Mapping Properties of the Nonlinear Fourier Transform in Dimension Two

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1 Communications in Partial Differential Equations, 3: 59 60, 007 Copyright Taylor & Francis Group, LLC ISSN print/ online DOI: 0.080/ Mapping Properties of the Nonlinear Fourier Transform in Dimension Two M. LASSAS, J. L. MUELLER, AND S. SILTANEN Institute of Mathematics, Helsini University of Technology, Espoo, Finland Department of Mathematics, Colorado State University, Fort Collins, Colorado, USA A class of compactly supported Schrödinger potentials in dimension two is given for which the inverse scattering method related to the Noviov Veselov evolution equation is well-defined. There is no smallness assumption on the initial potential. Regularity results are proven for the direct and inverse scattering transforms, also called nonlinear Fourier transforms. Keywords Complex geometrical optics; Evolution equation; Exponentially growing solution; Fourier transform; Inverse scattering; Noviov Veselov equation; Schrödinger equation. Mathematics Subject Classification Primary 8U40, 46T0; Secondary 4B0, 35Q58, 35J0, 35K55.. Introduction Let q L p for some <p< and consider the Schrödinger equation + q = 0 (.) where \0 is a parameter. As in Faddeev (966) and Nachman (996), we loo for exponentially growing solutions of (.) with asymptotic behavior x e ix. More precisely, the solution is characterized by e ix x L p L for fixed \0 (.) where / p = /p /. Here and throughout the paper a point x = x x in will be identified with x = x + ix. Soexpix = expi + i x + ix with and x. Received August 9, 005; Accepted May 3, 006 Address correspondence to S. Siltanen, Institute of Mathematics, Helsini University of Technology, Box 00, Espoo, FIN-005, Finland; samuli.siltanen@tut.fi 59

2 59 Lassas et al. Exponentially growing solutions do not necessarily exist for all. A point is called a non-exceptional point of q if there is a unique solution of (.) satisfying (.). Otherwise is called an exceptional point of q. If a potential q does not have exceptional points, one can define the scattering map q t, taing the potential q to its scattering transform t defined by t = e i x qxx dx (.3) Under suitable assumptions the potential q can be recovered from its scattering transform t via the inverse scattering map t q defined by tx = i t x e xx d (.4) where x, d denotes Lebesgue measure, x = + i and e x x x = e ix+ x. The functions x in (.4) are determined by solving the equation x = t 4 e xx (.5) with fixed x and assuming large asymptotics x L L r for some <r<. We remar that the maps and are often called the direct and inverse nonlinear Fourier transforms. See the wor of Beals and Coifman (980, 985, 986, 989), Ablowitz and Nachman (986), Nachman and Ablowitz (984), and Henin and Noviov (988) for early references on the method. Note also that a formula equivalent to (.4) is given by Boiti et al. (987, formula (4.0)). Let us define a special class of potentials. Definition.. A compactly supported potential q C0 is of conductivity type if q = / / for some real-valued C satisfying x c>0 for all x and x for all x \. Nachman (996) shows in that q is well-defined for conductivity type potentials. We extend his result in Theorems. and. by proving that the apparently singular functions q and q belong to the Schwartz class. The crucial point of the proof is to use boundary integral equations arising from electrical impedance tomography to avoid the log singularity of Faddeev s Green s function. Also, we mae use of a relationship found by Barceló et al. (00) (see also Knudsen, 00) between and the well-nown scattering map of the Davey Stewartson (DS) II equation. Then we can prove the following new result. Theorem.. Let q C 0 be of conductivity type. Then q = q. Further, we study mapping properties of the inverse scattering map. Theorem.. Let t satisfy t/ and t/. Then the function t given by (.4) is well-defined and continuous. Furthermore, tx Cx (.6)

