Research Article  Open Access
The Solution of Embedding Problems in the Framework of GAPs with Applications on Nonlinear PDEs
Abstract
The paper presents a special class of embedding problems whoes solutions are important for the explicit solution of nonlinear partial differential equations. It is shown that these embedding problems are solvable and explicit solutions are given. Not only are the solutions new but also the mathematical framework of their construction which is defined by a nonstandard function theory built over nonstandard algebraical structures, denoted as “GAPs”. These GAPs must not be neither associative nor division algebras, but the corresponding function theories built over them preserve the most important symmetries from the classical complex function theory in a generalized form: a generalization of the CauchyRiemannian differential equations exists as well as a generalization of the classical Cauchy Integral Theorem.
1. Introduction
Except some small areas, at time the nonlinear world is inaccessible for analytic methods, that is, for methods without using numerical algorithms, we only know some explicit solutions of a few nonlinear differential equations (see e.g., [1–13]), and we know precious few about the “embedding’’ of nonlinear structures without any symmetry in higher dimensional structures with symmetry ([14–16], etc.).
This fact legitimates the creation of a new mathematical theory, presented in [17], which allows an “analytic access’’ on wider regions of nonlinearity. It was shown that this theory succeeds in solving nonlinear partial differential equations (concretely the Einstein equations from General Relativity) if the problem defining parameters (concretely the stress energy tensor) have a certain symmetry which can be seen as a wide generalization of the classical symmetry of holomorphy. This new symmetry—in the following denoted as “symmetry’’ or “GAPsymmetry’’ allows the explicit solution of Einstein equations and also of a broad variety of other partial differential equations.
This leads to the following question as considered here: is it possible to embed an arbitrary dimensional nonlinear partial differential equation (which only symmetry is given by smoothness of all coefficient functions) in an dimensional nonlinear partial differential equation with symmetry, . If this holds, we can solve the given dimensional system by embedding this system into the dimensional system, which is solvable on the base of symmetry. Since embedding of an equation means embedding of the coefficient functions (which usually are given by vector fields (tensor fields)), our question above boils down to the following question:
is it possible to embed an arbitrary dimensional smooth vector field (tensor field) into a “symmetrical’’ dimensional tensor field,
This question will be answered positively here.
For a better understanding of the practical meaning of this result, some analogies to the classical situation of holomorphy are given (see the following section and also more details in [17]).
(i)The simplest GAP is defined by the twodimensional field of the complex numbers. In this case, the symmetry is defined by the classical symmetry of holomorphy. It is a wellknown fact, that an arbitrary smooth realvalued function , , can be “embedded’’ in a complex valued function , in the sense that the real part of for values in the region is identically with . Our Embedding Theorem is a generalization of this classical result, by generalizing the twodimensional field to an dimensional GAP (generally a nonassociative, noncommtative, nondivision algebra).(ii)The classical symmetry of holomorphy leads immediately to related (partly equivalent) symmetries: the wellknown CauchyRiemannian differential equations and the wellknown Cauchy Integral Theorem. The GAPformalism generalizes these symmetries: it is shown that in GAPs, analogies of the CauchyRiemannian differential equations exist (in the following denoted as Pseudo CauchyRiemannian differential equations or shorter: “PCRE’’) as well as analogies of the Cauchy Integral Theorem (in the following denoted as Pseudo Cauchy Integral Theorem or shorter: “PCIT’’).(iii)The classical symmetries: CauchyRiemannian differential equations and Cauchy’s Integral Theorem lead to a solution theory of the twodimensional linear Laplace equation. In a similar way, their analogies—the PCRE and the PCIT—lead to a solution theory of dimensional nonlinear partial differential equations.After these considerations, some remarks to the structure of the paper are presented as follows.
