On the remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the Hopf-Cole transformation

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by  A. I. Zenchuk

On the remarkable relations among PDEs integrable by
the inverse spectral transform method, by the method of
characteristics and by the Hopf-Cole transformation
A. I. Zenchuk1,§ and P. M. Santini2,§
1 Center of Nonlinear Studies of the Landau Institute for Theoretical Physics
(International Institute of Nonlinear Science)
Kosygina 2, Moscow, Russia 119334
2 Dipartimento di Fisica, Universit`a di Roma ”La Sapienza” and
Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1
8
Piazz.le Aldo Moro 2, I-00185 Roma, Italy
0
0 §e-mail: [email protected] , [email protected]
2

n February 3, 2008
a
J
Abstract

5
2
We establish deep and remarkable connections among partial differential equations

] (PDEs) integrable by different methods: the inverse spectral transform method, the
I
S method of characteristics and the Hopf-Cole transformation. More concretely, 1) we
. show that the integrability properties (Lax pair, infinitely-many commuting symmetries,
n
i large classes of analytic solutions) of (2+1)-dimensional PDEs integrable by the Inverse
l
n Scattering Transform method (S-integrable) can be generated by the integrability prop-
[
erties of the (1+1)-dimensional matrix Bu¨rgers hierarchy, integrable by the matrix Hopf-

1 Cole transformation (C-integrable). 2) We show that the integrability properties i) of
v
S-integrable PDEs in (1+1)-dimensions, ii) of the multidimensional generalizations of
5
4 the GL(M,C) self-dual Yang Mills equations, and iii) of the multidimensional Calogero
9 equations can be generated by the integrability properties of a recently introduced mul-
3
tidimensional matrix equation solvable by the method of characteristics. To establish
.
1
the above links, we consider a block Frobenius matrix reduction of the relevant matrix
0
8 fields, leading to integrable chains of matrix equations for the blocks of such a Frobenius
0 matrix, followed by a systematic elimination procedure of some of these blocks. The con-
:
v struction of large classes of solutions of the soliton equations from solutions of the matrix
Xi Bu¨rgers hierarchy turns out to be intimately related to the construction of solutions in
Sato theory. 3) We finally show that suitable generalizations of the block Frobenius ma-
r
a
trix reduction of the matrix Bu¨rgers hierarchy generates PDEs exhibiting integrability
properties in common with both S- and C- integrable equations.
1 Introduction
Integrable nonlinear partial differential equations (PDEs) can be grouped into different classes,
depending on their method of solution. We distinguish the following three basic classes.
1. Equations solvable by the method of characteristics [1], hereafter called, for the sake of
brevity, Ch-integrable, like the following matrix PDE in arbitrary dimensions [2]:
N
w + w ρ(i)(w)+[B,w]σ(w) = 0, (1)
t xi
i=1
X
1
where w is a square matrix and ρ(i)(·),σ(·) are scalar functions representable as positive
power series, or like the vector equations solvable by the generalized hodograph method
[3, 4, 5, 6, 7, 8].
2. Equations integrable by a simple change of variables, often called C-integrable [9], like
the matrix Bu¨rgers equation [10]
w −Bw −2Bw w +[w,B](w +w2) = 0, (2)
t xx x x
whereB isanyconstantsquarematrix, linearizablebythematrixversion oftheHopf-Cole
transformation Ψ = wΨ [11].
x
3. Equations integrable by less elementary methods of spectral nature, the inverse spectral
transform (IST) [12, 13, 14, 15, 16] and the dressing method [17, 18, 19, 20, 16], often
called S-integrable [9] or soliton equations. Within this class of equations, we distinguish
four different subclasses, depending on the nature of the associated spectral theory.
(a) Soliton equations in (1+1)-dimensions like, for instance, the Korteweg-de Vries
(KdV) [21, 12] and the Nonlinear Schr¨odinger (NLS) [22] equations, whose inverse
problems are local Riemann-Hilbert (RH) problems [13, 15].
(b) Their (2+1)-dimensional generalizations, like the Kadomtsev-Petviashvily (KP) [23]
and Davey-Stewartson (DS) [24] equations, whose inverse problems are nonlocal RH
[25, 26] or ∂¯ - problems [27].
(c) The self-dual Yang-Mills (SDYM) equation [28, 29] and its generalizations in arbi-
trary dimensions.
(d) Multidimensional PDEs associated with one-parameter families of commuting vector
fields, whose novel IST, recently constructed in [30, 31], is characterized by nonlinear
RH [30, 31] or ∂¯ [32] problems. Distinguished examples are the dispersion-less KP
equation, theheavenly equationofPlebanski [33]andthefollowing integrablesystem
of PDEs in N +4 dimensions [30]:
N N
~v −~v + (~v ·∇ )~v − ~v ·∇ ~v =~0, (3)
t1z2 t2z1 z1 ~x z2 z2 ~x z1
i=1 i=1
X X
where ~v is an N-dimensional vector and ∇ = (∂ ,..,∂ ).
~x x1 xN
Each one of the above methods of solution allows one to solve a particular class of PDEs
and is not applicable to other classes.
Recently, several variants of the classical dressing method have been suggested, allowing
to unify the integration algorithms for C - and S - integrable PDEs [34], for C- and Ch-
integrable PDEs [35], and for S- and Ch- integrable PDEs [36]. In particular, the relation
between the matrix PDE (1), integrable by the method of characteristics, and the GL(M,C)
SDYM equation has been recently established in [37]. As a consequence of this result, it was
shown that the SDYM equation admits an infinite class of lower-dimensional reductions which
are integrable by the method of characteristics.
Inthispaperweextendtheresultsof[37],showingtheexistence ofremarkablydeeprelations
among S-, C- and Ch- integrable systems. More precisely, we do the following.
2
1. We show (in § 2) that the integrability properties (Lax pair, infinitely-many commuting
symmetries, large classes of analytic solutions) of the C-integrable (1+1)-dimensional ma-
trix Bu¨rgers hierarchy can be used to generate the integrability properties of S-integrable
PDEs in (2+1) dimensions, like the N-wave, KP, and DS equations; this result is achieved
using ablockFrobeniusmatrixreductionoftherelevant matrixfieldofthematrixBu¨rgers
hierarchy, leading to integrable chains of matrix equations for the blocks of such a Frobe-
nius matrix, followed by a systematic elimination procedure of some of these blocks. The
construction of large classes of solutions of the soliton equations from solutions of the ma-
trix Bu¨rgers hierarchy turns out to be intimately related to the construction of solutions
in Sato theory [38, 39, 40, 41]. On the way back, starting with the Lax pair eigenfunctions
of the derived S-integrable systems, we show that the coefficients of their asymptotic ex-
pansions, for large values of the spectral parameter, coincide with the elements of the
above integrable chains, obtaining an interesting spectral meaning of such chains. It
follows that, compiling these coefficients into the Frobenius matrix, one constructs the
C-integrable matrix Burgers hierarchy and its solutions from the eigenfunctions of the
S-integrable systems.
2. We show (in§3) that the integrability properties of themultidimensional matrixequation
(1), solvable by the method of characteristics, can be used to generate the integrability
properties of
(a) S-integrablePDEsin(1+1)dimensions, liketheN-wave, KdV,modifiedKdV(mKdV),
and NLS equations (in §3.2);
(b) S-integrable multidimensional generalizations of the GL(M,C) SDYM equations
(in §3.3); this derivation from the simpler and basic matrix equation (1), allows
one to uncover for free two important properties of such equations: a convenient
parametrization, given in terms of the blocks of the Frobenius matrix, allowing one
to reduce by half the number of equations, and the existence of a large class of
solutions describing the gradient catastrophe of multidimensional waves.
(c) S-integrable multidimensional Calogero equations [42, 43, 44, 45, 46] (in §3.4).
As before, these results are obtained considering a block Frobenius matrix reduction,
leading to integrable chains, followed by a systematic elimination procedure of some of
their elements. Vice-versa, such chains are satisfied by the coefficients of the asymptotic
expansion, for large values of the spectral parameter, of the eigenfunctions of the soliton
equations.