3 Inverse Scattering Method 593 Our motivation for the study of the above scattering problem is two-fold. First, the problem is formally related to the Noviov Veselov evolution equation that generalizes the KdV equation to dimension +. Second, the scattering transform t can be used to solve the inverse conductivity problem that has many practical applications. Let us discuss these motivations in more detail. Noviov and Veselov (986) and Veselov and Noviov (984) (see also Nizhni, 980) introduce the following evolution equation in a periodic setting: q = 3 x q 3 x q + 3 x q v + 3 x q v (.7) where 0 and x v = x q. Boiti et al. (987) discuss equation (.7) in the nonperiodic case and show that if q evolves in according to (.7) and does not have exceptional points, then the scattering data evolves as q = e i3 + 3 q 0 Tsai (993, 994) assumes the absence of exceptional points and gives a formal derivation of a hierarchy of evolution equations parametrized by n = 3 5 and containing the non-periodic version of (.7) as the case n = 3. Thus equation (.7) can be solved formally using the inverse scattering method: q = expi q 0 (.8) However, problems caused by exceptional points prevent the rigorous use of formula (.8) for general potentials q 0. We remar that the relationship between the Noviov Veselov equation (.7) and a scattering problem analogous to (but different from) the above is studied by Grinevich (986, 000), Grinevich and Manaov (986), and Grinevich and Noviov (985, 995). The inverse conductivity problem is formulated by Calderón (980) as follows: given a regular domain d with d> and a strictly positive function L, define the Dirichlet-to-Neumann (DN) map by f = u (.9) where u is the unique H solution of the Dirichlet problem u = 0in and u = f on. The DN map represents all static voltage-to-current density measurements on the boundary. Calderón ass whether is uniquely determined by and if so, how to reconstruct from boundary measurements. This problem has a practical imaging application called electrical impedance tomography (EIT) used for medical imaging, geophysics, nondestructive testing, and process tomography. See Cheney et al (99), Siltanen et al. (000), Mueller and Siltanen (003), and Isaacson et al. (004) for more details on EIT. What is the relation between EIT and inverse scattering? Assume that the conductivity is smooth, strictly positive, and in a neighborhood of. Then we can extend as one outside and define q= / / C0. Exponentially growing solutions corresponding to such conductivity type potential q can then be used to solve the inverse conductivity problem, as shown

4 594 Lassas et al. by Sylvester and Uhlmann (988), Nachman (988, 996), and Noviov (988). Our starting point is Nachman (996), where Nachman shows that conductivity type potentials in dimension two do not have exceptional points, allowing us to show that formula (.8) is well-posed for such potentials. Also, an EIT algorithm based on Nachman s proof is presented in Siltanen et al. (000), Mueller and Siltanen (003), Knudsen et al. (004), Isaacson et al. (004), readily providing numerical inverse scattering algorithms. The two-dimensional isotropic inverse conductivity problem is solved for once wealy differentiable conductivities by Brown and Uhlmann (997), and further, for L conductivities by Astala and Päivärinta (006). Both approaches are based on the use of a equation. An EIT algorithm based on Brown and Uhlmann (997) is given in Knudsen (00, 003) and Knudsen and Tamasan (004). We remar that Brown and Uhlmann (997) is related to the DS II equation, and it is not nown if Astala and Päivärinta (006) is related to some evolution equation. Choose a positive odd integer n, and set for all m n = exp i n n + n (.0) Note that m n =. Given q L p with no exceptional points, one can formally define a function q x at time >0 by multiplying the initial scattering transform q by m n and applying the inverse scattering map : t m n t (.) q q The smoothness results for q and q and Theorem. can be used to show that the inverse scattering scheme (.) is well-defined for conductivity type initial potentials: Corollary. Let q C0 be of conductivity type. Fix a positive odd integer n, let 0 and define q by q = m n q Then q x is continuous in x and belongs to L p for any <p<. Proof. The infinitely smooth function m n and its derivatives grow at most polynomially at infinity. Thus by Theorems. and. m n t and mn t and the claim follows from Theorem.. Note that there is no smallness assumption for the initial data in the corollary.

5 Inverse Scattering Method 595. The Scattering Map This section is devoted to proving that if q C 0 is of conductivity type, then q is a well-defined, smooth function whose derivatives decay when, and the apparently singular functions q/ and q/ actually have bounded derivatives of all orders at = 0. Theorem.. Let q C0 be of conductivity type. Then the functions q and q/ belong to the Schwartz class. Proof. By Nachman (996, Thm.) the potential q does not have exceptional points. Thus t = q is well-defined using formula (.3) for nonzero and setting t0 = 0. To analyze derivatives of t, let us discuss scattering data introduced by Beals and Coifman (988) for the Davey Stewartson (DS) II equation. By assumption we have q = / / for some smooth, strictly positive function satisfying C0. Define q = x / / Consider the equation D Q = 0 (.) fora matrix x depending on x and, where ] [ ] [ x 0 0 q D = Q = 0 x q 0 Brown and Uhlmann (997) showed that there are exponentially growing solutions of equation (.) of the form [ ] e ix 0 x = mx 0 e i x where m is uniquely specified by the requirement that each element of the matrix m I belongs to L r for any r>. The DS scattering data is the matrix [ ] S = i 0 e ix qx x dx e i x qx x 0 Based on Barceló et al. (00), Knudsen (00, Thm 3.5.) showed that the scattering transforms t of q and S of q are related by t = S (.)