(1)In Section 2.1 the concept of “analyticity’’ is introduced: we will define the socalled analytic tensor fields by demanding that these fields allow a representation as generalized power series in very general algebras.(2)In Section 2.2 we will specialize these algebras to the socalled “PAkstructures’’ by introducing symmetries which can be seen as wide generalizations/modifications of the classical associativity symmetry. In the function theories built up over special PAkstructures (socalled “pseudorings’’), there exist generalizations of the wellknown CauchyRiemannian differential equations as well as generalizations of the wellknown Cauchy Integral Theorem.(3)In Section 2.3 a further generalization is given by generalizing the PAkstructures to the socalled “GAPs.’’ It is shown that also in this wide framework the main results of Section 2.2 hold.(4)Section 2.4 solves the problem of calculating GAPs explicitly.(5)Section 3 presents applications of the GAPformalism on famous partial differential equations from Theoretical Physics: new explicit solutions of Einstein equations from General Relativity and NavierStokes equations are given. The request of a further generalization of these results (which would be important from a physical point of view) shows the necessity of “embedding low dimensional unsymmetrical structures into higher dimensional symmetrical structures,’’ that is, the necessity of an Embedding formalism, which is developed in the following Section 4.(6)Section 4 presents the Embedding Theorem: it is shown that an arbitrary dimensional smooth vector field always can be embedded into a special dimensional smooth vector field ) restricted by special GAPsymmetries. In other words, The world of GAPs is wide enough to allow embedding for rather general (smooth) structures. It is remarkable that the Embedding Theorem not only shows the possibility of Embedding but also endows the tools for practical applications. A simple example is given.Finally some remarks to the style of the presented paper, which is given by accentuation of constructiveness, are presented like it is demanded by the concrete problems: not only are statements to existence and uniqueness desired and given but furthermore the explicit construction of a wide variety of solutions. For this reasons, this paper is not written for pure mathematicians but for physicists and applied mathematicians.
2. Elements of GAPTheory
In this chapter the most important mathematical concepts will be presented as necessary for our solution method. Some of these concepts have been presented in [17], where the interested reader can find the proofs as missing here. The larger part of concepts is new; the corresponding propositions of course all will be proved in the following.
2.1. The Concept of Analyticity in General Algebras
We start by remembering on same elementary terms of algebra theory: let denote an arbitrary vector space built over the field of real numbers, and an arbitrary binary distributive operation. The algebraic structure defined by and will be denoted here as . The operation on vectors can be described by means of an arbitrary base , , dim according to , where denote the structure constants of the algebra with respect to the base . Instead of ,dim, we will write in the future The associated baseindependent object to the structure constants , is the structure constant tensor of the algebra: , with , where denotes the dual space of and the dual base of , defined by . By means of the product can be developed as follows: In the future we will denote the quadratic matrix by and the quadratic matrix by or shorter in formal denotation and . If for a fixed element , then equation has a nonzero solution and if for a fixed element , then equation has a nonzero solution . We remember that in the special case of a ring operation a nonzero element with another nonzero element, is called a zero divisor, and a ring without zero divisors is called a division ring. In this work we will use the denotation “zero divisor’’ also for more general algebras in the sense above.
If has a left unit element (shorter: “left unit”), we denote this element as defined by If has a right unit element (shorter: “right unit”), we denote this element as defined by If left unit and right unit are identical we will write The existence of a right unit or a left unit allows to define the following generalizations of the inverse element conception: Here the symbol denotes the ight inverse of in respect of the right unit (see the superscripts , denotes the eft inverse of in respect of the ight unit (see the superscripts and so forth. The existence of a unit element allows the definition of an inverse element for : . The extension in will help us to distinguish an inverse vector from the inverse of a quadratic matrix .
The classical algebra symmetries (anti)commutativity, associativity, Jacobisymmetry, existence of a left unit/right unit can be described by means of the structure constants as shown in Table 1.

The symmetries follow immediately from representing the vectors in respect of a vector base and by calculating the base products , and so forth, by the structure constants.
