Commun. Theor. Phys. 68 27 425 438 Vol. 68, No. 4, October, 27 Initial-Boundary Value Problem for Two-Component Gerdjiov Ivanov Equation with 3 3 Lax Pair on Half-Line Qiao-Zhen Zhu 朱巧珍,, En-Gui Fan 范恩贵,, and Jian Xu 徐建 2, School of Mathematical Sciences, Fudan University, Shanghai 2433, China 2 College of Science, University of Shanghai for Science and Technology, Shanghai 293, China Received July 6, 27; revised manuscript received August 7, 27 Abstract The Foas unified method is used to analyze the initial-boundary value problem of two-component Gerdjiov Ivanonv equation on the half-line. It is shown that the solution of the initial-boundary problem can be expressed in terms of the solution of a 3 3 Riemann Hilbert problem. The Dirichlet to Neumann map is obtained through the global relation. PACS numbers: 2.3.I, 2.3.Rz, 3.65.N DOI:.88/253-62/68/4/425 Key words: two-component Gerdjiov Ivanov equation, initial-boundary value problem, Foas unified method, Riemann Hilbert problem Introduction The Inverse Scattering Transform IST is a powerful tool to investigate the initial value problem for integrable evolution equations on the line, which is discovered in 967. [ While the IST method can only solve the pure initial value problems. In many situations, we need to consider the initial-boundary value IBV problems instead of pure initial value problems. In 997, Foas announced a unified transformation method to analyze the IBV problems for linear and nonlinear integrable PDEs. [2 4 The Foas method is a generalization of the IST formalism from initial value to IBV problems. Based on the simultaneous spectral analysis, one can yield the solution in terms of the solution of a matrix Riemann Hilbert RH problem formulated in the complex plane, where is the spectral parameter of the Lax pair. The jump matrix of this RH problem depends on the variables x, t and it is uniquely defined in terms of some functions of called the spectral functions. These functions can be expressed by the boundary values. However, for a well-posed initialboundary value problem IBV, only part of the boundary ψ x = ψ t = iλ 2 + i q 2 /2 λq λq iλ 2 i q 2 /2 ψ, 2iλ 4 + i q 2 λ 2 + qq x q q x /2 + i q 4 /4 2qλ 3 + iq x λ values can be prescribed, but they are in general related. The most difficult step in the solution of boundary value problems is the determination of those spectral functions. This can be achieved by analysing the so-called global relation. The Foas method was usually used to analyze the IBV problem for integrable PDEs with 2 2 Lax pair. [5 In 22, Lenells extended this method to the IBV problem of integrable systems with 3 3 Lax pair on the halfline. [2 After that, several important integrable equations with 3 3 Lax pair have been investigated, including the Degasperis Procesi equation, Sasa Satsuma equation, Ostrovsy Vahneno equation etc. [3 5 The Gerdjiov Ivanov GI equation q t = iq xx + q 2 q x + i 2 q3 q 2, is a celebrated derivative nonlinear Schrodinger DNLS equation first presented by Gerdjiov and Ivanov. [6 The GI equation is an integrable system since it admits a 2 2 Lax pair 2qλ 3 + iq x λ 2iλ 4 i q 2 λ 2 qq x q q x /2 i q 4 /4 ψ, 2 Supported by grants from the National Science Foundation of China under Grant No. 6795, National Science Foundation of China under Grant No. 5365, Shanghai Sailing Program supported by Science and Technology Commission of Shanghai Municipality under Grant No 5YF48, and the Hujiang Foundation of China B45 E-mail: qiaozhenzhu3@fudan.edu.cn Corresponding author, E-mail: faneg@fudan.edu.cn E-mail: jianxu@usst.edu.cn c 27 Chinese Physical Society and IOP Publishing Ltd http://www.iopscience.iop.org/ctp http://ctp.itp.ac.cn
