The NLDE package

latest update: May 2024

### Introduction

NLDE stands for NonLinear Algebra and Differential (also Difference) Equations, yes, there is a missing "A"? Not really. NLDE is a mathematical software for symbolic computations with solutions of NonLinear (ordinary and partial) Differential (also Difference) Equations (NLDE). These equations are algebraic in their indeterminates, i.e., the differential equations are polynomials in the independent variables and some (partial) derivatives of the dependent variables. We call them algebraic differential equations (ADEs), and their solutions are D-algebraic functions.

For instance, the Korteweg-De Vries (KDV) equation is the algebraic PDE:

Suppose we want to find an ADE for the following transformation

This means we want to find an algebraic PDE (of the lowest order possible) whose solutions u(t,x) are expressed in terms of solutions of the KDV equation by the above relation. One finds

One can perform these computations automatically with NLDE. It is also possible to compute ADEs for exponentiation, sums, products, and ratios of D-algebraic functions.

NLDE is implemented in Maple 2022 and works with recent versions (Maple 2018 - 2023).

The easiest way to use NLDE in Maple is by putting the downloaded file, NLDE.mla, in your working directory and including the lines

> restart;

> libname:=currentdir(), libname:

> with(NLDE):

at the beginning of your worksheet.

However, to avoid carrying the file NLDE.mla in all your working directories, you can just put it in a directory that you defined for libraries. For more details about this, please have a look at the Maple help page of libname.

### Source, help page, and user guide

If you want to know more about NLDE, please visit D-Algebraic Functions in Maple.

### New (April 2024): HoloToSimpleRatrec

James Worrel and I wrote the paper '' On Rational Recursion for Holonomic Sequences ''. The paper proves that every holonomic sequence satisfies a rational recursion in which each term depends rationally on some preceding terms (the index variable does not occur in the recursion). Such a relation between terms of a sequence is often desired in Computer science, mainly when the sequence in study encodes a characteristic property of a formal language.

The implementation enables the automatic generation of somos-like sequences, which are integral sequences defined by rational relations. For instance, the rational recursion below defines an integral sequence for any integral initial values s(0), s(1), s(2) (nonzero), and s(3).