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Differential equations over octonions

Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is used. Such technique is applied to linear and non-linear partial differential equations in real variables. Possible areas of applications of these results are outlined.


Introduction.
Differential equations are very important not only for mathematics, but also for their many-sided applications in different sciences including physics, mechanics, other natural sciences, techniques, economics, etc. [2,7,13,34,16,27,39]. Mainly differential equations are considered over fields such as real, complex, or with non-archimedean norms. Recently they are also begun to be studied over Clifford algebras [9,10,11].
This article is devoted to differential equations of octonion variables for octonion valued functions. It opens new possibilities for finding solutions or integrating differential equations not only ordinary of real of complex variables, but also some types of partial differential equations including nonlinear and their systems. For this purpose previous results of the author on non-commutative line integration over Cayley-Dickson algebras are used [22,21,23]. In these works the super-differentiable functions of the Cayley-Dickson variable z were investigated as well. Such functions were also called z-holomorphic or z-differentiable, or shortly holomorphic or differentiable, when the Cayley-Dickson algebra is outlined.
As it is well-known, quaternions were first introduced by W.R. Hamilton in the second half of the 19-th century. Hamilton had planned to use them for problems of mechanics and mathematics [12,33]. Later on Cayley had developed their generalization known as the octonion algebra. Then Dickson had investigated their generalizations known now as the Cayley-Dickson algebras [1,4,14].
Octonions and quaternions are widely used in physics, but mainly algebraically. Maxwell had utilized quaternions to derive his equations of electrodynamics, but then he had rewritten them in real coordinates. Already Yang and Mills had used them in quantum field theory, but theory of functions over octonions and quaternions in their times was not sufficiently developed to satisfy their needs. They have formulated the problem of developing analysis over octonions and quaternions [8]. This is natural, because quantum fields are frequently non-abelian [36].
Dirac had used complexified bi-quaternions to solve spin problems of quantum mechanics. He had solved the Klein-Gordon hyperbolic partial differential equation with constant coefficients. The technique of this article developed below permits to integrate also partial differential equations of elliptic and hyperbolic types with variable coefficients. The latter equations are important in many applications, for example, in physics, mechanics, quantum mechanics, in economics for describing processes in the exchange stock market as well.
Each Cayley-Dickson algebra A r+1 is obtained from the preceding A r with the help of the so called doubling procedure [1,14,17]. This gives the family of embedded algebras: A r → A r+1 → .... For a unification of notation it is convenient to put: A 0 = R for the real field, A 1 = C for the complex field, A 2 = H denotes the quaternion skew field, A 3 = O is the octonion algebra, A 4 denotes the sedenion algebra. The quaternion skew field is associative, but non-commutative. The octonion algebra is the alternative division algebra with the multiplicative norm. The sedenion algebra and Cayley-Dickson algebras of higher order r ≥ 4 are not division algebras and have not any non-trivial multiplicative norm. Each equation of the form ax = b with non-zero octonion a and any octonion b can be resolved in the octonion algebra: x = a −1 b, but it may be non-resolvable in Cayley-Dickson algebras of higher order r ≥ 4 because of divisors of zero. Therefore, in this article differential equations are considered with octonion or quaternion variables for octonion or quaternion valued functions. If a given differential equation permits a generalization for Cayley-Dickson algebras of higher order it will be indicated.
We recall the doubling procedure for the Cayley-Dickson algebra A r+1 from A r , because it is frequently used. Each Cayley-Dickson number z ∈ A r+1 is written in the form z = ξ + ηl, where l 2 = −1, l / ∈ A r , ξ, η ∈ A r . The addition of such numbers is componentwise. The conjugate of any Cayley-Dickson number z is given by the formula: (1) z * := ξ * − ηl.
The technique developed below permits to integrate ordinary differential equations of different types and their systems of ordinary differential equations up to 8 equations. An octonion variable is 8 dimensional over the real field, so (super-)differential equations over octonions encompass some partial differential equations of up to 8 real variables. This gives effective ways for their integration with the help of a non-commutative non-associative line integral over octonions. The quaternion skew field is the subalgebra of the octonion algebra and the line integral over quaternions is already associative.
Section 2 of this article contains the theorem about an existence of solutions of ordinary and partial differential equations over octonions.
Differential equations over octonions treated below in Section 3 include R-linear octonion additive differential operators, some types of non-linear equations including that of Bernoulli. Moreover, differential equations in complete differentials and their integration multipliers are studied. Differential equations of n-th order are investigated as well in Section 3. Examples illustrating this new technique for ordinary and some types of partial differential equations are given. It is shown, that the new technique over octonions enlarges possibilities of the integration of ordinary and partial linear or nonlinear differential equations in comparison with the real and complex fields.
Main results of this paper are obtained for the first time.
