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Nonlinear integrable evolution equations in 1 + 1 dimensions arise from constraints of the 2 + 1-dimensional hierarchies associated with the Kadomtsev–Petviashvili (KP) equation, the modified KP equation and the Dym equation, respectively. Links of Backlund type and of reciprocal type are known to exist between the 2 + 1-dimensional systems. The corresponding links between the constrained flows are discussed in a general framework. In particular, squared eigenfunction symmetries generated by solutions of the associated linear scattering problems are considered. The links between the soliton hierarchies are extended to these symmetries.

The integrable Kadomtsev–Petviashvili (KP) hierarchy is compatible with generalized k‐constraints of the type (Lk)−=∑i qi∂−1xri. A large class of solutions—among them solitons—can be represented by Wronskian determinants of functions satisfying a set of linear equations. In this paper we shall obtain additional conditions for these functions imposed by the constraints.

A three-by-three matrix spectral problem for AKNS soliton hierarchy is proposed and the corresponding Bargmann symmetry constraint involved in Lax pairs and adjoint Lax pairs is discussed. The resulting nonlinearized Lax systems possess classical Hamiltonian structures, in which the nonlinearized spatial system is intimately related to stationary AKNS flows. These nonlinearized Lax systems also lead to a sort of involutive solutions to each AKNS soliton equation.

Integrable systems in 1+1 dimensions arise from the KP hierarchy as symmetry reductions involving square eigenfunctions. Exploiting the residual gauge freedom in these constraints new integrable systems are derived. They include generalizations of the hierarchy of the Kundu–Eckhaus equation and higher‐order extensions of the Yajima–Oikawa and Melnikov hierarchies. Constrained modified KP flows yield further integrable equations such as the hierarchies of the derivative NLS equation, the Gerdjikov–Ivanov equation, and the Chen–Lee–Liu equation.

Modifications of the standard Poisson brackets associated with differential scattering operators are considered. A linear bracket originates from a non-standard r-matrix on the algebra of pseudo-differential operators. Two quadratic brackets are investigated which provide Hamiltonian formulations for various reductions of the modified KP hierarchy.

The bi-Hamiltonian structure of an integrable dynamical system introduced by Melnikov (1986) is studied. This equation arises as a symmetry constraint of the KP hierarchy via squared eigenfunctions and can be understood as a Boussinesq system with a source. The standard linear and quadratic Poisson brackets associated with the space of pseudo-differential symbols are used to derive two compatible Hamiltonian operators. A bi-Hamiltonian formulation for the Drinfeld-Sokolov system is derived via reduction techniques.

Generalized Lax equations are considered in the spirit of Sato theory. Three decompositions of an underlying algebra of pseudo-differential operators lead, in turn, to three different classes of integrable nonlinear hierarchies. These are associated with Kadomtsev-Petviashvili, modified Kadomtsev-Petviashvili and Dym hierarchies in 2 + 1 dimensions. Miura- and auto-Bäcklund transformations are shown to originate naturally from gauge transformations of the Lax operators. General statements on reciprocal links between these hierarchies are established, which, in particular, give rise to novel reciprocal auto-Bäcklund transformations for the Dym hierarchy. These links are formulated as Darboux theorems for the associated Lax operators.

Generalizations of the classical Darboux theorem are established for pseudo-differential scattering operators of the form L = ∑ i=0 N u i ∂ i + Σ i=1 m Φ i ∂ −1 Ψ i † i. Iteration of the Darboux transformations leads to a gauge transformed operator with coefficients given by Wronskian formulas involving a set of eigenfunctions of L. Nonlinear integrable partial differential equations are associated with the scattering operator L which arise as a symmetry reduction of the multicomponent KP hierarchy. With a suitable linear time evolution for the eigenfunctions the Darboux transformation is used to obtain solutions of the integrable equations in terms of Wronskian determinants.

Three different decompositions of the algebra of pseudo-differential operators and corresponding r-matrices are considered. Three associated classes of nonlinear integrable equations in 1+1 and 2+1 dimensions are discussed within the framework of generalized Lax equations and Sato's approach.

The bi-Hamiltonian structure of integrable supersymmetric extensions of the Korteweg-de Vries (KdV) equation related to theN=1 and theN=2 superconformal algebras is found. It turns out that some of these extensions admit inverse Hamiltonian formulations in terms of presymplectic operators rather than in terms of Poisson tensors. For one extension related to theN=2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical KdV. They can be explained by a factorization of the corresponding Lax operator. All the bi-Hamiltonian formulations are derived in a systematic way from the Lax operators.

Basic invariants, such as conserved quantities, symmetries, mastersymmetries, and recursion operators are explicitly constructed for the following nonlinear lattice systems: The modified Korteweg–de Vries lattice, the Ablowitz–Ladik lattice, the Brusci–Ragnisco lattice, the Ragnisco–Tu lattice and some cases of the class of integrable systems introduced by Bogoyavlensky. The algorithmic basis for obtaining these quantities is described and the interrelation between the underlying mastersymmetry approach and the Lax pair analysis is discussed. By explicit presentation of the higher‐order members of the corresponding hierarchies new completely integrable lattice flows are found. For all systems, multi‐Hamiltonian formulations are given.

Following Sato's famous construction of the KP hierarchy as a hierarchy of commuting Lax equations on the algebra of microdifferential operators, it is shown that n-reduction leads to a recursive scheme for these equations. Explicit expressions for the recursion operators and the Hamiltonian operators are obtained.

