Abstract
We establish deep and remarkable connections among partial differential equations (PDEs) integrable by different methods: the inverse spectral transform method, the method of characteristics and the HopfCole transformation. More concretely, 1) we show that the integrability properties (Lax pair, infinitelymany commuting symmetries, large classes of analytic solutions) of (2+1)dimensional PDEs integrable by the Inverse Scattering Transform method (integrable) can be generated by the integrability properties of the (1+1)dimensional matrix Bürgers hierarchy, integrable by the matrix HopfCole transformation (integrable). 2) We show that the integrability properties i) of integrable PDEs in (1+1)dimensions, ii) of the multidimensional generalizations of the selfdual Yang Mills equations, and iii) of the multidimensional Calogero equations can be generated by the integrability properties of a recently introduced multidimensional matrix equation solvable by the method of characteristics. To establish the above links, we consider a block Frobenius matrix reduction of the relevant matrix fields, leading to integrable chains of matrix equations for the blocks of such a Frobenius 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 matrix Bürgers 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 matrix reduction of the matrix Bürgers hierarchy generates PDEs exhibiting integrability properties in common with both  and  integrable equations.
On the remarkable relations among PDEs integrable by
the inverse spectral transform method, by the method of characteristics and by the HopfCole transformation
A. I. Zenchuk and P. M. Santini
Center of Nonlinear Studies of the Landau Institute for Theoretical Physics
(International Institute of Nonlinear Science)
Kosygina 2, Moscow, Russia 119334
Dipartimento di Fisica, Università di Roma ”La Sapienza” and
Istituto Nazionale di Fisica Nucleare, Sezione di Roma 1
Piazz.le Aldo Moro 2, I00185 Roma, Italy
email: zenchuk@itp.ac.ru , paolo.santini@roma1.infn.it
July 27, 2019
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.

Equations solvable by the method of characteristics [1], hereafter called, for the sake of brevity, integrable, like the following matrix PDE in arbitrary dimensions [2]:
(1) where is a square matrix and 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].

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 integrable [9] or soliton equations. Within this class of equations, we distinguish four different subclasses, depending on the nature of the associated spectral theory.

Multidimensional PDEs associated with oneparameter 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 dispersionless KP equation, the heavenly equation of Plebanski [33] and the following integrable system of PDEs in dimensions [30]:
(3) where is an dimensional vector and .
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  and  integrable PDEs [34], for  and  integrable PDEs [35], and for  and  integrable PDEs [36]. In particular, the relation between the matrix PDE (1), integrable by the method of characteristics, and the 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 lowerdimensional reductions which are integrable by the method of characteristics.
In this paper we extend the results of [37], showing the existence of remarkably deep relations among ,  and  integrable systems. More precisely, we do the following.

We show (in 2) that the integrability properties (Lax pair, infinitelymany commuting symmetries, large classes of analytic solutions) of the integrable (1+1)dimensional matrix Bürgers hierarchy can be used to generate the integrability properties of integrable PDEs in (2+1) dimensions, like the Nwave, KP, and DS equations; this result is achieved using a block Frobenius matrix reduction of the relevant matrix field of the matrix Bürgers hierarchy, leading to integrable chains of matrix equations for the blocks of such a Frobenius 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 matrix Bürgers 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 integrable systems, we show that the coefficients of their asymptotic expansions, 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 integrable matrix Burgers hierarchy and its solutions from the eigenfunctions of the integrable systems.

We show (in 3) that the integrability properties of the multidimensional matrix equation (1), solvable by the method of characteristics, can be used to generate the integrability properties of

integrable PDEs in (1+1) dimensions, like the Nwave, KdV, modified KdV (mKdV), and NLS equations (in §3.2);

integrable multidimensional generalizations of the 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.
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. Viceversa, 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.


We show (in 4) that a proper generalization of the block Frobenius matrix reduction of the matrix Bürgers hierarchy can be used to construct the integrability properties of nonlinear PDEs exhibiting properties in common with both  and  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 , together with the block Frobenius matrix reduction (13), have been used in [47] to construct some explicit solutions of the linear Schrödinger 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 Bürgers hierarchy, the corresponding formalism turns out to be intimately related to the construction of solutions of soliton equations in Sato theory.
Fig. 1 The remarkable relations among PDEs integrable by the inverse spectral transform method, by the method of characteristics and by the HopfCole transformation.
2 Relation between  and integrability
Usually  and  integrable systems are considered as completely integrable systems with different integrability features. In this section we show the remarkable relations between them.
2.1 integrable PDEs
It is well known that the hierarchy of integrable systems associated with the matrix Hopf  Cole transformation
(4) 
can be generated by the compatibility condition between equation (4) and the following hierarchies of linear commuting flows (the hierarchy generated by higher derivatives and its replicas):
(5) 
where and are square matrix functions and are constant commuting square matrices. The integrability conditions yield the following hierarchy of integrable equations and its replicas:
(6) 
where
(7) 
and is the identity matrix.
The first three examples, together with their commuting replicas, read:

: a integrable wave equation in (1+1)dimensions:
(8) 
: the matrix Bürgers equation:
(9) 
: the 3rd order matrix Bürgers equation:
(10)
The way of generating solutions of the  integrable PDEs (6) is elementary: take the general solution of equations (5):
(11) 
where is an arbitrary contour in the complex plane, is an arbitrary measure and is an arbitrary matrix function of the spectral parameter , and is the vector of all independent variables: . Then
(12) 
solves (6).
2.2 Block Frobenius matrix structure, integrable chains and integrable PDEs
It turns out that the integrable hierarchy of (1+1)dimensional PDEs (6), including the wave, Bürgers and third order Bürgers equations (8)(10) as distinguished examples, generates a corresponding hierarchy of integrable (2+1)dimensional PDEs, including the celebrated 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 :
(13) 
where and are the identity and zero matrices, , and are matrix functions. This block structure of is consistent with eqs.(4) and (5) (and therefore with the whole integrable hierarchy (6)) iff matrix is a block Wronskian matrix:
(14) 
and
(15) 
where the blocks are matrices, and are constant commuting matrices. In equations (13)(15), the matrices , and 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 (810), one obtains the following (by construction) integrable infinite chains of PDEs, for :
(16) 
(17)  
(18)  
From these chains, whose spectral nature will be unveiled in 2.4, one constructs, through a systematic elimination of some of the blocks , the target integrable PDEs. Here we consider the following basic examples.
(2+1)dimensional wave equation.
DStype equation.
Choosing in equations (16), in eq.(17), in both equations, and simplifying the notation as follows:
(21) 
one obtains the following complete system of equations for , :
(22)  
(23)  
(24)  
Using eqs.(22) and (23), one can eliminate and from eq.(24). In the case ( is a scalar), this results in the following equation for :
(25)  
In the simplest case of square matrices (), with ( is a scalar constant), this equation reduces to the DS system:
(26)  
where
(27) 
If , this system admits the reduction :
(28)  
becoming DSI and DSII if and respectively.
Kp.
To derive the celebrated KP equation, choose , take eqs.(17) with , and eq. (18) with , , , where is a scalar parameter, obtaining:
(29)  
where we have set and used again the notations (21).
After eliminating and , one obtains the scalar potential KP for , , :
(30) 
KPI and KPII correspond to and respectively.
2.3 Lax pairs for the integrable systems
Also the Lax pairs for the 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
(31) 
Due to the block Frobenius structure of , it is convenient to work with the duals of equations (4) and (31):
(32)  
(33) 
Substituting (13) and (15) into equations (3233), one obtains a system of linear chains for the blocks of matrix . The first few equations involving the blocks of the first row read:
(34)  
(35)  
(36)  
(37)  
where and is the ()block of matrix .
Lax pair for the wave equation.
Setting into eqs.(34,35) and eliminating , one obtains (the dual of) the Lax pair for the wave equation (20):
(38) 
where . The dual of it, is the wellknown Lax pair of the wave equation (20):
(39) 
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 in the integrable chains, and use the notation (21). The first equation of the dual of the Lax pair is eq. (38) with . To derive the second equation, we set into eq. (36), and eliminate the fields , , using eq. (34) with . In this way one obtains the dual of the Lax pair for DStype equations:
(40)  
Therefore the Lax pair reads:
(41) 
The compatibility conditions of equations (40) or (41) yield a nonlinear system equivalent to the system (2224).
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 and . To write the second equation of the Lax pair for KP, we must take the scalar version of eq.(37) with , and eliminate , using eq.s(36) for . As a result, the dual of the Lax pair reads
(42)  
and the Lax pair is
(43)  
2.4 From the Lax pairs of integrable PDEs to integrable PDEs
As usual in the IST for (2+1)dimensional soliton equations, one introduces the spectral parameter into the Lax pairs (39),(41) and (43) as follows
(44) 
obtaining, respectively, the following spectral systems for the new eigenfunction :
(45) 
(46)  
(47)  
(48)  
It is now easy to verify that the coefficients of the  large expansion of the eigenfunction satisfy the infinite chains (1618):
(49) 
Therefore we have obtained the spectral interpretation of such chains. In addition, since the infinite chains (1618) for the ’s are equivalent, via the Frobenius structure (13), to integrable systems, we have also shown how to go backward, from  to  integrability.
2.5 Construction of solutions and Sato theory
In order to construct solutions of the Sintegrable PDEs generated in §2.2 from the elementary solution scheme (11),(12) of the matrix Burgers hierarchy, we consider the matrices and to be finite matrices consisting of blocks (this can be done assuming that ), where is an arbitrary positive integer greater than the number of blocks involved in the Sintegrable PDE under consideration. Taking into account the structures of and given by eqs.(13) and (14) respectively, we have that
(50) 
We remark that the blocks , of are defined, via (11), by equations
(51) 
in terms of the arbitrary spectral functions , while the remaining blocks are constructed through the equations . Then, via (12), the components of the blocks are expressed in terms of through the compact formula
(52) 
3 Relation between  and integrability
Following the same strategy illustrated in §2, in this section we establish the deep relations between the matrix PDE (1), recently introduced in [2] and integrated there by the method of characteristics, and i) (1+1)dimensional integrable soliton equations like the KdV and NLS equations; ii) the  SDYM equation and its multidimensional generalizations; iii) the multidimensional Calogero systems [42, 43, 44, 45, 46]. In this section, matrix must be diagonalizable.
3.1 Matrix equations integrable by the method of characteristics
Consider the following matrix eigenvalue problem
(56) 
for the matrix , where is the diagonal matrix of eigenvalues, is a suitably normalized matrix of eigenvectors, and associate with it the following flows for :
(57) 
where are constant commuting matrices as in 2 and is the vector of all independent variables: . The compatibility between the flows (57) implies the following commuting quasilinear PDEs for the eigenvalues:
(58) 
the additional compatibility with the eigenvalue problem (56) implies the following nonlinear PDEs: