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On a complex differential Riccati equation

We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrodinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation as, e.g., the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical"one-dimensional"results we discuss new features of the considered equation like, e.g., an analogue of the Cauchy integral theorem.


Introduction
The ordinary differential Riccati equation has received a great deal of attention since a particular version was first studied by Count Riccati in 1724, owing to both its specific properties and the wide range of applications in which it appears. For a survey of the history and classical results on this equation, see for example [7] and [25]. This equation can be always reduced to its canonical form (see, e.g., [6], [10]), and this is the form that we will consider. One of the reasons for which the Riccati equation has so many applications is that it is related to the general second order homogeneous differential equation. In particular, the one-dimensional Schrödinger equation is related to (2) by the easily inverted substitution This substitution, which as its most spectacular application reduces Burger's equation to the standard one-dimensional heat equation, is the basis of the well-developed theory of logarithmic derivatives for the integration of nonlinear differential equations [19]. A generalization of this substitution will be used in this work. A second relation between the one-dimensional Schrödinger equation and the Riccati equation is as follows. The one-dimensional Schrödinger operator can be factorized in the form if and only if (2) holds. This observation is a key to a vast area of research related to the factorization method (see, e.g., [22] and [23]) and to the Darboux transformation (see, e.g., [20], [24] and [26]). In the present work we consider a result similar to (4) but already in a two-dimensional setting. Among the peculiar properties of the Riccati equation stand out two theorems of Euler, dating from 1760. The first of these states that if a particular solution y 0 of the Riccati equation is known, the substitution y = y 0 + z reduces (2) to a Bernoulli equation which in turn is reduced by the substitution z = 1 u to a first order linear equation. Thus given a particular solution of the Riccati equation, it can be linearized and the general solution can be found in two integrations. The second of these theorems states that given two particular solutions y 0 , y 1 of the Riccati equation, the general solution can be found in the form where k is a constant. That is, given two particular solutions of (2), the general solution can be found in one integration. Other interesting properties are those discovered by Weyr and Picard ( [29], [7]). The first is that given a third particular solution y 3 , the general solution can be found without integrating. That is, an explicit combination of three particular solutions gives the general solution. The second is that given a fourth particular solution y 4 , the cross ratio is a constant. Thus the derivative of this ratio is zero, which holds if and only if the numerator of the derivative is zero, that is, if and only if Dividing by (y 1 − y 2 )(y 3 − y 4 )(y 1 − y 4 )(y 3 − y 2 ), we see that Picard's theorem is equivalent to the statement In the present work we study the following equation where z is a complex variable, ∂ z = 1 2 (∂ x + i∂ y ), Q is a complex valued function of z and v is a real valued function. Note that this equation is different from the complex Riccati equation studied in dozens of works where it is supposed to have the form (1) with complex analytic coefficients p, q and r and a complex analytic solution u (see, e.g., [9]). We do not suppose analyticity of the complex functions involved and show that equation (7) unlike the equation considered in [9] is related to the two-dimensional stationary Schrödinger equation in a similar way as the ordinary Riccati and Schrödinger equations are related in dimension one. Moreover, we establish generalizations of the Euler and Picard theorems and obtain some other results which are essentially two-dimensional, for example, an analogue of the Cauchy integral theorem for solutions of the complex Riccati equation (7). Equation (7) first appeared in [14] as a reduction to a two-dimensional case of the spatial factorization of the stationary Schrödinger operator which was studied in a quaternionic setting in [1], [3], [11], [12], [13], [14] and later on using Clifford analysis in [2] and [8].
The ordinary Riccati equation is at the heart of many analytical and numerical approaches to problems involving the one-dimensional Schrödinger and Sturm-Liouville equations. Here we furnish a complete structural analogy between dimensions one and two regarding the relationship between the Schrödinger and the Riccati equations. Besides, the deep similarity between the ordinary Riccati equation and (7) strongly suggests that many known applications of the ordinary Riccati equation can be generalized to the twodimensional situation and many new aspects such as theorem 12 will become manifest.