6 596 Lassas et al. Since q is compactly supported and N times continuously differentiable, a result of Sung (994, Thm 4.4) shows that + S (.3) is continuous and vanishes at infinity for + N and arbitrary 0. A combination of (.) and (.3) yields the claim. Theorem.. Let q C 0 be of conductivity type. Then q/ C. Proof. Denote t = q. By Theorem. we now that t is infinitely smooth in and all its derivatives vanish at infinity. Thus we only need to prove that all derivatives of t/ are bounded in a neighborhood of 0. With no loss of generality we can assume that q and are supported in the open unit disc = D0. Namely, replacing q by qx = qx and by x = x with large enough >0 yields supp q while preserving the equality q = / /. By Theorem 3.3 of Siltanen et al. (000) we have t = t, so the claim holds for t/ if and only if it does for t/. Let H / H / be the DN map of. Denote by the DN map of the homogeneous conductivity. It is nown that is a classical pseudodifferential operator and if near then is an infinitely smoothing map, see Sylvester and Uhlmann (988) and Lee and Uhlmann (989). We can thus extend as a continuous map L C and write = yydy (.4) where dy is Lebesgue measure on the unit circle and is some C function on the torus. Further, we have = 0 and d = 0 (.5) for any H /. Of course, (.5) holds as well when is replaced with. Consider the following representation of t given by Nachman (996) (originally introduced by Noviov, 988 and Nachman, 988): t = e i x d \0 (.6) Use equations (.4) (.6) to write for any 0 t = e i x dx e i x ( ) = x yy dy dx = x ye x y dxdy (.7)

7 Inverse Scattering Method 597 where we denote ei x E x = y = y The use of Fubini s theorem in (.7) is justified since y is continuous in y by Nachman (996, Thm.) and Sobolev imbedding, and E x is continuous in x. Use (.7) to compute formally for any 0 + ( ) t = x y =0 =0 c E x ydxdy (.8) where c are constants and we use the following notation: E x = + E x x = + y We will prove that the maps E and are well-defined and continuous from to L for any 0 and 0. (We remar that a proof is given also in Astala and Päivärinta, 006; for the reader s convenience we present a constructive proof here.) Then differentiation under the integral sign in (.8) is justified by repeated applications of Lebesgue s dominated convergence theorem. Further, we show that E L C (.9) L C (.0) uniformly for 0 <. Combining (.9) and (.0) with (.8) yields the claim. The function E x has the following power series expansion: E x = ei x = n i x n (.) n! Thus the map E is well-defined and continuous from to L for any 0 and 0. Moreover, since lim 0 E x = i x, we see that (.9) holds. Let us discuss -smoothness of y. Define a single-layer operator S by S x = G x yydy where G is Faddeev s Green function G x = e ix g x satisfying G =. The function g is the following fundamental solution: g x = n= e ix d (.) + i i + The function g satisfies 4i g =. According to Theorem 5 of Nachman (996), the operator I + S is invertible in H /. Since the single layer

8 598 Lassas et al. operator S (as well as ) is a classical pseudodifferential operator, we see by Hörmander (994, Thm. 9..) that the pseudodifferential operator I + S is invertible also as an operator L L. Thus we can write for 0 y = I + S e ix (.3) Let us rewrite (.3) following Siltanen et al. (000). Set G 0 = log x and denote H x = G x G 0 x 0 Now H = 0 and so H x is infinitely smooth in x. Changes of variables in the integral in (.) show that g satisfies and we see that G x = G x and Write now g x = g x = g x = e xg x (.4) H x = H x log 0 I + S = A + H where A = I + S 0 and the operator H is given by H x = H x yydy here H x = H x H 0 and the constants H 0 and log were annihilated by (.5). Note that H is real-valued (Siltanen et al., 000), and we abuse notation by using interchangeably H x = H x + ix = H x x. So (.3) taes the form y = I + A H A e ix (.5) where A is invertible in H / by the proof of Theorem 3. in Siltanen et al. (000). As before, we see using Hörmander (994, Thm. 9..) that A is invertible also in L. For differentiating (.5) we analyze the -differentiability of H in the strong operator topology of LL. Denote for any 0 H x = + H x x x x Define an operator D 0 LL by D 0 fx = x y H 0 x yy zfzd + x y H 0 x yy zfzd