3. We show (in § 4) that a proper generalization of the block Frobenius matrix reduction of
the matrix Bu¨rgers hierarchy can be used to construct the integrability properties of non-
linear PDEs exhibiting properties in common with both S- and C- integrable equations.
Figure 1 below shows the diagram summarizing the connections discussed in § 2 and § 3.
We end this introduction mentioning previous work related to our main findings. i) The
matrix Burgers equation (2) with B = I, together with the block Frobenius matrix reduction
(13), have been used in [47] to construct some explicit solutions of the linear Schr¨odinger
and diffusion equations. ii) As already mentioned, once the connections illustrated in §2 are
exploited to construct large classes of solutions of soliton equations from simpler solutions of
the matrix Bu¨rgers hierarchy, the corresponding formalism turns out to be intimately related
to the construction of solutions of soliton equations in Sato theory.
3
(1+1) − dimensional
(2+1) − dimensional
C−integrable matrix PDES
S−integrable matrix PDES
block
(N−wave, Burgers, 3rd order
Frobenius elimination
integrable chains (N−wave, DS, KP, ..)
Burgers, ...) for matrix w
(j)
matrix for the w ’s (j)
of some w ’s
Lax pairs, symmetries
reduction Lax pairs, symmetries
and solutions
and solutions
w(1) w(2) w(3)
block (1+1) − dimensional
I 0 0
Frobenius w= S−integrable matrix PDES
matrix w 0 I 0 (N−wave, NLS, KdV, ..)
elimination Lax pairs, symmetries
integrable chains
for the w (j)’s (j) and solutions
of some w ’s
N − dimensional
ρ =0
j
matrix PDEs integrable by
Calogero systems
the method of characteristics:
block
in multidimensions
N−1 Frobenius integrable chains elimination
w +Σ w ρ (w)+[B,w] σ ( w ) =0
t x j (j)
j=1 j matrix for the w ’s (j) Lax pairs, symmetries
of some w ’s
reduction
Lax pairs, symmetries and solutions
and solutions σ=0
elimination
integrable chains GL(M,C) − SDYM
for the w (j)’s (j) and its (2N) − dimensional
of some w ’s
generalizations
Lax pairs, symmetries
and solutions
Fig. 1 The remarkable relations among PDEs integrable by the inverse spectral transform
method, by the method of characteristics and by the Hopf-Cole transformation.
2 Relation between C- and S-integrability
Usually C- and S- integrable systems are considered as completely integrable systems with
different integrability features. In this section we show the remarkable relations between them.
2.1 C-integrable PDEs
It is well known that the hierarchy of C-integrable systems associated with the matrix Hopf -
Cole transformation
Ψ = wΨ (4)
x
can be generated by the compatibility condition between equation (4) and the following hi-
erarchies of linear commuting flows (the hierarchy generated by higher x-derivatives and its
4
replicas):
Ψ = B(nm)∂nΨ, n,m ∈ N , (5)
tnm x +
where Ψ and w are square matrix functions and B(nm), n,m ∈ N are constant commuting
+
square matrices. The integrability conditions yield the following hierarchy of C-integrable
equations and its replicas:
w +[w,B(nm)W(n)]−B(nm)W(n) = 0, (6)
tnm x
where
W(n) = W(n−1) +W(n−1)w, n ∈ N ,
x + (7)
W(0) = I, W(1) = w, W(2) = w +w2, W(3) = w +2w w+ww +w3,....
x xx x x
and I is the identity matrix.
The first three examples, together with their commuting replicas, read:
1. n = 1: a C-integrable N-wave equation in (1+1)-dimensions:
w −B(1m)w +[w,B(1m)]w = 0, (8)
t1m x
2. n = 2: the matrix Bu¨rgers equation:
w −B(2m)w −2B(2m)w w +[w,B(2m)](w +w2) = 0; (9)
t2m xx x x
3. n = 3: the 3-rd order matrix Bu¨rgers equation:
w −B(3m)w −3B(3m)w w+[w,B(3m)](w +ww +2w w +w3)−
t3m xxx xx xx x x (10)
3B(3m)w (w +w2) = 0.
x x
The way of generating solutions of the C- integrable PDEs (6) is elementary: take the
general solution of equations (5):
Ψ(~x) = ekx+j,mP≥1B(jm)tjmkjΨˆ(k)dΩ(k), (11)
ZΓ
where Γ is an arbitrary contour in the complex k-plane, Ω(k) is an arbitrary measure and
Ψˆ(k) is an arbitrary matrix function of the spectral parameter k, and ~x is the vector of all
independent variables: ~x = {x,t ; n,m ∈ N }. Then
nm +
w = Ψ Ψ−1 (12)
x
solves (6).
5
2.2 Block Frobenius matrix structure, integrable chains and S-integrable
PDEs
It turns out that the C-integrable hierarchy of (1+1)-dimensional PDEs (6), including the N-
wave, Bu¨rgers and third order Bu¨rgers equations (8)-(10) as distinguished examples, generates
a corresponding hierarchy of S-integrable (2+1)-dimensional PDEs, including the celebrated
N-wave, DS and KP equations respectively. This is possible, due to the remarkable fact that
eqs. (4) and (5) are compatible with the following block Frobenius matrix structure of the
matrix function w:
w(1) w(2) w(3) ···
I 0 0 ···
w =  M M M , (13)
0 I 0 ···
M M M
 ... ... ... ... 
 
 
where I and 0 are the M × M identity and zero matrices, M ∈ N , and w(j), j ∈ N
M M + +
are M × M matrix functions. This block structure of w is consistent with eqs.(4) and (5)
(and therefore with the whole C-integrable hierarchy (6)) iff matrix Ψ is a block Wronskian
matrix:
Ψ(11) Ψ(12) Ψ(13) ···
∂−1Ψ(11) ∂−1Ψ(12) ∂−1Ψ(13) ···
Ψ =  x x x , (14)
∂−2Ψ(11) ∂−2Ψ(12) ∂−2Ψ(13) ···
x x x
 ... ... ... ... 
 
 
and
B(im) = diag(B˜(im),B˜(im),···), (15)
where the blocks Ψ(ij), i,j ∈ N are M × M matrices, and B˜(im), i ∈ N are constant
+ +
commuting M×M matrices. In equations (13)-(15), the matrices w, Ψ and B(im) are chosen to
be ∞×∞ square matrices containing an infinite number of finite blocks; only in dealing with
the construction of explicit solutions, it is convenient to consider a finite number of blocks.
Substituting the expressions (13) and (15) into the nonlinear PDEs (8-10), one obtains the
following (by construction) integrable infinite chains of PDEs, for n,m ∈ N :
+
w(n) −B˜(1m)w(n) +[w(n+1),B˜(1m)]+[w(1),B˜(1m)]w(n) = 0, (16)
t1m x
w(n) −B˜(2m)w(n) −2B˜(2m)w(n+1) −2B˜(2m)w(1)w(n) + (17)
t2m xx x x
[w(1),B˜(2m)](w(1)w(n) +w(n) +w(n+1))+[w(2),B˜(2m)]w(n) +[w(n+2),B˜(2m)] = 0,
x
w(n) −B˜(3m) w(n) +3(w(1)w(n) +w(n+1))+3w(1)(w(1)w(n) +w(n+1) +w(n))+ (18)
t3m xxx xx xx x x
(cid:16)
3w(2)w(n) +3w(n+2) +[w(1),B˜(3m)] w(n) +2(w(1)w(n) +w(n+1))+w(1)(w(n) +
x x xx x x x
(cid:17) (cid:16)
w(1)w(n) +w(n+1))+w(2)w(n) +w(n+2) +[w(2),B˜(3m)] w(n) +w(1)w(n) +w(n+1) +
x
[w(3),B˜(3m)]w(n) +[w(n+3),B˜(3m)] = 0.(cid:17) (cid:16) (cid:17)
From these chains, whose spectral nature will be unveiled in § 2.4, one constructs, through
a systematic elimination of some of the blocks w(j), the target S-integrable PDEs. Here
we consider the following basic examples.