Some preliminary results on the stationary Schrödinger equation and a class of pseudoanalytic functions
We need the following definition. Consider the equation ∂ z ϕ = Φ in the whole complex plane or in a convex domain, where Φ = Φ 1 + iΦ 2 is a given complex valued function whose real part Φ 1 and imaginary part Φ 2 satisfy the equation then we can reconstruct ϕ up to an arbitrary real constant c in the following way where (x 0 , y 0 ) is an arbitrary fixed point in the domain of interest. By A we denote the integral operator in (9): Note that formula (9) can be easily extended to any simply connected domain by considering the integral along an arbitrary rectifiable curve Γ leading from (x 0 , y 0 ) to (x, y), Thus if Φ satisfies (8), there exists a family of real valued functions ϕ such that ∂ z ϕ = Φ, given by the formula ϕ = A[Φ]. In a similar way we introduce the operator which is applicable to complex functions satisfying the condition and corresponds to the operator ∂ z . Let f denote a positive twice continuously differentiable function defined in a domain Ω ⊂ C. Consider the following Vekua equation where the subindex z means the application of the operator ∂ z (analogously, with the aid of the subindex z we will denote the application of the operator ∂ z ), W is a complex-valued function and W = C[W ] is its complex conjugate function. As was shown in [14], [15] (see also [16], [17], [18]) equation (11) is closely related to the second-order equation of the form where ν = (∆f ) /f and u are real-valued functions. In particular the following statements are valid.
Theorem 1 [15] Let W be a solution of (11). Then u = Re W is a solution of (12) and v = Im W is a solution of the equation Theorem 2 [15] Let u be a solution of (12) in a simply connected domain Ω. Then the function v ∈ ker ∆ + ν − 2 |∇f | f 2 such that W = u + iv be a solution of (11), is constructed according to the It is unique up to an additive term cf −1 where c is an arbitrary real constant. Given can be constructed as follows up to an additive term cf.
Thus, the relation between (11) and (12) is very similar to that between the Cauchy-Riemann system and the Laplace equation. Moreover, choosing ν ≡ 0 and f ≡ 1 we arrive at the well known formulas from classical complex analysis. Note that the potential η has the form of a potential obtained after the Darboux transformation but already in two dimensions.
For a Vekua equation of the form where a and b are arbitrary complex-valued functions from an appropriate function space [28], a well developed theory of Taylor and Laurent series in formal powers has been created (see [4], [5]). We recall that a formal power Z (n) (a, z 0 ; z) corresponding to a coefficient a and a centre z 0 is a solution of the Vekua equation satisfying the asymptotic formula For a rigorous definition we refer to [4], [5]. The theory of Taylor and Laurent series in formal powers among other results contains the expansion and the Runge theorems as well as more precise convergence results [21] and a recently obtained simple algorithm [17] for explicit construction of formal powers for the Vekua equation of the form (11). In section 4 we show how this theory is applied for generalizing the second Euler theorem. For this we need the expansion theorem from [4]. For the Vekua equation of the form (11) this expansion theorem reads as follows (for more details we refer the reader to [17]).
Theorem 3 Let W be a regular solution of (11) defined for |z − z 0 | < R.
Then it admits a unique expansion of the form W (z) = ∞ n=0 Z (n) (a n , z 0 ; z) which converges normally for |z − z 0 | < R.
Another result which will be used in the present work (section 4) is a Cauchy-type integral theorem for the stationary Schrödinger equation. It was obtained in [14] with the aid of the pseudoanalytic function theory.
Theorem 4 (Cauchy's integral theorem for the Schrödinger equation) Let f be a nonvanishing solution of (12) in a domain Ω and u be another arbitrary solution of (12) in Ω. Then for every closed curve Γ situated in a simply connected subdomain of Ω,

The two-dimensional stationary Schrödinger equation and the complex Riccati equation
Consider the complex differential Riccati equation where for convenience the factor 1/4 was included. We recall that ν is a real-valued function. Together with this equation we consider the stationary Schrödinger equation (12), where u is real-valued. Both equations are studied in a domain Ω ⊂ C.
Theorem 5 Let u be a solution of (19). Then its logarithmic derivative is a solution of (18).

Remark 6 Any solution of equation
Theorem 7 Let Q be a solution of (18). Then the function is a solution of (19).
Proof. Equation (19) can be written in the form Taking u in the form (21) where Q is a solution of (18) and using the observation that Observe that this theorem means that if Q is a solution of (18) then there exists a solution u of (19) such that (20) is valid. Theorems 5 and 7 are direct generalizations of the corresponding facts from the one-dimensional theory.
The following statement is a generalization of the one-dimensional factorization (4).
Theorem 8 Given a complex function Q, for any real valued twice continuously differentiable function ϕ the following equality is valid if and only if Q is a solution of the Riccati equation (18).
Proof. Consider from which it is seen that (22) is valid iff Q is a solution of (18). The second equality in (22) is obtained by applying C to both sides of the first equality.

Generalizations of classical theorems
In this section we give generalizations of both Euler's theorems for the Riccati equation as well as of Picard's theorem.
Theorem 9 (First Euler's theorem) Let Q 0 be a bounded particular solution of (18). Then (18) reduces to the following first order (real-linear) equation in the following sense. Any solution of (18) has the form and vice versa, any solution of (23) can be expressed via a corresponding solution Q of (18) as follows Proof. Let Q 0 be a bounded solution of (18). Then by theorem 7 we have that there exists a nonvanishing real valued solution f of (19) such that Q 0 = f z /f . Hence (23) has the form (11). Now, let Q be any solution of (18). Then again Q = u z /u where u is a solution of (19). According to theorem 2, u is a real part of a solution W of (11). Thus we have proved the first part of the theorem. Let W = u + iv be any solution of (23) (u = Re W ). Then u is a solution of (19) and by theorem 5 it can be represented in the form u = e A[Q] where Q is a solution of (18). Then by theorem 2 (formula (14)), W has the form (24).
Thus the Riccati equation (18) is equivalent to a main Vekua equation of the form (11).
In what follows we suppose that in the domain of interest Ω there exists a bounded solution of (18).
where {Q n } ∞ n=0 is the set of particular solutions of the Riccati equation (18) obtained as follows Q n (z) = ∂ z Re Z (n) (a n , z 0 , z) Re Z (n) (a n , z 0 , z) , Z (n) (a n , z 0 , z) are formal powers corresponding to equation (23) and both series in (25) converge normally for |z − z 0 | < R.
Proof. By the first Euler theorem we have where W is a solution of (23). From theorem 3 we obtain Re Z (n) (a n , z 0 ; z) .
Every formal power Z (n) (a n , z 0 ; z) corresponds to a solution of (18): Q n (z) = ∂ z Re Z (n) (a n , z 0 ; z) Re Z (n) (a n , z 0 ; z) or Re Z (n) (a n , z 0 ; z) = e A[Qn](z) from where we obtain (25).
In the next statement we give a generalization of Picard's theorem in the form (6).

Cauchy's integral theorem
Theorem 12 (Cauchy's integral theorem for the complex Riccati equation) Let Q 0 and Q 1 be bounded solutions of (18) in Ω. Then for every closed curve Γ lying in a simply connected subdomain of Ω, Proof. From theorem 7 we have that f = e A[Q 0 ] and u = e A[Q 1 ] are solutions of (19). Now, applying theorem 4 we obtain for every closed curve Γ situated in a simply connected subdomain of Ω, which gives us the equality From this we obtain the result. As a particular case let us analyze the Riccati equation (18) with ν ≡ 0 which is related to the Laplace equation. If in (26) we assume that Q 0 ≡ 0, then (26) takes the following form Γ Q 1 e A[Q 1 ] dz = 0.