9 Inverse Scattering Method 599 Tae any f L L and write ( ) lim H+h H h 0 h f D 0 f L = lim H + hx y H x y h 0 h fydy D 0 fx dx = 0 (.6) where we used Fubini s theorem once and dominated convergence twice; this is possible since H and are smooth. Thus as a mapping to L has a differential 0 y = ( I + A H ) A e ix = I + A I + A + I + A H H A H A e ix H A e ix implying that is in C L. The above analysis can be extended to any higher partial derivative + / in a similar manner. 3. Properties of Solutions of the Equation Fix <r< and x. Assume given a function t satisfying t/ and consider the equation x = t 4 e xx (3.) for satisfying the asymptotic condition Denote the solid Cauchy transform by = x L r L (3.) d (3.3) where d denotes the Lebesgue measure of. Note that and are inverses of each other (modulo analytic functions). Further, define a real-linear operator T x x = t 4 e xx (3.4)

10 600 Lassas et al. Nachman (996) proved that the operator I T x L r L r is invertible and that T x L r. Now the equation (3.) together with the asymptotic condition (3.) can be written in the convenient form or in the long form = + T x (3.5) x = + t 4 e x x d (3.6) The solution of equation (3.5) is given by x = + I T x T x (3.7) The following lemma can be found in Veua (96). Lemma 3.. Let f L p L p, where p and /p + /p =. Then f d d [ 8f L pf L p p /p p /p ] / The following lemma establishes some properties of the function T x appearing in (3.7). Lemma 3.. Let t satisfy t/. Define for any x gx = t 4 e x d (3.8) Then for any 0 there is a constant M = M >0, independent of x and, such that the following estimate holds: + x gx x M minx for all x (3.9) where x = +x / and = + /. Moreover, the maps x gx and x x j gx are continuous from to L r for any <r< and j =. Proof. Denote = t/. Choose some <p< and note that the norms p and p are bounded by Schwartz seminorms of. Thus by Lemma 3. we have for any x gx [ 8 p p p /p p /p ] / = M (3.0) where M < does not depend on x.

11 Inverse Scattering Method 60 The norm is bounded by Schwartz seminorms of. Thus we can estimate for 0 4 gx = + e x + d + d = leading to gx 4 = M for x \0 (3.) with M < independent of x and. There are positive constants C C and R depending only on the Schwartz seminorms of such that ˆ = C <, and ˆ C for R. Here ˆ denotes the Fourier transform of. Note that g can be written in the form gx = ( ) e 4 x Fourier transforming g with respect to yields ĝx = 4 e xˆ = 4 ˆ ê x = ˆ + x 4 We want to estimate the L norm of ĝx for large x. Fix x with x >R. 4 ˆ + x ˆ + x ĝx = d + d x x x C x + C ˆ + xd + C x + C x x x x = C + C x 0 x x x d r rdr ˆ x d Since the inverse Fourier transform is continuous from L to L we have gx M 3 x for x >R (3.) where M 3 < does not depend on x. A combination of (3.0), (3.), and (3.) now yields the estimate (3.9) for the case = = 0. Next we show the continuity of the map x gx. Write gx gy = 4 e x e y d = 4 e x e x y d

12 60 Lassas et al. Clearly lim y x ( e x y L = 0 for any since is rapidly decaying. Thus replacing gx by gx gy and by e x y in estimates (3.0) and (3.) shows that lim y x gx gy r = 0 implying continuity of x gx from to L r. Let now 0. Then by repeated applications of Lebesgue s dominated convergence theorem we can write + x x gx = 4 i i e x d Estimate (3.9) follows from repeating the above proof with replaced by. Finally, continuity of the map x /x j gx follows analogously by repeating the above proof with replaced by j. Now Lemma 3. and formula (3.7) yield the following estimate: Lemma 3.3. Let t satisfy t/ and tae <r<. Let be the unique solution of (3.5) satisfying x L r L. Then the map x x is continuous from to L r and x r Cx (3.3) Proof. By Lemma 3. we now that T x L r depends continuously on x. Let f L r with f r = and write for any y ( ) t T x f T y f = 4 e x y e x f By the proof of Nachman (993, Lemma 4.) we now that af r c 0 a f r = c 0 a (3.4) Thus we see that the map x T x is continuous from to LL r : lim T x f T y f r c 0 y x 4 lim y x t( ex y ) = 0 Since the operator I T x is invertible for all x, the map x I T x is continuous from to LL r as well. Hence the right hand side of (3.7) depends continuously on x. It remains to prove estimate (3.3). Lemma 3. shows that /r T x r M (x r d) r Cx (3.5)

13 Further, by Lemma.. of Liu (997) we have Inverse Scattering Method 603 I T x LL r C (3.6) where C does not depend on x. Now (3.3) follows from (3.7), (3.5) and (3.6). Now we are ready to prove the crucial estimate on the x-derivatives of. Lemma 3.4. Let t satisfy t/ and tae <r<. Let be the unique solution of (3.5) satisfying x L r L. Then the map x x /x j is continuous from to L r and x x Cx for j = (3.7) j r Proof. Let us write formally ( ) x = I + T x x T j x x I + T x T x + I + T x T j x x j (3.8) By Lemma 3. we now that the maps x T x j x with j = are continuous from to L r. We need to prove that (i) The operator T x is differentiable with respect to x in the strong operator topology of LL r, and (ii) The maps x T x j x with j = are continuous from to LL r. Then (i), (ii) and (3.8) together yield the continuity of the map x x /x j. Denote T x f= i t e x f d (3.9) We will show that lim h 0 ( ) Tx+h f T x f = T h x f (3.0) in the L r topology uniformly in f. Let >0 and tae f L r with f r =. With c 0 as in (3.4), choose such a h 0 > 0 that e h t L < c 0 for all 0 <h <h 0. Compute using the mean value theorem and estimate (3.4) (( ) T x+h f T x f T h x f = e h + i r h e x t ) 4 f r = ( 4 i e h + i e x t ) r f

14 604 Lassas et al. = e h e x t f c 0 e h t < where 0 <h <h<h 0 and h = h. This proves (3.0), which in turn implies (i) for j =. The proof for j = is analogous, and claim (ii) can be proved using a similar argument. Finally, to prove estimate (3.7) tae L r norm of both sides of equation (3.8). Using Lemma 3. we see similarly to (3.5) that T x x Cx (3.) j r Using (3.9) and (3.4) we get the following uniform bound: T x x C (3.) j LL r r Now (3.5), (3.6), (3.), and (3.) yield the estimate (3.7). 4. Proof of Theorem. Let q be an infinitely smooth, compactly supported potential of conductivity type. By Theorems. and. we see that t = q is well-defined and t t (4.) It is shown in Nachman (996) that for any \0 the equation x x + ix = q x (4.) 4 has a unique solution satisfying L p L for any < p <. Furthermore, it is shown in Nachman (996) that the functions are also the unique solutions of the equation (3.) with asymptotic condition (3.) with any <r<. Define f= x x + i. Use (4.) and (3.7) to write 0 = f q 4 x q 4 = f q 4 I T x T x q 4 Because q is real-valued and does not depend on, we can write f I T x q T x 4 q = 0 (4.3) 4 Applying I T x to both sides of (4.3) yields 0 = f q 4 T q xf + T x 4 T q x 4 = f q 4 T xf (4.4)

15 Inverse Scattering Method 605 Compute the commutator of T x with the operator x x + i: x x + i T x x = x x + i t 4 e x t 4 e x x x + i = t ( x e x x e x x x i ) 4 = t ( e x x i x e x x x i ) = 0 (4.5) 4 Tae 0 and compute similarly for the operator : x x + i = x x + i = i x = i x x d x x + i x d x d i x x d x d (4.6) Differentiation outside the integral sign in (4.6) is justified for continuous x L. Applying the commutator relations (4.5) and (4.6) to equation (4.4) gives x x + i = T x x x + i + q 4 Further, from (4.7) and (3.5) Thus, as required. = x x + it x + q 4 = x x + it x + q 4 i x T x x d (4.7) q 4 i x T x x d = x x + i ( + T x T x ) = 0 q = 4i t x 4 e xx d = i t x e xx d = t