Oft wird man ein Computeralgebra-System wie MuPAD interaktiv bedienen, d.h., man gibt eine Anweisung wie z.B. die Multiplikation zweier Zahlen an das System und wartet dann, bis MuPAD das Ergebnis berechnet hat und auf dem Bildschirm ausgibt. Wahrend einer laufenden Rechnung konnen keine weiteren Eingaben gemacht werden, die Rechnung kann aber zu jedem Zeitpunkt vom Benutzer unterbrochen werden.

Alle MuPAD-Objekte bestehen aus Operanden (Abschnitt 4.1). Eine wichtige Eigenschaft eines Computeralgebra-Systems besteht in der Moglichkeit, diese Bausteine durch neue Werte zu ersetzen. MuPAD stellt hierzu die Funktionen subs (englisch: substitute), subsex (englisch: substitute expression) und subsop (englisch: substitute operand) zur Verfugung.

Every input to MuPAD yields a result after evaluation by the system. The computed objects are stored internally in a history table. Note that the result of every statement is stored, even if it is not printed on the screen. You can use it later by means of the function last. The command last(1) returns the previous result, last (2) the last but one, and so on. Instead of last(i), you may use the shorter notation %i. Moreover, % is short for %1 or last (1).

Das meiste mathematische Wissen MuPADs ist in so genannten Bibliotheken enthalten. Eine Bibliothek (englisch: library) besteht aus einer Sammlung von Funktionen zur Losung von Problemen eines speziellen Gebietes wie etwa der linearen Algebra, der Zahlentheorie, der Numerik usw. Die Funktionen einer Bibliothek sind in der MuPAD-Programmiersprache geschrieben. Sie werden in gleicher Weise benutzt wie Kernfunktionen, ohne dass eine Kenntnis der MuPAD-Programmiersprache erforderlich ist.

MuPAD bietet einige Zufallszahlengeneratoren, mit denen viele Experimente durchgefuhrt werden konnen.

Most of MuPAD’s mathematical knowledge is organized in libraries. Such a library comprises a collection of functions for solving problems in a particular area, such as linear algebra, number theory, numerical analysis, etc. Library functions are written in MuPAD’s programming language. You can use them in the same way as kernel functions, without knowing anything about the programming language.

The function solve solves systems of equations. This routine can handle various different types of equations. Besides “algebraic” equations, also certain classes of differential and recurrence equations can be solved. Further, there is a variety of specialized solvers that only handle special classes of equations. Many of these algorithms are actually called by the “universal” solve once it has identified the type of the equations. The user can also call the specialized solvers directly. With ? solvers, you can request a survey of all the solvers that are available in MuPAD.

Jede Eingabe an MuPAD liefert nach Auswertung durch das System ein Ergebnis. Unabhangig von der Ausgabe kann mit der Punktion last spater auf diesen Wert zugegriffen werden, da die berechneten MuPAD-Objekte intern in einer sogenannten Ergebnistabelle (im englischen Sprachgebrauch: history table) gespeichert werden. Der Befehl last (1) liefert dann das letzte Ergebnis zuruck, last (2) das vorletzte Ergebnis, usw. Statt last(i) kann man auch das aquivalente kurzere Symbol %i benutzen. Zusatzlich existiert die Abkurzung % fur %1 bzw. last(l). Damit kann die folgende Eingabe » f:= diff(ln(ln(x)), x): int(f, x); auch so dem System ubergeben werden: » diff(ln(ln(x)), x): int(%, x)

Branching instructions are an important element of every programming language. Depending on the value or the meaning of variables, different commands are executed. The simplest variant in MuPAD is the if statement:» for i from 2 to 4 do if isprime(i) then print(expr2text(i).″ is prime″) else print(expr2text(i).″ is not prime″) end_if end_for: ″2 is prime″ ″3 is prime″ ″4 is not prime″

You can use MuPAD’s random number generators random, frandom and stats::xxxRandom to perform many experiments in MuPAD.

Nicht alle von MuPAD berechneten Ergebnisse werden auf dem Bildschirm dargestellt. Typische Beispiele sind die Befehle in for-Schleifen (Kapitel 16) oder innerhalb von Prozeduren (Kapitel 18): nur das Endergebnis (das Ergebnis des letzten Befehls) wird ausgegeben, die Ausgabe der Zwischenergebnisse wird unterdruckt. Man hat aber durchaus die Moglichkeit, sich die Zwischenschritte anzeigen zu lassen oder die Art der Ausgabe zu andern.

Ein wichtiges Element einer Programmiersprache sind so genannte Verzweigungen, wo je nach Wert oder Bedeutung von Variablen unterschiedliche Befehle auszufuhren sind. MuPAD stellt dazu als einfachste Variante ein if-Konstrukt zur Verfugung: >> for i from 2 to 4 do if isprime(i) then print(expr2text(i).“ ist eine Primzahl”) else print(expr2text(i).“ ist keine Primzahl”) end_if end_for: “2 ist eine Primzahl” “3 ist eine Primzahl” “4 ist keine Primzahl”

MuPAD does not display all computed results on the screen. Typical examples are the commands within for loops (Chapter 16) or procedures (Chapter 18): only the final result (i.e., the result of the last command) is printed and the output of intermediate results is suppressed. Nevertheless, you can let MuPAD print intermediate results or change the output format.

Computer algebra systems such as MuPAD are often used interactively. For example, you can enter an instruction to multiply two numbers and wait until MuPAD computes the result and prints it on the screen.