6
(2+1)-dimensional N-wave equation. Fixing n = 1 in eqs.(16), and choosing m = 1,2,
one obtains the following complete system of equations for w(i), i = 1,2:
w(1) −B˜(1m)w(1) +[w(1),B˜(1m)]w(1) +[w(2),B˜(1m)] = 0, m = 1,2. (19)
t1m x
Eliminating w(2) from equations (19) one obtains the classical (2+1)-dimensional S-integrable
N-wave equation:
[w(1),B˜(12)]−[w(1),B˜(11)]−B˜(11)w(1)B˜(12) +B˜(12)w(1)B˜(11) + (20)
t11 t12 x x
[[w(1),B˜(11)],[w(1),B˜(12)]] = 0.
DS-type equation. Choosing n = 1,2 in equations (16), n = 1 in eq.(17), m = 1 in both
equations, and simplifying the notation as follows:
t = t , B˜(j) = B˜(j1), j ∈ N , (21)
j j1 +
one obtains the following complete system of equations for w(i), i = 1,2,3:
w(1) −B˜(1)w(1) +[w(2),B˜(1)]+[w(1),B˜(1)]w(1) = 0, (22)
t1 x
w(2) −B˜(1)w(2) +[w(3),B˜(1)]+[w(1),B˜(1)]w(2) = 0, (23)
t1 x
w(1) −B˜(2)w(1) −2B˜(2)w(2) −2B˜(2)w(1)w(1) + (24)
t2 xx x x
[w(1),B˜(2)](w(1)w(1) +w(1) +w(2))+[w(2),B˜(2)]w(1) +[w(3),B˜(2)] = 0.
x
Using eqs.(22) and (23), one can eliminate w(3) and w(2) from eq.(24). In the case B˜(2) = αB˜(1)
(α is a scalar), this results in the following equation for w(1):
[w(1),B˜(1)]+α w(1) −B˜(1)w(1)B˜(1) +[[w(1),B˜(1)],w(1)]+ (25)
t2 t1t1 xx t1
(cid:16)
B(1)w [B(1),w]−[B(1),w]w B(1) = 0.
x x
(cid:17)
In the simplest case of square matrices (M = 2), with B˜(1) = β diag(1,−1) (β is a scalar
constant), this equation reduces to the DS system:
1 1
β˜q − (q + q )−2(ϕq +2rq2) = 0, (26)
t2 2 xx β2 t1t1
1 1
−β˜r − (r + r )−2(ϕr+2qr2) = 0,
t2 2 xx β2 t1t1
1
ϕ − ϕ +4(rq) = 0,
xx β2 t1t1 xx
where
1
q = w(1), r = w(1), ϕ = (w(1) +w(1)) , β˜ = . (27)
12 21 11 22 x αβ
If β˜ = i, this system admits the reduction r = q¯:
1 1
iq − (q + q )−2(ϕq+2q¯q2) = 0, (28)
t2 2 xx β2 t1tu1
1
ϕ − ϕ +4(q¯q) = 0,
xx β2 t1t1 xx
becoming DS-I and DS-II if β2 = −1 and β2 = 1 respectively.
7
KP. To derive the celebrated KP equation, choose M = 1, take eqs.(17) with n = 1,2, and
eq. (18) with n = 1, B˜(2) = β, B˜(3) = −1, where β is a scalar parameter, obtaining:
w(1) −β w(1) +2w(1)w(1) +2w(2) = 0, (29)
t2 xx x x
(cid:16) (cid:17)
w(2) −β w(2) +2w(2)w(1) +2w(3) = 0,
t2 xx x x
(cid:16) (cid:17)
w(1) +w(1) +3 (w(1))2w(1) +(w(1))2 +w(1)w(1) +
t3 xxx x x xx
(cid:16) (cid:17)
3 w(2) +w(2)w(1) +w(1)w(2) +3w(3) = 0,
xx x x x
(cid:16) (cid:17)
where we have set m = 1 and used again the notations (21).