16 606 Lassas et al. 5. Proof of Theorem. By Lemma 3.4 the map x x is differentiable from L r with the derivative defined in the norm topology. Thus x + he lim j x h 0 x h x j where e = 0 e = 0. Let r be the exponent conjugate to r: Then in the weighted L r -norm with weight w = + 3 we have L r = 0 ( e x x + he lim j e x x h 0 ) e h x x w j L r = 0 + =. r r Thus, x e x w is an L -valued continuously differentiable function, and as t w L, we see that t e t xx d e x x x d j ( lim t e x+hej x + he j e x x h 0 h ) e x x x d j x j = = 0 Thus we can differentiate under the integral sign below to obtain t tx = i x e xx d t = i x e t xx d + i x e xd t = i x e x x d t + i e t x x x d + i ie x d t = te x x d + i e x x x d (5.) + te x d (5.) Now (5.) is rapidly decaying since it is the Fourier transform of a Schwartz function. By Lemmas 3.3 and 3.4 the L r norms of x and x x are finite and depend continuously on x. Thus the fact that t/ together with Hölder s inequality shows that (5.) is well-defined and continuous in x.

17 Inverse Scattering Method 607 To prove estimate (.6) for term (5.) write the equation (3.) in the form x = t 4 e xx + t 4 e x (5.3) Now integration by parts yields for x 0 x ( = ix + ix t te x x d + i te x x d e x x x d t e x x x d) + i = t e x t e x d + i x x x d [ ] = e x tx + t x d ( ) t i e x x x d (5.4) Maing use of equation (5.3) we have that (5.4) equals [ e x tx ( t +t ( ) t i e x x x 4 e xx + t + t e x x ( t 4 e xx + t 4 e x )] 4 e x d ) d After some simplification we have that (5.4) equals ] [e x tx + t t x + d (5.5) 4 4 [ i e x ( t ) x x + t 4 e x x e x x (5.6) ] + t 4 e xie x d (5.7)

18 608 Lassas et al. The third term in (5.5) cancels with the term (5.7). Expanding the x derivative in (5.6) allows us to estimate (5.4) as follows: [ e x t ] t x + x d 4 [ i e x ( t ) x x + t ix 4 ] + t 4 xx d t x r + t r r r x ( ) + t x x r + t r 4 x x r r ( ( ) ) t C + t r + t t r x r where C<. Thus we can conclude that (5.4) is bounded by C x, and the proof is complete. r + Acnowledgments The authors than Bernard Deconinc, Atsushi Katsuda, Kim Knudsen, Adrian Nachman, Petri Ola, Lassi Päivärinta and Alexander Veselov for valuable discussions. Also, the authors than the anonymous referee for constructive suggestions that improved the paper. The wor of M. L. was supported by the Academy of Finland (grants 7434 and 075). This material is based upon wor supported by the National Science Foundation under Grant No (J. Mueller) and in part by the Japan Society for the Promotion of Science (Grant-in- Aid for JSPS Fellows No ) (S. Siltanen). References Ablowitz, M. J., Nachman, A. I. (986). Multidimensional nonlinear evolution equations and inverse scattering. Physica D 8:3 4. Astala, K., Päivärinta, L. (006). Calderón s inverse conductivity problem in the plane. Annals of Mathematics 63: Barceló, J. A., Barceló, T., Ruiz, A. (00). Stability of the inverse conductivity problem in the plane for less regular conductivities J. Differ. Equations 73:3 70. Beals, R., Coifman, R. R. (980). Scattering, transformations spectrales et equations d evolution non lineaires I, II. Seminaire Goulaouic-Meyer-Schwartz , exp., 98 98, exp., Palaiseau: Ecole Polytechnique. Beals, R., Coifman, R. R. (985). Multidimensional inverse scattering and nonlinear partial differential equations. AMS 43: Beals, R., Coifman, R. R. (986). The d-bar approach to inverse scattering and nonlinear evolution equations. Physica 8:4 49. Beals, R., Coifman, R. R. (988). The spectral problem for the Davey-Stewartson and Ishimori hierarchies. Nonlinear Evolution Equations: Integrability and Spectral Methods. Manchester University Press, pp. 5 3.

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