426 Communications in Theoretical Physics Vol. 68 where the asteris denotes complex conjugate, the subscripts t and x denote partial differentiations, and λ is a spectral parameter. In recent years, there has been much wor on the GI equation, including Darboux transformation, [7 Hamiltonian structures, [8 algebrogeometric solutions, [9 rouge wave and breather soliton, [2 envelope bright and dar soliton solution. [2 Recently, Zhang, Cheng and He proposed the following two-component GI equation [22 q t =iq xx + q q q x + q 2 q 2x + i 2 q q 4 + q 2 4 + i q q 2 2 q, q 2t = iq 2xx + q 2 q q x + q 2 q 2x + i 2 q 2 q 4 + q 2 4 + i q q 2 2 q 2, 3 which is a two-component generalization of the classical GI equation. The system 3 is also integrable and but different from GI equation since it is related to a 3 3 matrix Lax pair where Ψ x = i 2 Λ + U Ψ, Ψ t = 2i 4 Λ + V Ψ, 4 U = QΛ + i 2 Q2 Λ, V = 2 3 QΛ + i 2 ΛQ 2 + iq x 2 [Q x, Q + i 4 Q4 Λ. 5 q q 2 Λ =, Q = q. 6 q 2 In Ref. [22, the N-soliton solutions obtained by Riemann Hilbert method, where discrete spectral λ and initial problem of q x = qx, t = in Schwartz space are considered. Inspired by our recent wor on the IBV problem of two-component NLS equation, [23 we would lie to consider the general IBV problem of two-component GI equation 3 on the half-line x under the following boundary conditions q x, t = = q x, q 2 x, t = = q 2 x, q x =, t = g t, q 2 x =, t = g 2 t, q x x =, t = g t, q 2x x =, t = g 2 t, 7 where q and q 2 lie in Schwartz space. We show that the solution of the initial-boundary problem can be expressed in terms of the solution of a 3 3 Riemann Hilbert problem. Organization of this paper is as follows. In the following Sec. 2, we perform the spectral analysis of the associated Lax pair for the two-component GI equation 3. Then in Sec. 3, we formulate the main Riemann Hilbert problem for the IBV problem of two-component GI equation 3. Finally, we also get the map between the Dirichlet and the Neumann boundary problem through analysising the global relation in Sec. 4. 2 Spectral Analysis 2. The Closed One-Form Suppose that q x, t and q 2 x, t are sufficiently smooth and decay functions as x in the half-line domain Ω = { x <, < t < T }. Maing a transformation Ψ = µ e iλ2 x 2iΛ 4t, 8 then we find that µx, t, satisfies a new Lax pair µ x + i 2 [Λ, µ = U µ, µ t + 2i 4 [Λ, µ = V µ. 9 Letting  denotes the operators which acts on a 3 3 matrix X by ÂX = [A, X, then one can rewrite the equations in Eq. 9 in differential form as d e i2 x+2i 4 tˆλµ = W, where the closed one-form W x, t, is defined by W = e i2 x+2i 4 tˆλu dx + V dtµ. Equations 9 are the coordinate form of Eq. ; However, for the implementation of Foas method, it is often convenient to use Eq.. 2.2 The Eigenfunctions µ j s We define three eigenfunctions {µ j } 3 of Eq. 9 by the Volterra integral equations µ j x, t, = I + e i 2 x+2i 4 tˆλw j x, t,, γ j j =, 2, 3, 2 where W j is given by Eq. with µ replaced with µ j, and the contours {γ j } 3 are showed in Fig.. Fig. The three contours γ, γ 2, and γ 3 in the x, t- domain. The first, second and third column of the matrix equation 2 involves the exponentials [µ j : e 2i2 x x +4i 4 t t, e 2i2 x x +4i 4 t t, [µ j 2 : e 2i2 x x 4i 4 t t, [µ j 3 : e 2i2 x x 4i 4 t t. 3 The contours {γ j } 3 can be given by the following inequalities: γ : x x, t t, γ 2 : x x, t t, γ 3 : x x. 4
No. 4 Communications in Theoretical Physics 427 So, these inequalities imply that the functions {µ j } 3 are bounded and analytic for C such that belongs to µ : D 2, D 3, D 3, µ 2 : D, D 4, D 4, µ 3 : D 3 D 4, D D 2, D D 2, 5 where {D n } 4 denote four open, pairwisely disjoint subsets of the complex plane showed in Fig. 2. And the sets {D n } 4 admit the following properties: D = { C Re l < Re l 2 = Re l 3, Re z < Re z 2 = Re z 3 }, D 2 = { C Re l < Re l 2 = Re l 3, Re z > Re z 2 = Re z 3 }, D 3 = { C Re l > Re l 2 = Re l 3, Re z < Re z 2 = Re z 3 }, D 4 = { C Re l > Re l 2 = Re l 3, Re z > Re z 2 = Re z 3 }, where l i and z i are the diagonal entries of matrices i 2 Λ and 2i 4 Λ, respectively. In fact, for x =, µ, t, has enlarged domain of boundedness: D 2 D 4, D D 3, D D 3, and µ 2, t, has enlarged domain of boundedness: D D 3, D 2 D 4, D 2 D 4. Fig. 2 The sets D n, n =,..., 4, which decompose the complex -plane. 2.3 The Spectral Functions M n s For each n =,..., 4, a solution M n x, t, of Eq. 9 can be defined by the following system of integral equations: M n ij x, t, = δ ij + e i2 x+2i 4 tˆλw n x, t, ij, D n, i, j =, 2, 3, 6 γij n where W n is given by Eq. with µ replaced with M n, and the contours γij n, n =,..., 4, i, j =, 2, 3 are defined by γ if Re l i < Re l j and Re z i Re z j, γij n = γ 2 if Re l i < Re l j and Re z i < Re z j, for D n. 7 γ 3 if Re l i Re l j According to the definition of the γ n, one finds that γ 3 γ 2 γ 2 γ 3 γ γ γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ = γ 3 γ 3 γ 3, γ 2 = γ 3 γ 3 γ 3, γ 3 = γ γ 3 γ 3, γ 4 = γ 2 γ 3 γ 3. 8 γ 3 γ 3 γ 3 γ 3 γ 3 γ 3 γ γ 3 γ 3 γ 2 γ 3 γ 3 The following proposition ascertains that the M n s defined in this way have the properties required for the formulation of a Riemann Hilbert problem. Proposition For each n =,..., 4, the function M n x, t, is well-defined by Eq. 6 for D n and x, t Ω. Moreover, M n admits a bounded and continous extension to D n and M n x, t, = I + O,, D n. 9 Proof Analogous to the proof provided in Ref. [2. Remar Of course, for any fixed point x, t, M n is bounded and analytic as a function of D n away from a possible discrete set of singularities { j } at which the Fredholm determinant vanishes. The bounedness and analyticity properties are established in appendix B in Ref. [2. 2.4 The Jump Matrices The spectral functions {S n } 4 can be defined by S n = M n,,, D n, n =,..., 4. 2 Let M denote the sectionally analytic function on the Riemann plane which equals M n for D n. Then M satisfies the jump conditions M n = M m J m,n, D n D m, n, m =,..., 4, n m, 2 where the jump matrices J m,n x, t, are given by J m,n = e 2i2 x+4i 4 tˆλs m S n. 22 2.5 The Adjugated Eigenfunctions As the expressions of S n will involve the adjugate
428 Communications in Theoretical Physics Vol. 68 matrix of {s, S} defined in the next subsection. We will also need the analyticity and boundedness properties of the minors of the matrices {µ j x, t, } 3. We recall that the adjugate matrix X A of a 3 3 matrix X is defined by m X m 2 X m 3 X X A = m 2 X m 22 X m 23 X, m 3 X m 32 X m 33 X where m ij X denote the ijth minor of X. It follows from Eq. 9 that the adjugated eigenfunction µ A satisfies the Lax pair µ A x i 2 [Λ, µ A = U T µ A, µ A t 2i 4 [Λ, µ A = V T µ A, 23 where V T denotes the transform of a matrix V. Thus, the eigenfunctions {µ A j }3 are solutions of the integral equations µ A j x, t, = I γ j e i 2 x x +2i 4 t t ˆΛ U T dx + V T dtµ A, j =, 2, 3. 24 Then we can get the following analyticity and boundedness properties: µ A : D 3, D 2, D 2, µ A 2 : D 4, D, D, µ A 3 : D D 2, D 3 D 4, D 3 D 4. 25 s S = s 2 m 33 s/s m 32 s/s, In fact, for x =, µ A, t, has enlarged domain of boundedness: D D 3, D 2 D 4, D 2 D 4, and µ A 2, t, has enlarged domain of boundedness: D 2 D 4, D D 3, D D 3. 2.6 The J m,n s Computation Let us define the 3 3 matrix value spectral functions s and S by Thus, µ 3 x, t, = µ 2 x, t, e i2 x+2i 4 tˆλs, µ x, t, = µ 2 x, t, e i2 x+2i 4 tˆλs. 26 s = µ 3,,, S = µ,,. 27 And we deduce from the properties of µ j and µ A j that s and S have the following boundedness properties: Moreover, s : D 3 D 4, D D 2, D D 2, S : D 2 D 4, D D 3, D D 3, s A : D D 2, D 3 D 4, D 3 D 4, S A : D D 3, D 2 D 4, D 2 D 4. M n x, t, = µ 2 x, t, e i2 x+2i 4 tˆλs n, D n. 28 Proposition 2 The S n defined in Eq. 2 can be expressed in terms of the entries of s and S as follows: s 3 m 23 s/s m 22 s/s m 33 sm 2 S m 23 sm 3 S m 32 sm 2 S m 22 sm 3 S s s T S A s T S A m 33 sm S m 3 sm 3 S m 32 sm S m 2 sm 3 S S 2 = s 2 s T S A s T S A, m 23 sm S m 3 sm 2 S m 22 sm S m 2 sm 2 S s 3 s T S A s T S A S S T s 2 s 3 s A S s 2 s 3 2 m s S 3 = S T s 22 s 23, S s A 4 = s 22 s 23. 29 S 3 s 32 s 33 S T s 32 s 33 s A Proof Let γ X 3 denote the contour X, x, t in the x, t plane, here X > is a constant. We introduce µ 3 x, t, ; X as the solution of Eq. 2 with j = 3 and with the contour γ 3 replaced by γ X 3. Similarly, we define M n x, t, ; X as the solution of Eq. 6 with γ 3 replaced by γ X 3. We will first derive expression for S n ; X = M n,, ; X in terms of S and s; X = µ 3,, ; X. Then Eq. 29 will follow by taing the limit X. First, We have the following relations: M n x, t, ; X = µ x, t, e i2 x+2i 4 tˆλr n ; X, M n x, t, ; X = µ 2 x, t, e i2 x+2i 4 tˆλs n ; X, M n x, t, ; X = µ 3 x, t, e i2 x+2i 4 tˆλt n ; X.3 Then R n ; X and T n ; X are given as follows: R n ; X = e 2i4 T ˆΛM n, T, ; X, T n ; X = e i2 xˆλm n X,, ; X. 3