After eliminating w(2) and w(3), one obtains the scalar potential KP for u = w(1), y = t ,
2
t = t :
3
1 3 3
u + u + u2 + u = 0. (30)
t 4 xxx 2 x x 4β2 yy
(cid:16) (cid:17)
KP-I and KP-II correspond to β2 = −1 and β2 = 1 respectively.
2.3 Lax pairs for the S-integrable systems
Also the Lax pairs for the S-integrable systems derived in §2.2 can be constructed in a similar
way, from the system (4), (5). We first observe that, due to equation (4), equations (5) can be
rewritten as
Ψ = B(nm)W(n)Ψ. (31)
tnm
Due to the block Frobenius structure of w, it is convenient to work with the duals of equations
(4) and (31):
Ψ˜ = −Ψ˜w, (32)
x
Ψ˜ = −Ψ˜B(nm)W(n). (33)
tnm
Substituting (13) and (15) into equations (32-33), one obtains a system of linear chains for the
blocks of matrix Ψ˜. The first few equations involving the blocks of the first row read:
Ψ˜(1n) +Ψ˜(11)w(n) +Ψ˜(1(n+1)) = 0, (34)
x
Ψ˜(1n) +Ψ˜(11)B˜(1m)w(n) +Ψ˜(1(n+1))B˜(1m) = 0, (35)
t1m
Ψ˜(1n) +Ψ˜(11)B˜(2m)(w(n) +w(1)w(n) +w(n+1))+Ψ˜(12)B˜(2m)w(n) +Ψ˜(1(n+2))B˜(2m) = 0,(36)
t2m x
Ψ˜(1n) + Ψ˜(11)B˜(3m) w(n) +2(w(1)w(n) +w(n+1))+w(1)(w(n) +w(1)w(n) +w(n+1))+(37)
t3m xx x x x
(cid:16) (cid:16)
w(2)w(n) +w(n+2) +Ψ˜(12)B˜(3m) w(n) +w(1)w(n) +w(n+1) +Ψ˜(13)B˜(3m)w(n) +
x
Ψ˜(1(n+3))B˜(3m) = 0(cid:17), (cid:16) (cid:17)
where n ∈ N and Ψ˜(ij) is the (i,j)-block of matrix Ψ˜.
+
8
Lax pair for the N-wave equation. Setting n = 1 into eqs.(34,35) and eliminating Ψ˜(12),
one obtains (the dual of) the Lax pair for the N-wave equation (20):
ψ˜ −ψ˜ B˜(1m) +ψ˜[B˜(1m),w(1)] = 0, m = 1,2, (38)
t1m x
where ψ˜ = Ψ˜(11). The dual of it, is the well-known Lax pair of the N-wave equation (20):
ψ −B˜(1m)ψ −[B˜(1m),w(1)]ψ = 0, m = 1,2. (39)
t1m x
Of course, the compatibility condition of eqs.(38) and/or eqs. (39) yields the nonlinear system
(20).
Lax pair for DS. In this paragraph we set m = 1 in the integrable chains, and use the
notation (21). The first equation of the dual of the Lax pair is eq. (38) with m = 1. To derive
the second equation, we set m = n = 1 into eq. (36), and eliminate the fields Ψ˜(12), Ψ˜(13), using
eq. (34) with n = 1, 2. In this way one obtains the dual of the Lax pair for DS-type equations:
ψ˜ −ψ˜ B˜(1) +ψ˜[B˜(1),w(1)] = 0, (40)
t1 x
ψ˜ +ψ˜ B˜(2) +ψ˜ [w(1),B˜(2)]+ψ˜s˜= 0,
t2 xx x
s˜= [B˜(2),w(2)]+w(1)B˜(2) +B˜(2)w(1) +[B˜(2),w(1)]w(1).
x x
Therefore the Lax pair reads:
ψ −B˜(1)ψ −[B˜(1),w(1)]ψ = 0,
t1 x
ψ −B˜(2)ψ +[w(1),B˜(2)]ψ −s(y)ψ = 0, (41)
t2 xx x
s = [B˜(2),w(2)]+2B˜(2)w(1) +[B˜(2),w(1)]w(1).
x
The compatibility conditions of equations (40) or (41) yield a nonlinear system equivalent to
the system (22-24).
Lax pair for KP. In this paragraph we use the notations (21) as well. The first equations
of the dual of the Lax pair for KP are the scalar versions of eq.(40b) and eq.(41b) respectively,
with B˜(2) = β and B˜(3) = −1. To write the second equation of the Lax pair for KP, we must
take the scalar version of eq.(37) with m = n = 1, and eliminate Ψ˜(1i), i = 2,3,4 using eq.s(36)
for n = 1,2. As a result, the dual of the Lax pair reads
1
ψ˜ +ψ˜ +2ψ˜u = 0, (42)
β t2 xx x
3 u
ψ˜ +ψ˜ +3ψ˜ u − ψ˜ t2 −u = 0,
t3 xxx x x 2 β xx
(cid:18) (cid:19)
and the Lax pair is
1
ψ −ψ −2u ψ = 0, (43)
β t2 xx x
3 u
ψ +ψ +3u ψ + t2 +u ψ = 0.
t3 xxx x x 2 β xx
(cid:18) (cid:19)
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2.4 From the Lax pairs of S-integrable PDEs to C-integrable PDEs
As usual in the IST for (2+1)-dimensional soliton equations, one introduces the spectral pa-
rameter λ into the Lax pairs (39),(41) and (43) as follows
λxI+ P B˜(im)timλi
ψ(λ;~x) = χ(λ;~x)e i,m≥1 , (44)
obtaining, respectively, the following spectral systems for the new eigenfunction χ:
χ −B˜(1m)χ −λ[B˜(1m),χ]−[B˜(1m)w(1)]χ = 0, (45)
t1m x
χ −B˜(1)χ −λ[B˜(1),χ]−[B˜(1)w(1)]χ, (46)
t1 x
χ −B˜(2)χ +λ2[χ,B˜(2)]−2λB˜(2)χ +[w(1),B˜(2)](χ +λχ)−sχ = 0, (47)
t2 xx x x
s = [B˜(2),w(2)]+2B˜(2)w(1) +[B˜(2),w(1)]w(1).
x
χ −β 2λχ +χ +2χw = 0, (48)
t2 x xx x
(cid:16) (cid:17) 3 w
χ +3λχ +3λ2χ +χ +3λχw + χ t2 +w = 0.
t3 xx x xxx x 2 β xx
(cid:16) (cid:17)
It is now easy to verify that the coefficients of the λ - large expansion of the eigenfunction
χ satisfy the infinite chains (16-18):
w(n)(~x)
χ(λ;~x) = I − . (49)
λi
n≥1
X
Therefore we have obtained the spectral interpretation of such chains. In addition, since
the infinite chains (16-18) for the w(n)’s are equivalent, via the Frobenius structure (13), to
C-integrable systems, we have also shown how to go backward, from S- to C- integrability.
2.5 Construction of solutions and Sato theory
In order to construct solutions of the S-integrable PDEs generated in §2.2 from the elementary
solution scheme (11),(12) of the matrix Burgers hierarchy, we consider the matrices w and Ψ to
be finite matrices consisting of n ×n blocks (this can be done assuming that Ψ(1(n0+1)) = 0),
0 0
where n is an arbitrary positive integer greater than the number of blocks w(j)′s involved in
0
the S-integrable PDE under consideration. Taking into account the structures of w and Ψ given
by eqs.(13) and (14) respectively, we have that
w(1) w(2) ··· ··· w(n0)
I 0 ··· ··· 0 Ψ(11) Ψ(12) ··· Ψ(1n0)
M M M
w =  0M ... ... ... ... ,Ψ =  ∂x−1Ψ. (11) ∂x−1Ψ. (12) ··.· ∂x−1Ψ.(1n0) .
 ... ... ... ... ...   .. .. .. .. 
 ... ... 0 I 0   ∂x−n0+1Ψ(11) ∂x−n0+1Ψ(12) ··· ∂x−n0+1Ψ(1n0) 
M M M  
 
  (50)
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