has a zero eigenvalue of multiplicity eight: two associated with phase invariance, four associated with translation invariance, and the remaining two associated with constant power In the full lattice equation (6.38), the phase invariance persists, but the translation invariances are broken, and the power of gap solitons is no longer a constant (see Figs 6.12 and 6.13) As a result, in the linearization spectrum of a gap soliton near the band edge, the zero eigenvalue only has multiplicity two (induced by phase invariance), and the other six multiplicities of the zero eigenvalue have to bifurcate out The four multiplicities of the zero eigenvalue associated with spatial translations bifurcate out along the imaginary axis for on-site solitons, thus not creating instabilities (for off-site solitons, this bifurcation is along the real axis and hence symmetry-breaking instability is generated) These translation-induced eigenvalue bifurcations are analogous to the 1D case, and the bifurcated eigenvalues are exponentially small in the soliton amplitude The bifurcation of the remaining two multiplicities of the zero eigenvalue associated with the envelopesoliton’s constant power is new and has no counterpart in the 1D case This eigenvalue bifurcation can create instabilities to on-site 2D solitons and will be calculated analytically below We will show that this bifurcation is along the real axis and creates instability when ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐ ✐ ✐ 2010/10/8 page 299 ✐ ✐ 6.4 Stability of 2D Gap Solitons Bifurcated from Bloch Bands 299 σ P (ω) > A surprising feature of this bifurcation is that the bifurcated eigenvalue is algebraically small, rather than exponentially small, in the soliton amplitude This motivates us to use a power series expansion to calculate this eigenvalue This calculation follows Shi et al (2008) (except a notational change of ω = −µ) For simplicity, we consider real-valued gap solitons that are single-Bloch-wave packets near band edges; i.e., the A2 term in Eq (6.49) is absent These solitons include the ones bifurcating from edges A and B, as well as the C1 family bifurcating from the edge C, among others (see Figs 6.12 and 6.13) Near the band edge ω0 , this single-Bloch-wave packet soliton is given by u = u0 + u1 + u2 + ω = ω0 + τ u3 + · · · , , (6.64) (6.65) where is the small amplitude of the soliton, p1 (x)p2 (y) is a Bloch wave at edge ω0 , the leading-order term u0 is a slowly varying packet of this single Bloch wave, u0 = A(X, Y )p1 (x)p2 (y), (6.66) X = (x − x0 ), Y = (y − y0 ), and (x0 , y0 ) is the center position of the envelope function For on-site solitons, (x0 , y0 ) = (0, 0) In (6.66), we denoted the envelope function as A(X, Y ) rather than A1 (X, Y ) of Eq (6.49) for simplicity, and the other notations are the same as those in Sec 6.3.2 Here the 1D Bloch functions p1 (x) and p2 (x) can be the same such as at edges A and B They can also be different such as at edges C and D For real-valued gap solitons u, the envelope function A(X, Y ) is also real and satisfies the stationary 2D NLS equation D1 ∂ 2A ∂ 2A + D + τ A + σ αA3 = ∂X ∂Y (6.67) When p1 (x) = p2 (x), then D1 = D2 ; hence this equation reduces to (6.61) When p1 (x) = p2 (x), this equation is a reduction of the coupled envelope equations (6.54)–(6.55) by taking A2 = 0, which is possible at band edges where γ = (such as edges C and D) The envelope equation (6.67) admits a positive and single-humped ground-state solution A(X, Y ) > 0, which induces families of gap solitons such as those bifurcated from edges A and B in Fig 6.12 and the C1 family bifurcated from the edge C in Fig 6.13 Stability of such gap solitons near band edges will be analyzed below Before we conduct the stability analysis of such real-valued 2D gap solitons, we need some information on these solitons in their asymptotic expansion (6.64) To obtain such information, we first define the following linear operator: L0 = ∂2 ∂2 + − [F (x) + F (y)] + ω + σ u2 (x, y) ∂x ∂y (6.68) ✐ ✐ Please purchase PDF Split-Merge on www.verypdf.com to remove this watermark ✐ ✐