Table of Contents

— *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.

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.

**n**is the number of iterations.**E**is**an LPDO**, that is a Linear Partial Differential Operator in the form

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.

**u**is optional, the dependent function.

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:

- and
**M arbitrary**LPDO. - L is a third order LPDO given in its normalized form, i.e. for L in one of those forms.
- with principal symbol

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

LPDO__set_vars(vars):

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

LPDO__set_vars([x,y,z,t]):

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

Declaration

LPDO__warnings:=true:

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

LPDO__warnings:=false:

LPDOs

where **J** is a multi-index,

, and

are kept as a one dimensional array of its coefficients.

- LPDO__create() returns an LPDO that multiplies by zero, an LPDO of order zero.
- LPDO__set_value(L::array,f,J::list) sets coefficient of LPDO
**L**equaled to**f**. - LPDO__create0(a00) returns an LPDO of multiplication by
**a00**, an LPDO of order zero. - LPDO__create1(a10,a01,a00) returns an LPDO

.

- LPDO__create2(a20,a11,a02,a10,a01,a00) returns an LPDO

.

- Create_LPDO_from_coeff(a::list) returns LPDO with the coefficients from list
**a**, in which they are

ordered lexicographically. E.g.

- Create_LPDO_from_coeff([a00,a10,a01,a20,a11,a02]) returns

- LPDO__add_value(L::array,f,J::list) adds
**f**to the coefficient with multi-index**J**.

- LPDO__degree(L::array) returns the order of LPDO
**L**.

- LPDO__simplify(L::array) applies
**simplify**to all the coefficients of LPDO**L**.

- LPDO__expand(L::array) applies
**expand**to all the coefficients of LPDO**L**.

- LPDO__factor(L::array) applies
**factor**to all the coefficients of LPDO**L**.

- LPDO__add(L1::array,L2::array) returns the sum of
**L1**and**L2**. - LPDO__inc(L1::array,L2::array) adds
**L1**and**L2**, and save the result as**L1**. - LPDO__minus(L1::array,L2::array) returns the difference of
**L1**and**L2**. - LPDO__mult(L1::array,L2::array) returns the composition of
**L1**and**L2** - LPDO__mult1(f,L::array) multiplies
**L**by function**f**on the left.

- LPDO__print(L::array) is the “pretty print”.
- LPDO__conj(L::array,g) returns the result of the gauge transformation

- LPDO__apply(L::array, f) applies
**L**to function**f**. - LPDO__value(L::array,J::list) returns the coefficient with index
**J**, that is at**D_J**in**L**. - LPDO__transpose(L::array) returns the formal adjoint of
**L**.

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:

- computes chain of the invariants of length
**n**into both directions - if we meet zero on the way, then the corresponding transformed LPDO is factorizable. So we find its kernel and compute the kernel of the initial LPDO using known formulas.

- Laplace_solver(n,E,u) has been described here.
- LPDO__Lapl_trans(i,L) returns
**L_i**, where**i=1**or**i=-1**. LPDO**L**must be of the form

or

- LPDO__Lapl_chain(n,L) returns the chain of Laplace invariants and their generalizations. The chain is output in the form

Here **n** is the number of iterations of Laplace transformations.

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.

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.

2015/02/16

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:

- Levy transformations (i.e. transformations based on Wronskian formulas)
- 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 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.

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.

- LPDO__XXY_DT_with_Dy(I1,I2,I3,I4,I5)
- LPDO__XXY_DT_with_Dx(I1,I2,I3,I4,I5,r) (case I2<>I3 only)
- LPDO__XYS_DT_with_Dx(I1,I2,I3,I4,I5)
- LPDO__XYS_DT_with_Dy(I1,I2,I3,I4,I5)
- LPDO__XYS_DT_with_Dx_Dy(I1,I2,I3,I4,I5)

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.

- LPDO__DT_of_type_I(C,M,f) performs a step of invertible Darboux transformation of type I. Returns list [N,L1,Minv,Ninv,A,G], where N,L1 are operators from the intertwining relation NL=L1M, and Minv,Ninv are

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).

- LPDO__XYS_chain_of_first_order_invertible_DTs(chain,I1,I2,I3,I4,I5) applies Darboux transformations as indicated in argument “chain” to an operator given by its gauge invariants. Argument chain should be of the form of the list, where each element indicates Darboux transformation of certain type. Thus, e.g. [1,1,2,3] indicates the sequence: Darboux transformations with principal symbol Dx, then with Dx, then with Dy, and then with Dx+Dy. The output is a list of lists: [[invariants after first Darboux tr.],[invariants after second Darboux tr.],…]
- LPDO__XYZ_Chart(b,list_of_invs) computes (including plotting) b applications of all possible Darboux transformations of order 1 of type I to an operator given by its gauge invariants. Returns list of lists: [L1,L2,L3]. L1: if used later inside command display(L1, scaling=constrained,axes=Normal), will plot the orbit, L2: list of points (D_y up, D_x right, D_x+D_y on the diagonal), L3: list of lists of invariants (in correspondence with L2)

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")

- LPDO__a_step_of_Eq_and_Shift_for_Type_I(C,M,f,S,T) performs a step of Equivalence-Shift procedure for L with symbol XXY for type I so for L=CM+f
- LPDO__a_step_of_Eq_and_Shift(L,M,N,L1,Minv,Ninv,A,G,S,T) performs a step of Equivalence-Shift procedure for L of arbitrarily form. Returns [Ln,Mn,Nn,L1n,Minvn,Ninvn,An,Gn].
- LPDO__a_step_of_Eq_and_Shift_basic(L,M,N,L1,S,T) performs a step of Equivalence-Shift procedure M -> M+S*L, N -> N+L1*S for L of arbitrarily form, does not compute the auxilary operators, just the change in L and M, L1, N. Returns [Ln,Mn,Nn,L1n]

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