Table of Contents

LPDO package

Ekaterina Shemyakova 2015/02/16 14:21

This is a package for computer algebra system MAPLE. It contains symbolic algorithms for Linear Partial Differential Operators (LPDOs) and for linear Partial Differential Equations (PDEs) with parametric coefficients. We consider arbitrary LPDOs in arbitrary many independent variables and of arbitrary orders and allow coefficients to be symbolic. Numerous tools are available with the focus on the Darboux, generalized Darboux and related invariant algorithms.


To use the package, download lpdo_.txt and type

read "/home/username/path_to_file/lpdo_.txt";

(for Linux) and

read `\\path_to_file\\lpdo_.txt`;

(for Windows) at the top of your Maple sheet.

Highlights of the Package


Solves E by Laplace method (not to be confused with Laplace integral method) if this is possible to do so by less or equal to n iterations. Works significantly better than existing in MAPLE PDEtools[Laplace], which fails in many examples.

or a PDE, a Partial Differential Equation in the form
Here a,b,c may depend on the independent variables. One can use any names for independent variables. When work with LPDOs, specify them using LPDO__set_vars at the top of the worksheet. For example, LPDO__set_vars([s,t]): would declare two independent variables s and t.

Demo: demonstration_laplace_transformations_ver2.mws

ginv(L) and ginv(L,M)

Procedure ginv(L) returns a set of generating differential invariants for L with respect to the gauge transformations
The same procedure but with two arguments, ginv(L,M) returns a set of generating joint differential invariants for L and M with respect to the gauge transformations. Works for the following operators and pairs of operators:

Demo: demonstration_invariants.mws

Basic Manipulations with LPDOs

Package can work with LPDOs of arbitrary orders and depended on arbitrary many independent variables. The independent variables are to be declared at the start of each worksheet using function


where vars is a list of independent variables. For example,


sets 4 independent variables x,y,z,t.



means that using some of the function available in the package, you may get some additional information (e.g. solutions of which PDEs must belong to the field). By default


where J is a multi-index,
, and
are kept as a one dimensional array of its coefficients.

Declaration of an LPDO and changing its coefficients.



ordered lexicographically. E.g.

Simplification of the coefficients

Algebraic operations



Laplace transformations


All differential invariants of operators

with respect to the gauge transformations

can be expressed in terms of so-called Laplace invariants

If h is not zero then define a +1 Laplace transformation by equality

If k is not zero then define a -1 Laplace transformation by equality

Thus until h or k are not zero, we have chain

There are some formulas relating those invariants, which implies that each new iteration of a Laplace transformation brings us only one new invariant (only h or k is new). So, basically, we have the following chain of invariants:

The algorithm works as follows:

Implemented procedures


Here n is the number of iterations of Laplace transformations.

Demo: demonstration_laplace_transformations_ver2.mws


The property of the existence of a factorization of an LPDO is invariant with respect to gauge transformations. Thus, differential invariants can be very useful for search of a factorization of an LPDO. For each type of factorization (e.g. for LPDOs with symbol (pX+qY)XY there are 12 different types of factorizations) return the conditions for the existence of a factorization for an LPDO are implemented for all bivariate LPDOs of third order in terms of generating invariants of this LPDO. More specialized notions as obstacles to factorizations are implemented too.


Darboux transformations

Classical Laplace transformations (do not confuse with Laplace transform involving integrals) are implemented. Namely, the funcitons return the resulting operators, as well as the chain of invariants for Laplace LPDOs in classical for Darboux form, that is with symbol XY, as well as for Laplace LPDOs in the form more familiar for Physicists, that is with symbol XX+YY. Construction of a Darboux transformation of order one (two) by one (two) linearly independent solutions.

Cancellation of the mixed derivatives in M from the left and from the right, for a pair (M,L) defining a Darboux transformation. Generating sets of invariants for pairs (M,L) defining a Darboux transformation unders the gauge transforamtiosn of the pairs and under the gauged evolutions. Expressions of the later in terms of the former. Invariant description of pairs (M,L) defining any at all Darboux transformation.

Invertible Darboux transformations


For the first ideas on efficient and new types of invertible Darboux transformations refer to [Programming 2015 paper, to appear (part I)] and this paper (part II).

Reading the literature on Darboux transformations, one can notice only two particular cases/types of Darboux transformation:

  1. Levy transformations (i.e. transformations based on Wronskian formulas)
  2. Laplace tranformations

We discovered that there exist other types of Darboux transformations too, and among them there are many transformations with the property of invertibility. We singled out a class of such transformations that are not only invertible, but the formulas for the inverses are particular good, which make the corresponding algorithms very efficient. We call them Darboux transformations of type I.

Darboux transformations of type I

Darboux transformations of type I are those Darboux transformations (M,N) with source L and target L1 for which it is true that L=CM+f where f is an element of the field of coefficients, and C is a linear partial differential operator.

Invertible Darboux transformations represented by transformations of the gauge invariants

Not all Darboux transformations of first order are invertible. But those that are invertible, must be of type I.

For the case of operators of order three and of two independent variables we can find a generating set of all gauge differential invariants I1,I2,I3,I4,I5 (see procedure ginv here). Then the following functions allow to compute the transformations of this invariants under Darboux transformations of type I of first orders. The output is the list of new invariants.

XXY indicates that the procedure is for the operators of the form

XYS indicates that the procedure is for the operators of the form

Dx, Dy, or Dx_Dy stand for Darboux transformations with M of the corresponding principal symbol.

Orbits of Darboux transformations of type I

Darboux transformations of type I are created to be particular efficient. Implementation of orbits for operators of arbitrary kinds is work in progress. Here we implement the analogue of Laplace invariants chain for operators of the form

In the same manner as for Laplace transformations, Darboux transformations of type I are invertible. Therefore, if one element in the orbit is solvable then all the others are solvable too. Here for brevity we say that linear partial operator is solvable meaning the corresponding equation is solvable in quadratures.

operators defining the intertwining relation for the inverse. That is Ninv*L1=L*Minv. Operators A,G are operators that relates M and N: MA=GN (this relation is only true for invertible Darboux transformations).

Maple 16 worksheet example 1 (infinite orbit)

Maple 16 worksheet example 2 (9 elements orbits, "square shape")

Maple 16 worksheet example 3 (infinite orbit in one direction, "kite shape")

Equivalence-Shift procedure