No. 4 Communications in Theoretical Physics 429 The relations 3 imply that s; X = S n ; X T n ; X, S = S n ; X R n ; X. 32 These equations constitute a matrix factorization problem which, given {s, S} can be solved for the {R n, S n, T n }. Indeed, the integral equations 6 together with the definitions of {R n, S n, T n } imply that R n ; X ij = if γ n ij = γ, S n ; X ij = if γ n ij = γ 2, T n ; X ij = δ ij if γ n ij = γ 3. 33 It follows that Eqs. 32 are 8 scalar equations for 8 unnowns. By computing the explicit solution of this algebraic system, we find that {S n ; X } 4 are given by the equation obtained from Eq. 29 by replacing {S n, s} with {S n ; X, s; X }. Taing X in this equation, we arrive at Eq. 29. 2.7 The Residue Conditions Since µ 2 is an entire function, it follows from Eq. 28 that M can only have sigularities at the points where the S ns have singularities. We denote the possible zeros by { j } N and assume they satisfy the following Assumption 3. We assume that i s has n possible simple zeros in D denoted by { j } n ; ii s T S A has n n possible simple zeros in D 2 denoted by { j } n n + ; iii S T s A has n 2 n possible simple zeros in D 2 denoted by { j } n 2 n ; + iv m s has N n 2 possible simple zeros in D 4 denoted by { j } N n 2 +; and that none of these zeros coincide. Moreover, we assume that none of these functions have zeros on the boundaries of the D n s. We determine the residue conditions at these zeroes in the following: Proposition 3 Let {M n } 4 be the eigenfunctions defined by Eq. 6 and assume that the set { j } N of singularities are as the above assumption. Then the following residue conditions hold: Res =j [M 2 = m 33s j ṡ j s 2 j e 2θ j [M j, j n, j D. 34a Res =j [M 3 = m 22s j ṡ j s 3 j e 2θj [M j, j n, j D, 34b Res =j [M 2 = m 33s j M 2 S j m 23 s j M 3 S j s T S A j s j Res =j [M 3 = m 32s j M 2 S j m 22 s j M 3 S j s T S A j s j Res =j [M = s 33 j S 2 j s 23 j S 3 j S T s A j m s j n + j n 2, j D 3, Res =j [M = s 33 j [M j 2 s 23 j [M j 3 ṁ s j m 2 s j e 2θ j [M j, n + j n, j D 2, 34c e 2θ j [M j, n + j n, j D 2, 34d e 2θ j [M j 2 + s 22 j S 3 j s 32 j S 2 j S T s A j m s j e 2θ j [M j 3 34e e 2θ j, n 2 + j N, j D 4, 34f where f = df/d, and θ is defined by θx, t, = i 2 x + 2i 4 t. 35 Proof We will prove Eqs. 34a, 34e, the other conditions follow by similar arguments. Equation 28 implies the relation M = µ 2 e θ ˆΛS, M 3 = µ 2 e θ ˆΛS 3. 36a 36b In view of the expressions for S and S 3 given in Eq. 29, the three columns of Eq. 36a read: [M = [µ 2 s + [µ 2 2 s 2 e 2θ + [µ 2 3 s 3 e 2θ, 37a [M 2 = [µ 2 2 m 33 s s [M 3 = [µ 2 2 m 32 s s + [µ 2 3 m 23 s s, 37b + [µ 2 3 m 22 s s, 37c while the three columns of Eq. 36b read: [M 3 = [µ 2 S S T s A + [µ 2 2 S 2 S T s A e 2θ + [µ 2 3 S 3 S T s A e 2θ, 38a [M 3 2 = [µ 2 s 2 e 2θ + [µ 2 2 s 22 + [µ 2 3 s 32, 38b [M 3 3 = [µ 2 s 3 e 2θ + [µ 2 2 s 23 + [µ 2 3 s 33. 38c We first suppose that j D is a simple zero of s. Solving Eq. 37a for [µ 2 2 and [µ 2 3, substituting the result into Eqs. 37b and 37c, we find [M 2 = m 33s[M e 2θ m 33s[µ 2 e 2θ s s 2 s 2 + m 3s[µ 2 3 s 2,
43 Communications in Theoretical Physics Vol. 68 [M 3 = m 22s[M e 2θ m 22s[µ 2 e 2θ s s 3 s 3 m 2s[µ 2 2 s 3. Taing the residue of this equation at j, we find the condition 34a and 34b in the case when j D. In order to prove Eq. 34e, we solve Eqs. 38b, 38c for [µ 2 2 and [µ 2 3, then substituting the result into Eq. 38a, we find [M 3 = [µ 2 m s + s 33S 2 s 23 S 3 m ss T s A [M 3 2 e 2θ + s 22S 3 s 32 S 2 m ss T [M 3 3 e 2θ. 39 s A Taing the residue of this equation at j, we find the condition 34e in the case when j D 3. 2.8 The Global Relation The spectral functions S and s are not independent but satisfy an important relation. Indeed, it follows from Eq. 26 that µ 3 x, t, = µ x, t, e i2 x+2i 4 tˆλs s, D 3 D 4, D D 2, D D 2. 4 Since µ, T, = I, evaluation at, T yields the following global relation: S s = e 2i4 T ˆΛcT,, D 3 D 4, D D 2, D D 2, 4 where ct, = µ 3, T,. 3 Riemann Hilbert Problem In Sec. 2, we introduce the sectionally analytic function Mx, t,. It satisfies a Riemann Hilbert problem which can be formulated in terms of the initial and boundary value of q x, t and q 2 x, t. By solving the Riemann Hilbert problem, we can recover the solution of Eq. 3 for all values of the independent variables x, t. Theorem Suppose that q x, t and q 2 x, t are a pair of solutions of Eq. 3 in the half-line domain Ω with sufficient smoothness and decays as x. Then q x, t and q 2 x, t can be reconstructed from the initial value {q x, q 2 x} and boundary values {g t, g 2 t, g t, g 2 t} defined as follows, q x = q x,, q 2 x = q 2 x,, g t = q, t, g 2 t = q 2, t, g t = q x, t, g 2 t = q 2x, t. 42 Use the initial and boundary data to define the jump matrices J m,n x, t,, m, n =,..., 4 as well as the spectral s and S by Eq. 26. Assume that the possible zeroes { j } N of the functions s, s T S A, S T s A and m s are as in Assumption 3. Then the solution {q x, t, q 2 x, t} is given by q x, t = 2i lim Mx, t, 2, q 2 x, t = 2i lim Mx, t, 3, 43 where Mx, t, satisfies the following 3 3 matrix Riemann Hilbert problem: i M is sectionally meromorphic on the Riemann plane with jumps across the contours D n D m, n, m =,..., 4, see Fig. 2. ii Across the contours D n D m, M satisfies the jump condition M n x, t, = M m x, t, J m,n x, t,, D n D m, n, m =, 2, 3, 4. 44 iii Mx, t, = I + O/,. iv The residue condition of M is showed in Proposition 3. Proof It only remains to prove Eq. 43 and this equation follows from the large asymptotics of the eigenfunctions, see Subsec. 4. for details. 4 Non-Linearizable Boundary Conditions The most difficulty of initial-boundary value problems is that some of the boundary values are unnown for a well-posed problem. On the other hand, all boundary values are needed for the definition of S, and hence for the formulation of the Riemann Hilbert problem. Our main result, Theorem 2, expresses the spectral function S in terms of the prescribed boundary data and the initial data via the solution of a system of nonlinear integral equations. 4. Asymptotics An analysis of Eq. 9 shows that the eigenfunctions {µ j } 3 have the following asymptotics as : µ j x, t, = I + + 4 µ 4 µ 4 µ 4 2 µ 4 µ µ µ 2 µ 2 µ 3 22 µ 23 µ 3 µ 32 µ 33 2 µ 4 3 22 µ 4 23 µ 4 3 µ 4 32 µ 4 33 + 2 + O 5 = I + µ 2 µ 2 2 µ 2 µ 2 2 µ 2 3 22 µ 2 23 µ 2 3 µ 2 32 µ 2 33 2i q 2i q 2i q 2 + 3 2i q 2 µ 3 µ 3 µ 3 2 µ 3 2 µ 3 3 22 µ 3 23 µ 3 3 µ 3 32 µ 3 33
No. 4 Communications in Theoretical Physics 43 + 2 + 3 + 4 x,t x j,t j dx + η dt x,t x,t 22 dx + η 22 dt 23 dx + η 23 dt x j,t j x j,t j x,t x,t 32 dx + η 32 dt 33 dx + η 33 dt x j,t j x j,t j 2i q µ 2 22 + q 2µ 2 32 + 4 q x + q 8i q 2 2i q µ 2 23 + q 2µ 2 33 + 4 q 2x + q2 2i q µ 2 4 q x + q 8i q 2 2i q 2µ 2 4 q 2x + q 2 8i q 2 µ 4 µ 4 22 µ 4 23 µ 4 32 µ 4 33 + O 5, 8i q 2 45 where q 2 = q 2 + q 2 2, 46 = i 8 q 4 4 q q x + q 2 q 2x, 22 = i 8 q 2 q 2 4 q q x, 23 = i 8 q q 2 q 2 4 q q 2x, 32 = i 8 q q 2 q 2 4 q x q 2, 33 = i 8 q 2 2 q 2 4 q 2q 2x. η = 8i q 6 4 q q x + q 2 q 2x q q x q 2 q 2x q 2 + 4i q x 2 + q 2x 2 4 q q t + q 2 q 2t, η 22 = 8i q 2 q 4 + 4 q 2 q q x q q x + 8 q q x q 2 2 q 2 q 2x q 2 + q 2 q 2x q 2 q q x q 2 2 + i 4 q x 2 4 q q t, η 23 = 8i q q 2 q 4 + 4 q 2q x q 2 q q 2x q 2 2 47a + 8 q 2q x q 2 2 q 2 q 2 q x + q q2 2 q 2x q q 2x q 2 + i 4 q xq 2x 4 q q 2t, η 32 = 8i q q 2 q 4 + 4 q q 2x q 2 2 q 2 q x q 2 + 8 q q 2x q 2 q q 2 2 q 2x + q 2 q 2 q x q 2 q x q 2 2 + i 4 q xq 2x 4 q 2q t, η 33 = 8i q 2 2 q 4 + 4 q 2 2 q 2 q 2x q 2 q 2x + 8 q q x q 2 2 q q x q 2 2 + q 2 q 2x q 2 q 2 q 2x q 2 + i 4 q 2x 2 4 q 2q 2t. 47b Remar 2 The explicit formulas of µ 4 and µ4 ij, i, j = 2, 3 are not presented in the following analysis, we do not need the asymptotic expressions of these functions. Next, we define functions {Φ ij t, } 3 i,j= by: Φ t, Φ 2 t, Φ 3 t, µ 2, t, = Φ 2 t, Φ 22 t, Φ 23 t,. 48 Φ 3 t, Φ 32 t, Φ 33 t, We partition matrix as the following, Φ Φ µ 2, t, =, j = 2, 3, 49 Φ j Φ where Φ denotes a 2 2 matrix, Φ denotes a 2 vector, Φ j denotes a 2 vector. From the asymptotic of µ j x, t, in Eq. 45 we have µ 2, t, = I + + 3 2i q 2i q 2i qt,t, + 2,t, dx + η dt,t 2 + η 2 dt,,t 2i qt, dx + η dt 4 q x + 8i qqt q 2 dx + η 2 dt + 4 qt x + q T q 8i qt
432 Communications in Theoretical Physics Vol. 68 + 4 µ 4 µ 4 + O 5, where q = q, q 2 T, is defined by first identities of Eq. 47a, η is defined by first identities of Eq. 47b, 2 and η 2 are 2 2 matrices defined as the following, 2 2 22 2 2 23 η = 2 32 2, η 2 22 η 2 23 = 33 η 2 32 η 2. 5 33 Also, we have where Φ t, = Φ t + Φ2 t 2 Φ t, = I + Φ t Φ t = 2i gt t, Φ 2 + Φ 3 t 3 + Φ2 t 2 t =,t, 5 + Φ 4 t 4 + O 5,, D 2 D 4, 52a + Φ3 t 3 + Φ4 t 4 + O 5,, D 2 D 4, 52b 2 dx + η 2 dt, Φ 3 t = 2i gt Φ 2 t + 4 gt + 8i gt g T ḡ, here g t and g t are vector boundary functions defined by the boundary data of Eq. 7 as g t = g t, g 2 t T and g t = g t, g 2 t T. In particular, we find the following expressions for the boudary values: g T t = 2iΦ t, From the global relation 4 and replacing T by t, we find 53a g T t = 4Φ 3 t + 2igT Φ 2 t + i 2 gt g T ḡ. 53b µ 2, t, e 2i4 tˆλs = ct,, D 3 D 4, D D 2, D D 2. 54 Then we can write the columns of the global relation, undering the matrix partitioned as Eq. 49, as Φ t, s s e 4i4t + Φ t, = c t, s, D D 2, 55a Φ j t, s s e 4i4t + Φ t, = c t, s, D D 2. 55b The functions c t,, c t, are analytic and bounded in D D 2 away from the possible zeros of s, s T S A and of order O/ 3 as. We will also need the asymptotic of c t, and c t,. Lemma The global relation 55 implies that the large behavior of c t, and c t, satisfy c t, s c t, s = Φ t + Φ2 t 2 Φ = I + t + Φ 3 t 3 + Φ2 t Proof Recalling the Lax pair equation 9, we have 2 + Φ 4 t 4 + O 5,, D D 2, 56a + Φ3 t 3 + Φ4 t 4 + O 5,, D D 2. 56b µ t + 2i 4 [Λ, µ = V µ. 57 From the second column of Eq. 57 we get Φ t + 4i 4 Φ = i 2 q T q 2 qt x q q T q x + i q 2 Φ 4 qt + 2 3 q T + iqx T Φ Φ Φ t = 2 3 q + i q x Φ + i 2 qq T 2 q xq T qq x T i q qq 4 qt T Φ, 58 where q = q, q 2 T is a 2 vector function. Suppose Φ s admit an expansion of the form Φ = α t+ α t + α 2t 2 + α 3t 3 + α 4t 4 + + β t+ β t + β 2t 2 where the coefficients α l t and β l t, l are column vectors which are independent of. coefficients, we substitute the above equation into Eq. 58 and use the initial conditions + β 3t 3 + β 4t 4 + e 4i4t, 59 α + β = 2 I T, α + β = 2 T, α 2 + β 2 = 2, T. To determine these
No. 4 Communications in Theoretical Physics 433 Then we get Φ Φ = 2 I + Φ [ Φ + Φ + 2 Φ 2 + 2 Φ 2 Φ j Φ + 3 Φ 3 + O Φ 3 From the first column of Eq. 57 we get Φ t = i 2 q T q 2 qt x q q T q x + i q 2 Φ 4 qt + 2 3 q T + iqx T Φ j, Suppose Φ = Φ j + O 4 4 e 4i4t. 6 Φ jt 4i 4 Φ j = 2 3 q + i q x Φ + i 2 qq T 2 q xq T qq x T i q qq 4 qt T Φ j. 6 ξ t + ξ t + ξ 2t 2 + ξ 3t 3 + ξ 4t 4 + + ν t + ν t + ν 2t 2 + ν 3t 3 + ν 4t 4 + e 4i4t, 62 where the coefficients ξ l t and ν l t, l are column vectors which are independent of. coefficients, we substitute the above equation into Eq. 6 and use the initial conditions Then we get ξ + ν =, 2 T, ξ + ν =, 2 T, ξ 2 + ν 2 =, 2 T. Φ Φ j = 2 [ + + Φ j Φ Φ j + 2 Φ 2 Φ 2 j + Φ Φ j 2 + 3 Φ 3 + O Φ 3 j + O 5 To determine these 4 e 4i4t. 63 Substituting this expansion and the asymptotic of s into Eq. 55a, we can derive the asymptotic behavior Eq. 56a of c t, s as. By the same way, one can show that the formula 56b also holds. 4.2 Dirichlet and Neumann Problems In what follows, we can derive the effective characterizations of spectral function S for the Dirichlet g prescribed, the Neumann g prescribed problems. Define the following new functions as Ωt, = Φ t, Φ t,, ˆΩt, = Φ t, Φ t,, ωt, = Φ t, + Φ t,, ωt, = Φ t, + Φ t,. 64 Theorem 2 Let T <. Let q x = q x, q 2 x, x, be a vector function of Schwartz class. For the Dirichlet problem it is assumed that the function g t, t < T, has sufficient smoothness and is compatible with q x at x = t =. For the Neumann problem it is assumed that the function g t, t < T, has sufficient smoothness and is compatible with q x at x = t =. Suppose that s has a finite number of simple zeros in D. Then the spectral function S is given by A e 4i4T B e 4i4T C S= e 4i4T D E F, 65 e 4i4T G H I where A = Φ 22 Φ 33 Φ 23 Φ 32, B = Φ 3 Φ 32 Φ 2 Φ 33, C = Φ 2 Φ 23 Φ 3 Φ 22, D = Φ 23 Φ 3 Φ 2 Φ 33, E = Φ Φ 33 Φ 3 Φ 3, F = Φ 2 Φ 3 Φ Φ 23, G = Φ 2 Φ 32 Φ 22 Φ 3, H = Φ 2 Φ 3 Φ Φ 32, I = Φ Φ 22 Φ 2 Φ 2, and the complex-value functions {Φ l3 t, } 3 l= satisfy the following system of integral equations: Φ 3 t, = t e 4i4 t t [ i 2 g 2 + g 2 2 2 g ḡ + g 2 ḡ 2 g ḡ g 2 ḡ 2 Φ 23 t, = + i 4 g 2 + g 2 2 2 Φ 3 + 2 3 g + ig Φ 23 + 2 3 g 2 + ig 2 Φ 33 t, dt, t [ 2 3 ḡ + iḡ Φ 3 + i 2 g 2 2 g ḡ ḡ g i 4 g 2 + g 2 2 g 2 Φ 23