Let ( times) denote the repeated tensor product of the vector space and ( times) the repeated tensor product of the dual space . With , we denote an arbitrary base, and with the corresponding dual base. Then a tensor (also called as “tensor of type ’’) has a representation with and a tensor (also called as “tensor of type ’’) has a representation with .
After these wellknown assumptions, we will start with GAPTheory by introducing two new product operations and as follows.
Definition 2.1. Given an arbitrary algebra with structure constant tensor Then theassociatedproduct of order : is defined by and thedual associated product of order q: by
It is necessary to write instead of , because later we will deal with different product operations , and with the corresponding associated operations . In the case , it holds by definition that ; in the case , we write also for Due to the linearity of the vector spaces and the products of the base vectors uniquely define products of the form and for arbitrary tensors , , and . For this we look at the following results:
Proposition 2.2. The components of products , , , and are given by The following algebraical symmetries hold:Relation (2.32a) shows a very simple characterization of the derivative by the Liederivative, but it is only possible in a pseudoring with a right unit.
After the partial derivatives, covariant derivatives and Liederivatives we will study the exterior derivatives of analytic forms. Let us denote the set of all in region smooth forms with and the subset of all in analytical forms with
Proposition 2.15. Given a pseudoring of first type, an arbitrary nontrivial form and arbitrary vector fields not constant (“Not constant” means of course not constant in the considered coordinate system), then it holds that
This proposition shows that some fundamental symmetries with practical importance only exist in commutative pseudorings: the product of analytical vector fields is analytic only for commutative pseudorings, and an analogue result holds for the product. Also the inverse element of a analytical vector field will be analytical only in a commutative pseudoring. Let us sum up the relations for the classical derivatives as follows:
PseudoCauchyRiemann equations:
Covariant derivatives:
Liederivative:
Exterior derivative: with pseudorings of first type, We sum up in the following paragraph.
Our aim was to study analytical tensor fields in the framework of pseudorings, which have been introduced here as a fundamental concept. It was shown that pseudoringsymmetry is an algebraic symmetry which also appears in the world of differential geometry, moreover: the geometrical world prefers the pseudoringsymmetry against the classical symmetry of associativity. For this the study of pseudorings is legitimized, and so the study of tensor fields built over pseudorings. In particular we have studied the partial derivatives of analytical tensor fields, their covariant derivatives, their Liederivatives and their exterior derivatives, all this in the frame of pseudoringsymmetry. It was shown, that all these fundamental derivatives can be written in terms of the socalled derivative, which has been introduced here as a fundamental concept, following from the symmetry of analyticity.
2.3. Generalizations. The World of GAPs
Until now we have dealt with PAkstructures and pseudorings , that is, with algebraic structures defined by maximal two product operations For some later applications, this framework will be too small and shall be generalized here (The most applications on mathematical physics only need some small parts of this section: the concept of a GAP, of a GAP characteristic, and of GAP Exponentials). Such an algebra generalization will lead us to the concept of GAPs (PAkstructures and pseudorings will be shown as the most simple GAPstructures) and will allow us to generalize the function theoretical concepts of analyticity, derivative, integration, and so forth, from the section above by replacing the underlying PAkstructures (pseudorings) by GAPs.
The aim of this section is to generalize the concepts of PAkstructures, analyticity, derivative, integration, Pseudo Cauchy Riemann equations, and so forth.
Let us denote as an algebraic structure with the operations an arbitrary natural number which also might be infinite. To generalize the concept of analyticity we have to generalize the concept of the power function from Section 2.1. For this we introduce the maps and as follows:
arbitrary constants, After these assumption we can introduce the concept of “chainanalyticity”.