2 Existence of solutions. 1. Remark. Suppose there is given a system of partial differential equations: (1) ∂ n j u j /∂t n j = F j (t, x 1 , ..., x n , u 1 , ..., u N , ..., ∂ k u l /∂t k 0 ∂x k 1 1 ...∂x k n n , ...), where j, l = 1, ..., N ; k 0 +k 1 +...+k n = k ≤ n l ; k 0 < n l , N, n ∈ N, 0 ≤ k ∈ Z, (x 1 , ..., x n ) ∈ U , x j ∈ K for each j, U is a domain in K n , where K = H is either the quaternion skew field or K = O is the octonion algebra. When a number t is real the F j are K-valued functions or functionals. In the case t ∈ K each F j is a mapping with values in the space L q (K ⊗n j ; K) of all R n j -poly-homogeneous K-additive operators with values in K. Each partial derivative ∂ k u l /∂t k 0 ∂x k 1 1 ...∂x k n n is the mapping with values in L q (K ⊗s ; K), where s = k − k 0 for t ∈ R and s = k for t ∈ K, K = H or K = O.
For t ∈ R there are given initial conditions:  We search a solution of the problem (1,2) in a neighborhood V of the given point (t 0 , x 0 1 , ..., x 0 n ) either in R × U or in K × U . For our convenience we introduce the notation: (3). Suppose that all functions F j are analytic by all variables and holomorphic by x 1 , ..., x n ; u 1 , ..., u N , also holomorphic by t, when t ∈ K, in a neighborhood Y of the point (t 0 , x 0 1 , ..., x 0 n , ..., φ 0 j,k 0 ,k 1 ,...,k n , ...), where analyticity by operators A l,k := ∂ k u l /∂t k 0 ∂x k 1 1 ...∂x k n n is defined relative to their suitable compositions and the convergence of a series of F j by {A l,k : l, k} is supposed to be relative to the operator norm, where has a holomorphic solution either by all variables (t, x 1 , ..., x n ) in some neighborhood of (t 0 , x 0 1 , ..., x 0 n ), when t ∈ K, or holomorphic by the variables (x 1 , ..., x n ) and (locally) analytic by t, when t ∈ R. This solution is unique in such class of functions.
Proof. To prove this theorem we introduce new functions, when n j > 1, so that v j,p := ∂ p u j /∂t p for each p = 1, ..., n j − 1, w l,k := ∂ k u l /∂t k 0 ∂x k 1 1 ...∂x k n n for each k 0 + k 1 + ... + k n = k ≤ n l and k 0 < n l . Then ∂v j,p /∂t = v j,p+1 for each p < n j and ∂v j,n j −1 /∂t = F j , also w j,k = ∂ k v l,k 0 /∂x k 1 1 ...∂x kn n for each (k 0 , ..., k n ) and l, where F j = F j (t, x 1 , ..., x n , u 1 , ..., u N , ..., w j,k , ...). The initial conditions now become: for each s = 1, ..., n j − 1 and each (x 1 , ..., x n ) ∈ U and every t ∈ K with Re(t) = Re(t 0 ) in the considered domain. We can take as a neighborhood of t 0 a ball in K with center at t 0 of a sufficiently small positive radius. In the real case of the variable t it would be an interval in R. For t ∈ R we automatically get initial conditions from these general.
.., x n ) for each (x 1 , ..., x n ) ∈ U and each t for the given domain, then Y due to analyticity of functions.
Therefore, this procedure of the introduction of new functions v j,s reduces the problem 1(1 − 3) to the problem, when n j = 1 for each j. It is possible to simplify further the problem, making substitutions: For new functions w j and w j,s the problem has the same form, but with zero initial conditions. Thus without loss of generality consider zero initial conditions and denote w j and w j,s again by u j and v j,s .
Consider at first the case when each F j is R-homogeneous and K-additive by each u j and each v j,s , where n j = 1 for each j. Without loss of generality consider t ∈ K, since when t ∈ R, then due to the local analyticity by t it is possible to take the extension of each F j to F j ∈ L q on a neighborhood of t 0 in K. In view of Proposition 2.18 and Remark 2.18.1 in [21,22] and Theorem 3.2.14 [25] if a solution of the problem exists in the considered class of holomorphic functions u j , j = 1, ..., N , then it is unique.
Consider now series of F j , u j , v j,s for each j and s by all their variables (t − t 0 , x 1 − x 0 1 , ..., x n − x 0 n ) in a neighborhood of (t 0 , x 0 1 , ..., x 0 n ). Expansion coefficients of u j and v j,s can be evaluated from the system of equations by induction in the order of increasing powers, since u j (t, x)| Re(t)=Re(t 0 ) = 0 and v j,s (t, x)| Re(t)=Re(t 0 ) = 0 for each j, s and each x = (x 1 , ..., x n ) ∈ U and every t with the prescribed real part Re(t) = Re(t 0 ) in the considered domain. Now prove the local convergence of the series.
It is possible to make the shift of variables such that one can consider Then there exists a constant S > 0 such that | k=(k 0 ,...,kn) {(a k , (t, x) k )} q(2|k|) | ≤ S for each k, |k| := k 0 + ... + k n . Therefore, this implies the inequality: for each β 0 ∈ K, ..., β n ∈ K with |β 0 | = 1,...,|β n | = 1 and each integers 0 ≤ k 0 , ..., k n < ∞, where vectors of associators q( * ) indicating on orders of brackets or multiplications are important only over O, a k = a k (g) (see Formulas 2.13(ii−iv) in [25]). Take the function Q(t, x) := S(1−t/|a 0 |) −1 ... (1− x n /|a n |) −1 . Then Q(t, x) is characterized by the condition: for each k and each β 0 , ..., β n in the algebra K with the natural restrictions |β 0 | = 1,...,|β n | = 1. The multiplications z → β p j z and z → zβ p j are pseudoconformal mappings from K onto K for each p ∈ N. Such holomorphic function Q is called the majorant of the holomorphic function g. In the case of variables A j,s ∈ L q consider majorants in the class of analytic functions by A j,s and their suitable compositions, where we choose b j,s ∈ K such that Let the problem I be the initial problem formulated in §1 and let the problem II be the problem, in which each participating function is substituted by its majorant. Let U j , V j,s denotes the solution of problem II. Then for each β 0 , ..., β n in K with |β 0 | = 1,...,|β n | = 1 and each j and s due to the initial conditions, where V j,0 := U j and v j,0 := u j . In the case k 0 > 0 coefficients a k (v j,s ) or a k (V j,s ) are obtained from coefficients a m (v j,s ) or a m (V j,s ) respectively with the help of addition and multiplication with numbers subjected to the conditions 0 ≤ m 0 < k 0 . Therefore, by induction these inequalities are true for each k. Thus if problem II has a solution, the problem I also has a solution.
Choose constants S > 0 and a > 0 such that the function S[1 − (t/α + x 1 + ... + x n )/a] −1 for some 0 < α < 1 would be the majorant for all terms of the system besides free terms, as well as take T [1 − (t/α + x 1 + ... + x n )/a] −1 as majorant for free terms, since such majorants with S = S j , a = a j , T = T j can be taken for each F j . Hence we can take S = max j=1,...,n S j , T := max j=1,...,n T j , a = min j=1,...,n a j , where α we shall choose below. For it problem II can be taken as: where C = T /S, while I ∈ L 1 is the constant unit operator. Seek a solution U j (t, x) = U (z), where z = t/α + x 1 + ... + x n . Then we get the equation: Choose a number α > 0 sufficiently large such that 1/α−N nS(1−z/a) −1 = 0 in a neighborhood of z = 0. Then the problem is: due to the alternativity of K. The integration of the latter equation gives the z-differentiable (holomorphic) solution due to Theorem 3.2.14 [25] and Remark 2.18.1 [21,22], since the mapping g is z-differentiable. Now estimate its expansion coefficients. For this consider an algebraic embedding θ of the complex field C into K. The restriction of U (z) on this copy of C has the form: also has nonnegative expansion coefficients by z and inevitably exp(s(z)) − 1 = s(z) + s 2 (z)/2 + ... and thus U (z) also have nonnegative expansion coefficients by z ∈ θ(C). But the algebra K is alternative and hence power associative, moreover, θ θ(C) = K. Thus expansion coefficients of U (x 1 + x 2 + ... + x n ) by degrees of x 1 ,...,x n are nonnegative, consequently, U (0, x 1 , ..., x n ) is the majorant of zero. Therefore, functions U j (t, x 1 , ..., x n ) = U (t/α+x 1 +...+x n ) compose the solution of problem II.
The problem I solved above can be reformulated in such a fashion, that there exists an inverse operator K −1 to the differential operator together with initial conditions which compose the operator equation Ku = g in the space of holomorphic vector-functions u, g.
Consider now the case, when F j are not R-homogeneous K-additive by u j or v j,s for some j, s. Denote the operator corresponding to this problem by M and the problem is M u = g in the space of holomorphic vector-functions.
Let X be a K-vector space.
Since F j are holomorphic by (t, x) and locally analytic by u j , v j,s for each j, s, then we can take power series expansions by these variables and consider a sufficiently small K-convex neighborhood of the point of zero initial conditions, where corrections of the second and higher order terms in u j , v j,s are much smaller.
This neighborhood W can be chosen such that The inverse of a holomorphic function is holomorphic (see also Theorem 2). That is functions defining M −1 are holomorphic by (x, u) also by t, when t ∈ K, moreover, they are locally analytic by others variables. In view of the proof above we can construct a solution u of Ku(t, x) = g for each (t, x) with |(t, x)| < R for some 0 < R < ∞ such that u ∈ W/2. Therefore, we get a solution of the problem M u = g also for each (t, x) in a sufficiently small neighborhood of 0. In view of the proof above this leads to the existence of a solution of the general problem.
3. Note. The problem 1 is the non-commutative holomorphic analog of Cauchy's problem, while Theorem 2 is the non-commutative analog of Kovalevsky's theorem (see its commutative classical form in §2 [30]). Instead of initial conditions at each t with Re(t) = Re(t 0 ) it is possible for t in a 2 r − 1-connected domain P in either the quaternion skew field H or in the octonion algebra O consider initial conditions for each t ∈ ∂P , when P in this section instead of U there satisfies conditions of Remark 2.1 below.
The proof is quite analogous in this case. This follows also by applying the non-commutative analog of the Riemann mapping Theorem 2.43 [25] establishing A r -holomorphic diffeomorphism between different domains in A r under definite rather mild conditions. For convenience of the readers we give below the detailed proof of the chain rule over the Cayley-Dickson algebra A r (though it is contained in [21,22] as well).

4.
Notation. If f : U → A r is either z-differentiable orz-differentiable at a ∈ U or on U , then we can write also Dz instead of ∂z and D z instead of ∂ z at a ∈ U or on U respectively in situations, when it can not cause a confusion, where U is open in A n r . 5. Proposition. Let g : U → A m r , r ≥ 2, and f : W → A n r be two differentiable functions on U and W respectively such that where f and g are simultaneously either (z,z), or z, orz-differentiable. Then the composite function f •g(z) := f (g(z)) is differentiable on U and for each z ∈ U and each h ∈ A k r , and hence f • g is of the same type of differentiability as f and g.
Proof. Theorems 2.11, 2.15, 2.16, 3.10 and Corollary 2.13 [21,22]  (that is in the z-representation). Indeed, the composition of two locally zanalytic functions f • g with such domains is the locally analytic function.
Since g is differentiable, the function g is continuous and g −1 (W ) is open in A k r . In view of Proposition 3 above if f and g are simultaneously either zdifferentiable orz-differentiable, then either ∂zf = 0 and ∂zg = 0 or ∂ z f = 0 and ∂ z g = 0 correspondingly on their domains.
Consider the increment of the composite function where η = g(z + h) − g(z), g(z + h) − g(z) = (Dg(z)).h + g (h)|h| (see §2.2 [21,22]). Since the derivative Df is A m r -additive and R-homogeneous (and continuous) operator on A m r , we have Suppose now that there are phrases corresponding to f n and g n denoted by µ n and ν n such that f n and g n uniformly converge to f and g respectively on each bounded canonical closed subset in W and U from a family W or U respectively, where W and U are coverings of W and U correspondingly. While D z µ n and D z ν n are fundamental sequences uniformly on each bounded canonical closed subset P ∈ W and Q ∈ U correspondingly relative to the operator norm (see Definitions in §2 [21,22]). Then the sequence µ n • ν n converges on each bounded canonical closed subset Q 1 such that where P ∈ W and Q ∈ U in U , when n tends to the infinity. Moreover, D y µ n (y,ỹ).(D z ν n (z,z))| y=ν n (z,z) and Dỹµ n (y,ỹ).(D zνn (z,z))| y=ν n (z,z) are the fundamental sequences of operators on each bounded canonical closed sub- where P ∈ W and Q ∈ U . The family Q := {Q 1 } specified above evidently is the covering of V .
Since the derivation operator D z by z is R-homogeneous and A r -additive, it is sufficient to verify this in the case D z [η • ψ(z,z)] locally in balls, where both series uniformly converge. Here the phrase is written as: j, a k,1 , ..., a k,p ∈ A r are constants, z 1 = z, z 2 =z, z is the Cayley-Dickson variable, a vector q indicates on an order of the multiplication, We have the identities where 1h := h and1h :=h for each h ∈ A r . Using shifts z → z − ζ we can consider that the series are decomposed around a zero point. If a series η is uniformly converging on a canonical closed subset Q 1 as above, then is also uniformly converging on Q 1 in accordance with our supposition about convergence of phrases and their derivatives.
In suitable Q 1 we deduce from Formula (4) that On the other hand, we have due to the Leibnitz rule, hence due to Formulas (3). Thus Formulas (5, 6) imply that Quite analogously or using the conjugation one deduces that Particularly, we get Combining Formulas (7,8) and taking into account the proof given above and applying Proposition 2.3 [21,22] we infer the chain rule, when both η and ψ are either the z orz or (z,z)-differentiable, that is in all three considered cases.
Proof. In accordance with §5 it is sufficient to prove Formula (1) for phrases η(z) and ψ(z), since each z-differentiable function is infinite z-differentiable (see §2 [21,22]). To derive Formula (1) the mathematical induction will be used below. For n = 1 Formula (1) coincides with that of Proposition 4. In accordance with the Leibnitz rule over the Cayley-Dickson algebra A r , 2 ≤ r, [21,22] the Formula is as usually symmetric relative to all transpositions of h 1 , ..., h n (see also [21,22]). Therefore, we deduce from the Leibnitz rule, that Suppose now that Formula (1) for d k η(ψ(z))/dz k is proved for each k = 1, ..., n. It is sufficient according to the continuation of the induction to demonstrate it for k = n + 1. From (1, 2) we infer the equality: for all Cayley-Dickson numbers h 1 , ..., h n+1 . This is the same expression as in Formula (1) for n + 1 instead of n after gathering together terms η (j) with the same j. .., h n = 1 when f (z), g(z) ∈ C for each z ∈ U C the deduced formula coincides with that of 5.2(1) in [32].
In this article some elementary facts about analytic functions z n , z 1/n , exp(z) and Ln(z) of the Cayley-Dickson variables are used. They were considered in details in previous works [22,21,25]. Recall that the exponential function is defined by the power series exp(z) := 1 + ∞ n=1 z n /n! converging on the entire Cayley-Dickson algebra A r . It has the periodicity property If A and B are two subsets in a complete uniform space X and θ : A → B is a continuous bijective mapping, the equivalence relation aΥb by When θ is uniformly continuous, θ has a uniformly continuous extension where cl(A) denotes the closure of the set A in X (see Theorem 8.3.10 in [5]). Certainly, the mapping θ can be specified by its graph We say, that A and B are glued (by θ), if B = θ(A) and the natural quotient mapping π : X → X/Υ is given, where X/Υ denotes the quotient space (see §2.4 [5]). For the uniformly continuous θ this means that the gluing is extended from A onto cl(A).
In the complex case to construct the Riemann surface of the logarithmic function one takes traditionally the complex plane C cut by the set Q 1 := {z = x + iy ∈ C : x < 0} and marking two respective points x 1 and x 2 of two edges Q 1,1 and Q 1,2 of the cut Q 1 arising from each given point Then one embeds C into either C × R or C × Ri and bents the obtained surface slightly along the perpendicular axis e 3 to C by neighborhoods of two edges of the cut Q 1 and gets the new surface C. Taking the countable infinite family of such surfaces C j with the edges of cuts Q j 1,1 and Q j 1,2 and gluing by respective points of the cuts Q j 1,2 with Q j+1 1,1 for each j one gets the Riemann surface of the logarithmic function, where j ∈ Z (see, for example, [18]).
Analogous procedure to construct the Riemann surface is in the cases r ≥ 2: one cuts A r by Q r and gets two edges Q r,1 and Q r,2 of the cut. This is described below.
The first simple construction for r ≥ 2 is the following. Take the set j=1 Ω j,r of subsets Ω j,r := {z ∈ Q r : z 0 = 0, ..., z j = 0} so that Ω j,r is contained in the boundary of the preceding set . Therefore, from each point z ∈ Q r two and only two different points z 1 and z 2 arise.
It is useful to embed A r either into A r × R 2 r −1 or into A r × I r , where I r := {z ∈ A r : Re(z) = 0}. Then one marks all pairs of respective points z 1 and z 2 arising from z ∈ Q r after cutting, slightly bents the cut copy of A r \{0} by (2 r − 1) axes perpendicular to A r by two neighborhoods of two edges of the cut Q r . Thus one gets the 2 r dimensional surface C r with two edges Q r,1 and Q r,2 of the cut. Taking the countable infinite family of such surfaces C j r with edges of the cuts Q j r,1 and Q j r,2 , j ∈ Z, and gluing respective points of edges Q j r,2 with Q j+1 r,1 for each j one gets the Riemann surface R r = R r,Ln of the logarithmic function Ln(z) : A r \ {0} → R r . Thus the latter mapping is already univalent with the image in the Riemann surface (see in details [22,21,25]).
For convenience we attach numbers 1 and 2 to faces in such manner that the winding around zero in the complex plane C M embedded into R r counterclockwise means the transition through the cut from Q j r,2 to Q j r,1 for each M in the connected set S + r . For the function z 1/n with n ∈ N its Riemann surface R r,z 1/n is obtained from n copies of surfaces C j r , j = 1, ..., n, by gluing the corresponding points of edges Q j r,2 with Q j+1 r,1 for j = 1, ..., n − 1 and of Q n r,2 with Q 1 r,1 (see also [18,22,21,25]).
Another more complicated construction is described below. Now we take the set Let z = z 0 + z be the Cayley-Dickson number with the negative real part z 0 < 0 and the imaginary part Im(z) = z , which can be written in the form z = |z | exp(πKe Lφ(z ) /2) and z = |z|e P ψ , where K, L and P are purely imaginary of the unit norm, φ and ψ ∈ R are reals. This gives the relations cos(ψ) = z 0 /|z| so that the parameter ψ is in the interval π/2 + 2πk < ψ < 3π/2 + 2πk for some integer number k. This means that for a continuous path γ contained in the set Q r the parameter ψ(γ(t)) is the continuous function of the real variable t ∈ R and remains in the same interval (π/2 + 2πk, 3π/2 + 2πk), consequently, Lnγ(t) preserves it branch along such path γ(t). This shows that after the first cut along Q r the obtained sets Q r,1 and Q r,2 need not be further cut. Thus the described reason simplifies the construction of the Riemann surface.
Each continuous path γ : [0, 1] → A r can be decomposed as the sum as well as the composition up to the homotopy satisfying the conditions of the homotopy theorem [22,21,26] also non-intersecting with Q r for all k < l ∈ Λ. Due to the homotopy theorem the logarithm Ln γ(t) does not change its branch along such continuous path γ(t), since this is the case for Ln γ k,l (t) for all k < l ∈ Λ.
For each z ∈ P j \ m,m =j P m cutting by Q r gives two points. If z ∈ (P k ∩ P j ) \ m,m =k,m =j P m with k < j cutting gives four points which can be organized into respective pairs after cutting of P k and then of P j . This procedure gives pairs (z 1,1 ; z 2,2 ) and (z 1,2 ; z 2,1 ). Then as above after cutting of A r by Q r one gets the 2 r dimensional surface C r with two edges Q r,1 and Q r,2 of the cut. For the countable infinite family of the surfaces C j r with edges of the cuts Q j r,1 and Q j r,2 , j ∈ Z, one glues respective points of edges Q j r,2 with Q j+1 r,1 for each j. Thus one gets the Riemann surface R r = R r,Ln of the logarithmic function Ln(z) : A r \ {0} → R r . Therefore, the latter mapping is univalent with the image in the Riemann surface (see in details [22,21,25]). For simplicity one can glue at first pairs of respective points of index l = 1 and extend gluing by continuity on points of index l > 1 (see above). The first construction for r ≥ 2 operates with points of index one only. That is why it is simpler than the second procedure.
The reader can lightly see that geometrically R r,f (for both types de- A Hausdorff topological space X is said to be n-connected for n ≥ 0 if each continuous map f : S k → X from the k-dimensional real unit sphere into X has a continuous extension over R k+1 for each k ≤ n (see also [38]).
A 1-connected space is also said to be simply connected.
It is supposed further, that a domain U in A m r has the property that (D1) each projection p j (U ) =: U j is (2 r − 1)-connected; (D2) π s,p,t (U j ) is simply connected in C for each k = 0, 1, ..., 2 r−1 , s = i 2k , p = i 2k+1 , t ∈ A r,s,p and u ∈ C s,p , for which there exists z = u + t ∈ U j , where e j = (0, ..., 0, 1, 0, ..., 0) ∈ A m r is the vector with 1 on the j-th place,  H(U, A r ). It is supposed that a locally analytic function f (x) is written in the x-representation ν(x), also denoted by ν = ν f . The latter is equivalent to the super-differentiability of f (see [22,21,23]). Each such f is supposed to be specified by its phrase ν.
For each super-differentiable function f (x) its non-commutative line in- A branch of the non-commutative line integral can be specified with the help of either the left or right algorithm (see [22,21,23]). We take further for definiteness the left algorithm if something another will not described. To rewrite a function from real variables z j in the z-representation the following identities are used: 2. The simplest differential equation of the first order has the form: Conditions 1(D1, D2). Here λ denotes the Lebesgue measure on A r induced from that of on the real shadow R 2 r .

pr h(β) β denotes the projection of β on h(β).
We have ψ(b) = ω(β) and ψ(a) = ω(α). Suppose that y 1 (ω(x)) and y 2 (ω(x) + φ 1 (x )) are two solutions of the differential equation (1), then their gives the general solution of the (super)-differential equation (1), since the . Its partial solution is defined, when the initial condition This means that the Cauchy problem can be formulated as given by Equations (1, 15) on the half-space for each x ∈ H α 0 . Therefore, the non-commutative or non-associative analog of the Cauchy problem is given by Equations (1, 15). In the previous papers [25,25] pseudoconformal mappings of the quaternion and octonion variables were investigated. Using the non-commutative or non-associative analog of the Riemann mapping theorem it is possible to formulate the non-commutative or nonassociative analog of the Cauchy problem on more general hyper-surfaces ). As usually cl(A) denotes the closure of a set A in A r , while Int(A) denotes the interior of A in A r (see [25,25]).
For such U = V the initial condition is 3. An R-linear differential equation of the first order.
Consider a differential equation, which is R-linear and A r -additive relative to an unknown function f of the first order: The octonion algebra is alternative, consequently, To solve this equation the exponential and logarithmic functions of quaternion and octonion variables can be used (see §3 in [22,21] and §2.7 above).
Since Ln(exp(x)) = x, we infer where I denotes the unit operator. Therefore, for each h 0 ∈ R, consequently, In accordance with §2 and Formula (7) Equation (3) has the general solution: when branches of Ln and are specified.
When b(t) ∈ R for each t, the equality and R is the center of the octonion algebra.
Using the variation method of the function C = φ Ln y (ω h (x)) in (9) and (3, 6, 7) one gets a solution of (1) analogously to the real case: since the octonion algebra is alternative and each equation bx = c with nonzero b has the unique solution x = b −1 c, (11) y 1 (ω(x)) = exp{− x α b(ω(t))dt}φ 1 (x ) is any solution of the homogeneous equation, where ω = ω h (x) (see also Formulas 2(5 − 7)). Here φ 1 (x ) is an octonion-valued function of x which can be specified by an initial condition 2(15) or more generally 2 (19). When φ 1 (x ) is arbitrary Formula (11) gives the general solution of the homogeneous differential equation. The second summand is the special solution of the inhomogeneous differential equation. Certainly, also for r = 1, U ⊂ A 1 = C, with b(x) and h(x) and Q(x) ∈ C for each x ∈ U Equation (1) has the general solution (10).
Symmetrically it is possible to solve the differential equation Another method of solving the inhomogeneous R-linear equation of the first order is like to that of Bernoulli. We present y as the product Its integration gives the special solution (see Formulas (3,9) above) where ω(x) = ω h (x) as in §2, branches of the indefinite integral b(ω(x))dx may differ on (super-)differentiable terms like φ(x ) as above. It remains Thus substituting u and v into the product decomposition (12) leads to: which is equivalent to (10), where y 1 (ω(x)) is given by Formula (11).
For example, if h(x) = x, then the particular solution of the differential

A differential equation with separated variables can be written
in the form: where U and W are domains satisfying Conditions 1(D1, D2). We seek a solution y so that y(U ) ⊂ W for suitable U and W . Integrating both sides of this equation with the help of the non-commutative line integral we deduce: for the equality (4) ((a n z n )b n )dz = d(a n z n+1 /(n + 1))b n is satisfied for each z ∈ A r , n = −1, a n ∈ A r , b n ∈ A r , (5) ((a n (1/z))b n )dz = (a n dLn z)b n in U not containing 0 with suitable branch of the logarithm Ln and the integral and with a n ∈ R (see in details the integration algorithms in [21,22,26]). Thus with the left algorithm we infer that (6) x α f (y)[(dy(t)/dt).h(t)]dt = t=x t=α f (y)dy(ω(t)) + φ f (Im y(ω(x)) for the locally analytic function f (y), consequently, the equation (Im y(ω(x))) gives the relation between the function y and the argument x (see §2 above).
When a mapping f and its phrase ν f are known, then a function g(y) = f (y)dy and its phrase ν g can be found easily with the help of the left algorithm. If an analytic function f is with real expansion coefficients, this integration reduces to the ordinary integration. Any other branch of the integral f (y)dy may differ from that of given by the left algorithm on a term like φ f (y − Re(y)) ∈ H. If f is a polynomial, then g is a polynomial. This means that y is a zero of the corresponding equation  [21,28,37]. Thus (9) y = g −1 (b(x)), but generally the inverse function g −1 need not be univalent [25] analogously to the complex case. The function φ s (x ) may be given with the help of the boundary condition 2 (19).

5.
Example. Consider the differential equation

Homogeneous differential equations.
Let us study the homogeneous differential equation of the form:  (ω h (x)), consequently, where we have substituted u = yx −1 . The second differential equation can be resolved symmetrically. Thus Equality (7) presents the general integral of the first differential equation. The function φ(x ) can be rewritten as xφ 1  When v(x) and (dv(x)/dx).h(x) commute for each x ∈ P α , this gives the following: The alternativity of the octonion algebra leads to: When l = 1/(k − m) it reduces to Therefore, for k = 1 Equation (1) is the octonion analog of the Bernoulli's equation and due to §3 we infer: where ω(x) is given by Equations 2 (5.2, 6), . To avoid any misunderstanding in our notations we mention that the function p from §2 is not related with the arbitrary function p of this section. An arbitrary function φ 1 ∈ H(U, A r ) is determined from the boundary condition 2(15), so that φ 1 (x ) = [η(x )] 1−m . Thus y = v 1/l is the solution of Cauchy's problem 6(1), 2(15).

Suppose that p(x), h(x) ∈ R \ {0}, h(x), p(x) and s(x) are x-differentiable
on U , open domains U and W in the Cayley-Dickson algebra A r satisfy Conditions 1(D1, D2), y(U ) ⊂ W . In view of Formulas 4(2 − 5) one gets the following: for v l = y over octonions with l = 1/(k − m). Thus as in §4 we deduce for s(x) = 0: Any other branch of the non-commutative line integral (4) may differ on a term like φ(v − Re(v)). Suppose that β(v) and its inverse function β −1 are calculated. To solve (2) consider C = −φ p as a function of x so that we seek v in the form: Differentiating both sides of (6) by x and using (5) we get: . The latter and 4(6) Formulas imply that Thus we infer the general solution in the form: Arbitrary functions φ 3 , φ s ∈ H(U, A r ) and φ 2 ∈ H(W, A r ) can be specified from the boundary condition 2 (19) choosing possibly simpler formulas for ψ and β: One can mention that in the particular case of f = 1 and k = 1 the reciprocal function β −1 = exp is the exponential function and Equations (9, 10) coincide with 3(10, 11) for h = 1.
8. The total differential equations.
The family of all R-homogeneous A r -additive operators on A m r with values in A n r is denoted by L q (A m r , A n r ) for natural numbers m, n ∈ N, 2 ≤ r. Suppose that there is given the differential 1-form over the Cayley-Dickson algebra A r :  B(α, τ ).dτ + φ B (x , y ). Thus where α, x ∈ U , β, y ∈ V . Therefore, Condition (4) is sufficient for the existence of the function F . A boundary condition can be specified as: Initially domains U and V can be taken as half-spaces {z : Re(z) ≥ τ 0 } with either τ 0 = α 0 or τ 0 = β 0 respectively and then they can be transformed with the help of pseudo-conformal or more generally z-(super-)differentiable diffeomorphisms into given domains in accordance with the octonion analog of the Riemann mapping theorem (see also §2 above).
9. Integration factor. Only in rare cases the differential A(x, y).dx + B(x, y).dy is complete. We consider the problem of an integration factor existence so that (1) w = µA.dx + µB.dy would be the complete differential, where µ ∈ H(U × V, A r ), 2 ≤ r ≤ 3. This means that
Let us consider the differential equation After the (super-)differentiation by x one obtains: the equality y = f (x, p) we infer: where (∂f /∂p) −1 denotes the inverse operator whenever it exists. The given relations (4, 5) between x, y and C provide the general integral of (1).