434 Communications in Theoretical Physics Vol. 68 Φ 33 t, = + + i 2 ḡ g 2 2 g 2ḡ ḡ g 2 i 4 g 2 + g 2 2 ḡ g 2 Φ 33 t, dt, t [ 2 3 ḡ 2 + iḡ 2 Φ 3 + i 2 g ḡ 2 2 g ḡ 2 ḡ 2 g i 4 g 2 + g 2 2 g ḡ 2 Φ 23 + i 2 g 2 2 2 g 2ḡ 2 ḡ 2 g 2 i g 2 + g 2 2 g 2 2 Φ 33 t, dt, 66 4 and {Φ l t, } 3 l=, {Φ l2t, } 3 l= satisfy the following system of integral equations: t [ Φ t, = + i 2 g 2 + g 2 2 2 g ḡ + g 2 ḡ 2 g ḡ g 2 ḡ 2 + i 4 g 2 + g 2 2 2 Φ Φ 2 t, = Φ 3 t, = + 2 3 g + ig Φ 2 + 2 3 g 2 + ig 2 Φ 3 t, dt, t [ e 4i4 t t 2 3 ḡ + iḡ Φ + i 2 g 2 2 g ḡ ḡ g i 4 g 2 + g 2 2 g 2 Φ 2 + t i 2 ḡ g 2 2 g 2ḡ ḡ g 2 i 4 g 2 + g 2 2 ḡ g 2 Φ 3 t, dt, [ e 4i4 t t 2 3 ḡ 2 + iḡ 2 Φ + i 2 g ḡ 2 2 g ḡ 2 ḡ 2 g i 4 g 2 + g 2 2 g ḡ 2 Φ 2 + i 2 g 2 2 2 g 2ḡ 2 ḡ 2 g 2 i 4 g 2 + g 2 2 g 2 2 Φ 3 t, dt, 67 t [ Φ 2 t, = e 4i4 t t i 2 g 2 + g 2 2 2 g ḡ + g 2 ḡ 2 g ḡ g 2 ḡ 2 Φ 22 t, = + Φ 32 t, = + i 4 g 2 + g 2 2 2 Φ 2 + 2 3 g + ig Φ 22 + 2 3 g 2 + ig 2 Φ 32 t, dt, + t + t [ 2 3 ḡ + iḡ Φ 2 + i 2 g 2 2 g ḡ ḡ g i 4 g 2 + g 2 2 g 2 Φ 22 i 2 ḡ g 2 2 g 2ḡ ḡ g 2 i 4 g 2 + g 2 2 ḡ g 2 Φ 32 t, dt, [ 2 3 ḡ 2 + iḡ 2 Φ 2 + i 2 g ḡ 2 2 g ḡ 2 ḡ 2 g i 4 g 2 + g 2 2 g ḡ 2 Φ 22 i 2 g 2 2 2 g 2ḡ 2 ḡ 2 g 2 i 4 g 2 + g 2 2 g 2 2 Φ 32 t, dt. 68 i For the Dirichlet problem, the unnown Neumann boundary value g t is given by g T t = 4 2 ωt, d + 2gT t π ˆΩt, d + i 2 gt tg T tḡ t + 8 s s e 4i4t d 4gT t π 2 Φ t, Φ j t, s s e 4i4t d. 69 ii For the Neumann problem, the unnown boundary value g t is given by g T t = 2 ωt, d + 4 Φ t, s s π π e 4i4t d. 7 Proof The representations 65 follow from the relation S = e 2i4 T ˆΛµ 2, T,. And the system 66 is the direct result of the Volteral integral equations of µ 2, t,. i In order to derive Eq. 69 we note that Eq. 53b expresses g in terms of Φ 2 and Φ3. Furthermore, Eq. 52 and Cauchy theorem imply 4 Φ2 t = Φ t, I Φ td = Φ t, I Φ td, D 2 D 4 4 Φ3 t = t Φ2 td = t Φ2 td. 2 Φ t, Φ D 2 2 Φ t, Φ D 4
No. 4 Communications in Theoretical Physics 435 Thus, 2 Φ3 t= + D 2 = + where It is defined by D 4 D [ 2 Φ t, Φ [ 2 Φ t, Φ [ 2 Φ t, + Φ It = 2 t Φ2 td = + D t Φ2 td + It = [ 2 Φ t, Φ D The last step involves using the global relation 55a to compute It [ 2 Φ t, Φ [ 2 Φ t, Φ t Φ2 td t Φ2 td t Φ2 td + It = 2 Φ t, + Φ t, d + It, 7 t Φ2 td. It = 2 [ 2 c t, s Φ t, s s e 4i4t Φ t Φ2 td. D 72 Using the asymptotic 56a and Cauchy theorem to compute the first term on the right-hand side of Eq. 72, we find It = 2 Φ3 t + 2 2 Φ t, s s e 4i4t d. 73 Equations 7 and 73 imply Φ 3 t = [ 2 Φ t, + Φ t, d + 2 2 Φ t, s s e 4i4t d. 74 Similarly, 2 Φ2 t = + [Φ t, I Φ td = + [Φ t, I Φ td D 2 D 4 D = D [Φ t, I d + Jt = [Φ t, Φ t, d + Jt, 75 where Jt is defined by Jt = 2 [Φ t, I d. D The last step involves using the global relation 55b to compute Jt Jt = 2 c t, s Φ jt, s s e 4i4t I d. D 76 Using the asymptotic 56b and Cauchy theorem to compute the first term on the right-hand side of Eq. 76, we find Jt = 2 Φ2 t 2 Φ j t, s s e 4i4t d. 77 Equations 75 and 77 imply Φ 2 t = [Φ t, Φ t, d 2 Φ j t, s s d. This equation together with Eqs. 53b and 74 yields Eq. 69. ii In order to derive the representations 7 relevant for the Neumann problem, we note that Eq. 53a expresses g in terms of Φ. Furthermore, Eq. 52a and Cauchy s theorem imply 4 Φ D t = Φ t, d = Φ t, d. 78 2 D 4 Thus, 2 Φ t = + Φ t, d = Φ t, d + 2 Φ t, d D D D = Φ t, + Φ t, d + 2 Φ t, d, D 79 and using the global relation, we have 2 Φ t, d = 2 D c t, s Φ t, s s e 4i4t d D
436 Communications in Theoretical Physics Vol. 68 Equations 53a, 79, and 8 yield 7. 4.3 Effective Characterizations Substituting into the system 66 the expressions = 2 Φ t + 2 Φ t, s s e 4i4t d. 8 Φ ij = Φ ij, + εφ ij, + ε 2 Φ ij,2 + ε 3 Φ ij,3 + ε 4 Φ ij,4 +, i, j =, 2, 3. g = εg + ε2 g 2 + ε3 g 3 + ε4 g 4 +, g 2 = εg 2 + ε2 g 2 2 + ε3 g 3 2 + ε4 g 4 2 +, g = εg + ε2 g 2 + ε3 g 3 + ε4 g 4 +, g 2 = εg 2 + ε2 g 2 2 + ε3 g 3 2 + ε4 g 4 2 +, 8 where ε > is a small parameter, we find that the terms of O give Φ 3, = Φ 23, = Φ 33, =, O : Φ, = Φ 2, = Φ 3, =, Φ 2, = Φ 22, = Φ 32, =. Moreover, the terms of Oε give the terms of Oε 2 give Φ 3,2 = Φ 23,2 = Φ 33,2 = Φ,2 = Oε 2 : Φ 2,2 = Φ 3,2 = Φ 2,2 = Φ 22,2 = Φ 32,2 = t t t t Φ 33, =, Φ 23, =, Φ 3, t, = Φ, =, Φ Oε : 2, = Φ 3, = t t t t e 4i4 t t 2 3 g 2 + ig 2 t dt, e 4i4 t t 2 3 ḡ + iḡ t dt, e 4i4 t t 2 3 ḡ 2 + iḡ 2 t dt, Φ 2, = e 4i4 t t 2 3 g + ig t dt, Φ 22, =, Φ 32, =, e 4i4 t t 2 3 g 2 2 + ig2 2 t dt, [ 2 3 ḡ + iḡ t Φ 3, t, + i 2 ḡ g 2 2 ḡ g 2 ḡ g 2 t dt, [ 2 3 ḡ 2 + iḡ 2 t Φ 3, t, + i 2 ḡ 2 2 2 ḡ 2 g 2 ḡ 2 g 2 t dt, [ i 2 g 2 + g 2 2 2 g ḡ + g 2 ḡ 2 g ḡ g 2 ḡ + 2 3 g + ig t Φ 2, t, + 2 3 g 2 + ig t t t t t e 4i4 t t 2 3 ḡ 2 + iḡ2 t dt, e 4i4 t t 2 3 ḡ 2 2 + iḡ2 2 t dt, e 4i4 t t 2 3 g 2 + ig2 t dt, 2 t 2 t Φ 3, t, dt, [ 2 3 ḡ + iḡ t Φ 2, t, + i 2 g 2 2 g ḡ g ḡ t dt, [ i 2 g ḡ 2 2 g ḡ 2 g ḡ 2 t dt. 82 83 84