Definition 2.16. Given an arbitrary algebraic structure and arbitrary tensors , we denote and , respectively, as chainanalyticin region if there exists an element , which allows the following representations of , respectively, : a constant of type a constant of type . The set of all in region chainanalytical tensors and for fixed is denoted as and
Consider that in the series above an infinite value of is allowed and an infinite set of operations is allowed to contain an infinite subset of identical operations. In the case (=finite or infinite), we will denote the structures above as analytical as in the earlier case of a single product operation. The denotation “chainanalytical” comes from the fact, that the series above are built by operations like the “elements of a chain”. We see, that for different operations the structure of chainanalytical tensor fields is much more general than the structure of simple analytical tensor fields. Now we will generalize the concept of PAkstructures for vector spaces with different dimensions.
Now I will present a concept, which allows a short overview about all symmetries—PAksymmetry as well as standard symmetries—of a given algebraic structure. For this we assign the algebraic structure a matrix denoted as the symmetry characteristic of the algebra as follows:
(“0” means: “no PAk symmetry is assumed” and not: “no PAksymmetry is allowed”. With other words: In the case “” it is possible that a PAksymmetry exists, but it is not ensured.)
Definition 2.17. An algebraic structure is called a GAP of order M, if the symmetry characteristic includes at least one value or In this case is denoted as the GAP characteristic of the GAP .
In other words a GAP is an algebraic structure which has at least one PA1symmetry or one PA2 symmetry. For further applications it will be advantageable, to generalize the GAPcharacteristic by giving an overview not only about the PAksymmetries but also about the ”standard symmetries” of all algebraic structures , that is, of its possible (anti)commutativity, Jacobisymmetry, associativity, existence of a right unit existence of a left unit and so forth. Since these symmetries all are defined for a single algebra operation, they only will appear in the main diagonal of Concretely we introduce the following denotations as subscripts of the elements :
As an example, we consider an algebraic structure with the following symmetry characteristic:
which means that the algebra has a right unit, the algebra is anticommutative and satisfies the Jacobisymmetry (i.e., is a Liealgebra), the algebra is commutative, the algebraic structure is a PA1structure, and the algebraic structure is a PA2structure. Per definition is a GAP.
Now some special types of GAPs will be introduced which allow a wide generalization of the functional theoretical concepts of analyticity, derivative, Pseudo Cauchy Riemann equations, integration, and so forth.
Definition 2.18. A PA1chain is a GAP with A PA1chain is denoted as a closed PA1chain, if additionally to the above it holds A closed PA1chain of Lietypeis defined as a closed PA1chain of order 3 with the additional restriction
The concept of a PA1chain allows a wide generalization of the earlier concept of the derivation, which will be presented now.
Definition 2.19. Given a PA1chain and chainanalytic tensors then the following series expansions uniquely define elements and which will be denoted as the derivative of tensor and the derivative of tensor .
We see that the tensors and do not contain the operation any longer and so we cannot define or Only and make sense, and also and and so forth. Thus we will define the operators
as the chain derivatives of order k. In the special case we write simplifying instead of From the definitions above it follows for :
and we see that
Theorem 2.20. Given a PA1chain and a chainanalytic tensor Then the following symmetries hold: (In the case the left side must be interpreted as ). An inverse relation exists, if the algebra has a right unit :
Theorem 2.21. Given a PA1chain and a chainanalytic tensor Then the following symmetries hold: (In the case the left side must be interpreted as ). An inverse relation exists, if the algebra has a right unit :
The proofs of Theorems 2.20 and 2.21 run the same lines as the proofs of Theorems 2.10 and 2.12. Relations (2.45) and (2.47) will be denoted as Pseudo Cauchy Riemann equations (PCRE) as in the simple case of a single vector space and a single product operation. (The expression must be interpreted as and as Now we will generalize the former concept of integration.
Definition 2.22. Given an arbitrary algebra and chainanalytic tensors and further let an algebra with the property Then the series with and arbitrary constants, are called the integrals or the antiderivatives of the tensors and .