With the notation (dy/dx).1 = p this equation is equivalent to y = xp 2 + p 2 .
We shall seek a solution in the case, when p = (dy/dx).1 and (dp/dx).1 commute with each other. This leads to the commutativity of p with (dp/dz).1, consequently, (2) (p 2 − p) + 2p(x + 1)(dp/dx).1 = 0 in accordance with §12. Then one reduces this equation to the linear relative to x: (3) (dx/dp).1 + (2/(p − 1))x = 2/(1 − p), when p 2 − p = 0. Its integration gives: and its substitution into the first equation produces Re(p)) and p commute, for example, when φ is real-valued or Im[φ(p − Re(p))] belongs to Im(p)R, where Im(p) := p − Re(p) denotes the imaginary part of p. Thus (4,5) is the solution in the parameter form. Excluding the parameter p from these two equations we obtain the solution It remains to consider the equation p 2 − p = 0. It has two solutions p = 0 and p = 1, which lead to two solutions of Equation (1): the singular solution y = 0 and the particular solution y = x + 1.
With the new function v Equation (10) becomes If any solution v is obtained, then Practically (1, 2, 3) and (10) in real coordinates correspond to the specific systems of 2 r partial differential equations of 2 r real variables with generally non-constant coefficients, because a k and h j,k and h j may be non-constant functions.
14. Note. It is important to mention simplifications for resolving differential equations over Cayley-Dickson algebras (1) j a j (x)y (1) (x).h j (x) = y (1) when a j (x) ∈ R and h j (x) ∈ A r for each x ∈ U , since the real field is the center Z(A r ) of the Cayley-Dickson algebra A r , 2 ≤ r. This means that the system of partial differential equation with generally non-constant coefficients a j : (2) j a j (x)y (1) (x).h j (x) + g(x, y) = 0 reduces to the ordinary differential equation over the Cayley-Dickson algebra

For integration of differential equations over the Cayley-Dickson algebra
A r the additivity of the non-commutative line integral is helpful: (4) γ n j=1 f j (x)dx = n j=1 γ f j (x), n ∈ N. Evidently one also has: (5) s;j 1 ,...,j n y (n) .(h 1,j 1 , ..., h n,j n )a s;1,j 1 (x)...a s;n,j n (x) = s y (n) (x).(h s 1 , ..., h s n ), where h s k (x) = j a s;k,j (x)h k,j (x), when a s;j,k (x) ∈ R and h s j,k (x) ∈ A r for all s, j, k and x ∈ U . The sum on the right of (5) may be simpler integrated with the help of (4). When s takes only one value the sum by s certainly gives only one term y (n) .(h 1 , ..., h n ).
If A n is a differential operator of the n-th order it may happen that there exists the first order differential operators κ 1 , ..., κ n such that (A n − κ n ...κ 1 ) is the differential operator A n−1 of order not greater than (n − 1).
Sometimes such reduction may serve for a simplification of integration of the corresponding differential equation (6) A n y = g putting (7) κ n ...κ 1 y = f and (8) A n−1 y = g − f . Equation (7) can be frequently integrated as the sequence of differential equations of the first order using their types presented above. Mention, that differential operators of the form (∇u, w) with real-valued u, ∇u = (∂u/∂z 1 , ..., ∂u/∂z m ) and real-valued (super-)differentiable functions w = w 1 (z),...,w m (z) is of the form given by Equation (1), where (∇u, w) := m j=1 w j (z)∂u(z)/∂z j . Therefore, equations of the form (9) ∂ k u(z)/∂z k 0 + (∇u(z), w(z)) = g(z) with k ∈ N reduce to that of considered above.
Suppose that either u(x) ∈ R or h 1 ∈ R or v(y) ∈ R. The second demand can be fulfilled with a change of variables if use ω h 1 (x). The first or the third condition means that y (1) is either right or left super-linear. Thus the second demand may happen to be the less restrictive between these three conditions. Take the case u ∈ R and h 1 = 1 = i 0 , consequently, x. To obtain a non-trivial solution we put δ/(θe −αt )] is independent of x. We put h = i 0 + s 1 i 1 + s 2 i 2 + s 3 i 3 and consider u as the function u = u(z) of the quaternion variable z ∈ H. Therefore, The equality ω h (y) = x gives the solution y j = x j − ts j for each j = 1, 2, 3, consequently, u(t, x) = u(x − ts).
One can take any sequence of z-differentiable functions g n with compact supports tending to δ(z), when n tends to the infinity. That is  Dirac had used the complexified bi-quaternion algebra to solve Klein-Gordon equation for quantum particles with spin. This partial differential equation is of the hyperbolic type with constant coefficients. Using the technique of this article it is possible to integrate subsequently hyperbolic partial differential equations with variable coefficients (see also § §13 and 14 above). This is planned to be written in details in subsequent papers together with differential equations of generalized functions of octonion variables, singular integrals (exceptional solutions) and examples of differential equations and their fundamental and general solutions.