No. 4 Communications in Theoretical Physics 437 the terms of Oε 3 give t [ Φ 3,3 = e 4i4 t t i 2 g 2 + g 2 2 2 g ḡ + g 2 ḡ 2 g ḡ g 2 ḡ 2 Φ 3, t, + 2 3 g + ig t Φ 23,2 t, + 2 3 g 2 + ig 2 t Φ 33,2 t, + 2 3 g 3 2 + ig3 2 t dt, t [ Φ 23,3 = 2 3 ḡ + iḡ t Φ 3,2 t, + 2 3 ḡ 2 + iḡ2 t Φ 3, t, + i 2 ḡ g2 2 + ḡ2 g 2 2 ḡ2 g 2 + ḡ g2 2 ḡ g2 2 ḡ2 g 2 t dt, t [ Φ 33,3 = 2 3 ḡ 2 + iḡ 2 t Φ 3,2 t, + 2 3 ḡ 2 2 + iḡ2 2 t Φ 3, t, Φ,3 = Φ 2,3 = Oε 3 : + i 2 ḡ 2 2 g 2 + ḡ 2 g2 2 2 ḡ2 2 g 2 + ḡ 2 g2 2 ḡ 2 g2 2 ḡ2 2 g 2 t t dt, [ i 2 g ḡ2 +g2 ḡ +g 2 ḡ2 2 +g2 2 ḡ 2 2 g ḡ2 +g2 ḡ +g 2 ḡ2 2 +g2 2 ḡ 2 g ḡ2 g 2 ḡ g 2 ḡ2 2 g2 2 ḡ 2 t + 2 3 g 2 + ig2 t Φ 2, t, + 2 3 g 2 2 + ig2 Φ 3, t, + 2 3 g + ig t Φ 2,2 t, + 2 3 g 2 + ig 2 t Φ 3,2 t, t e 4i4 t t [ 2 3 ḡ + iḡ t Φ,2 t, + 2 3 g 3 + ig3 2 g ḡ g ḡ Φ 2, t, + dt, 2 i 2 g 2 t + i 2 ḡ g 2 2 g 2 ḡ g 2 ḡ Φ 3, t, dt, 85 Φ 3,3 = Φ 2,3 = t e 4i4 t t [ 2 3 ḡ 2 + iḡ 2 t Φ,2 t, + 2 3 g 3 2 + ig3 2 g ḡ 2 g ḡ 2 Φ 2, t, + t i 2 g ḡ 2 2 t + i 2 g 2 2 2 g 2 ḡ 2 g 2 ḡ 2 Φ 3, t, [ e 4i4 t t i 2 g 2 + g 2 2 2 g ḡ + g 2 ḡ 2 g ḡ g 2 ḡ 2 Φ 2, t, + 2 3 g 3 + ig3 t + 2 3 g + ig t Φ 22,2 t, + 2 3 g 2 + ig 2 Φ 32,2t, t [ Φ 22,3 = 2 3 ḡ 2 + iḡ2 t Φ 2, t, + 2 3 ḡ + iḡ t Φ 2,2 t, + i 2 ḡ 2 g + ḡ g2 2 ḡ2 g + ḡ g2 ḡ g2 ḡ2 g t dt, t [ Φ 32,3 = 2 3 ḡ 2 + iḡ 2 t Φ 2,2 t, + 2 3 ḡ 2 2 + iḡ2 2 t Φ 2, t, + i 2 ḡ 2 2 g + ḡ 2 g2 2 ḡ2 2 g + ḡ 2 g2 ḡ 2 g2 ḡ2 2 g t dt. On the other hand, expanding 69 and assuming for simplicity that s has no zeros, we find g T t = 4 g 2T t = 4 g 3T t = 4 2 ω t, d + 8 2 ω 2 t, d + 2 π gt 2 ω 3 t, d + 2 π + 8 2 4 π dt, dt, 2 s s e 4i4t d, 86a ˆΩ t, d + 8 g T tˆω 2 t, + g 2T tˆω t, d 2 s 2 s2 e 4i4t d, 86b s 3 s3 + Φ,s 2 s2 + Φ,2s s e 4i4t d g T Φ j, s s e 4i4t d + i 2 gt g T ḡ. 86c
438 Communications in Theoretical Physics Vol. 68 where ω = εω + ε 2 ω 2 + Oε 3, s s = εs s + Oε 2. The Dirichlet problem can now be solved perturbatively as follows: assuming for simplicity that s has no zeros, we can use Eq. 83 to determine ω, then compute g T from Eq. 83. In the same way we can determine ˆΩ and ω 2 from Eqs. 83 and 84. Given g and g 2 2, then compute g2t from Eq. 86b. These arguments can be extended to the higher order and also can be extended to the systems 66, 67, and 68 thus yields a constructive scheme for computing S to all orders. Similarly, these arguments also can be used to the Neumann problem. That is to say, in all cases, the system can be solved perturbatively to all orders. References [ C. S. Gardner, J. M. Greene, M. D. Krusal, and R. M. Miura, Phys. Rev. Lett. 9 967 95. [2 A. S. Foas, Proc. R. Soc. Lond. A 453 997 4. [3 A. S. Foas, J. Math. Phys. 4 2 488. [4 A. S. Foas, Proc. R. Soc. Lond. A 457 2 37. [5 A. Boutet de Monvel, A. S. Foas, and D. Shepelsy, J. Inst. Math. Jussieu. 3 24 39. [6 A. S. Foas, A. R. Its, and L. Y. Sung, Nonlinearity 8 25 77. [7 A. S. Foas and A. R. Its, Theoret. Math. Phys. 92 992 388. [8 A. S. Foas and A. R. Its, Math. Comput. Simulation 37 994 293. [9 A. S. Foas and J. Lenells, J. Phys. A: Math. Theor. 45 22 952. [ J. Lenells and A. S. Foas, J. Phys. A: Math. Theor. 45 22 9522. [ J. Lenells and A. S. Foas, J. Phys. A: Math. Theor. 45 22 9523. [2 J. Lenells, Physica D 24 22 857. [3 J. Lenells, Nonlinear Anal. 76 23 22. [4 J. Xu and E. Fan, Proc. R. Soc. Ser. A 469 23 2368. [5 J. Xu and E. Fan, Math. Phys. Anal. Geom. 9 26 2. [6 V. Gerdjiov and I. Ivanov, Bulg. J. Phys. 983 3. [7 E. Fan, J. Phys. A 33 2 6925. [8 E. Fan, J. Math. Phys. 4 2 7769. [9 Y. Hou and E. Fan, J. Math. Phys. 54 23 7355. [2 S. Xu and J. He, J. Math. Phys. 53 22 6357. [2 L. Xing, W. Ma, and J. Yu, Nonlinear Dynam. 82 25 2. [22 Y. Zhang, Y. Cheng, and J. He, J. Nonlinear Math. Phys. 24 27 2. [23 J. Xu, J. Nonlinear Math. Phys. 23 26 67.