The denotation is well chosen because it is and We see, that
Proposition 2.23. If is a PA1chain, then the integrals and can be expressed by the classical integral conception as follows:
Theorem 2.24. a PA1chain and a smooth closed curve in region . Then it is
This Theorem generalizes the wellknown classical Cauchy Integral Theorem for GAPs and is denoted as Pseudo Cauchy Integral Theorem (PCIT). For the next step we remember on the classical denotation for a scalar differentiable function or formally It shall be shown now, that for the derivative of vectorvalued functions a similar result holds:
Proposition 2.25. Given a PA1chain and vectors∈ Then it holds
Proof. We start from the relation (2.52), with and Now we define from which follows as it is assumed above. Inserting and in (2.52) we obtain and analog:
We sum up in the following paragraph
In this section, the main earlier results have been generalized,along with the algebraic structures as well as the tensor fields built over these structures. In particular the PAkstructures have been generalized by the concept of the socalled “GAPs”. We have investigated special GAPs like PA1 chains and their specializations (closed PA1chains and closed PA1chains of Lietype), and have shown that the main results of the classical complex function theory (Cauchy Riemannian differential equations, Cauchy Integral Theorem) also hold in PA1chains.
2.4. The Explicit Construction of GAPs
The aim of this section is the explicit construction of a wide variety of GAPs.
In other words, we have to construct structure constant tensors of pseudorings, PAkstructures, and PA1chains explicitly. This problem makes sense, because without knowledge of concrete GAPs, the results of the sections above would be not applicable in practice. In the following, we introduce the maps , ( the set of integers, the set of natural numbers) with by
where denotes the smallest possible value, defined uniquely by the variable and the conditions above (For example, let us consider the case Then is uniquely defined by since it is only for Then it is ). From this definition it follows immediately and
Proposition 2.26. Given the constants andThen PAkstructures are given by the following structure constant tensors:
Proof. We show that the algebra (2.56a) satisfies the PA1condition Writing all sums explicitely we get because only delivers a nontrivial value for the index value and this index must be inserted in the term Since it is the right side above is symmetrical in indices and Thus the left side is symmetrical in indices and , that is, the PA1symmetry holds and statement (2.56a) has been proved. Now we have to show that algebra (2.56b) satisfies the PA2condition = = = = . The right side is symmetrical in indices and , and so the left side is, that is, the PA2symmetry holds and statement (2.56b) has been proved.
Proposition 2.27. Let be arbitrary. Then the structure constant tensor defines a pseudoring of first type . In the case , a unit element exists, given by and furthermore it hold s that
Because the PAkstructures above are defined by Kroneckersymbols, we will denote them as PAkstructures of Kroneckertype or shorter as Kronecker PAkstructures. According to the property (2.58a) a commutative Kronecker pseudoring is not a division algebra. It is easy to show, that this property also holds for noncommutative Kronecker pseudorings. Also the property (2.58a) can be generalized for noncommutative Kronecker pseudorings, where must be replaced by and so forth. (see the definition of in (2.2)). We see that a Kronecker algebra has a very simple structure, because the values of the structure constants are given by only two reals: 0 and 1. Now another type of PAkstructures will be presented which is more subtle than the Kronecker type.
Proposition 2.28. Given an algebra as follows: with , arbitrary nonzero constants. Then is a PAstructure if and only if has a right unit if and only if
The PAkstructures presented above are built by combinations of rank 2tensors and rank 1tensors. Therefore we will denote these structures as PAkstructures of splitting type or shorter as splitting PAkstructures. From this follows immediately the structure of splitting pseudorings. It is easy to show, that Kronecker PAkstructures generally cannot be transformed onto splitting PAkstructures by basistransformations in that is, these both structures are really different.
Splitting pseudorings have a large advantage to Kronecker pseudorings because it allow the explicit calculation of and so forth
Proposition 2.29. Given the structure constant tensor only restricted by Then the following statements hold:
Statement 1. The algebradefines an dimensional pseudoring of first type with right unit
Statement 2. In this algebra a vector has a right invers element