Vanishing and blow-up solutions to a class of nonlinear complex di ﬀ erential equations near the singular point

: A singular nonlinear di ﬀ erential equation

, , where > σ 1, is considered in a neighbourhood of the point = z 0 located either in the complex plane if σ is a natural number, in a Riemann surface of a rational function if σ is a rational number, or in the Riemann surface of logarithmic function if σ is an irrational number.It is assumed that ( ) = w w z , { } ∈ ⧹ a 0 , and that the function f is analytic in a neighbourhood of the origin in × .Considering σ to be an integer, a rational, or an irrational number, for each of the above-mentioned cases, the existence is proved of analytic solutions ( ) = w w z in a domain that is part of a neighbourhood of the point = z 0 in or in the Riemann surface of either a rational or a logarithmic function.Within this domain, the property ( ) = → w z lim 0 z 0 is proved and an asymptotic behaviour of ( ) w z is established.Several examples and figures illustrate the results derived.The blow-up phenomenon is discussed as well.

Introduction
A singular nonlinear differential equation where > σ 1, is considered in a neighbourhood of the point = z 0 either in the complex plane if { } ∈ ≔ σ 1, 2, … is a natural number, in a Riemann surface of a rational function if ∈ ⧹ σ is a rational number, or in the Riemann surface of logarithmic function if ∈ ≔ ⧹ σ is an irrational number.In (1), z is an independent variable, ( ) = w w z , and { } ∈ ⧹ a 0 .The function → f : is assumed to be analytic in a neighbourhood of the point ( ) ∈ × 0, 0 having the form: A proof is given of the existence of analytic solutions ( ) = w w z defined in a multiple connected domain in a neighbourhood of the singular point = z 0 and vanishing for → z 0 (the point = z 0 itself does not belong to this domain being its boundary point).Such a domain lies in the complex plane if ∈ σ , in a Riemann surface of a rational function if ∈ ⧹ σ , or in the Riemann surface of the logarithmic function if ∈ σ .Whether the complex plain or the Riemann surface is chosen is determined by the domain of the term z σ in (1).
Asymptotic analysis of equations in a neighbourhood of a singular point in the complex plane has a long history.In a pioneering article [3], a system of nonlinear differential equations and an initial problem are considered with a function h being holomorphic in a neighbourhood of ( ) 0, 0 and satisfying ( ) = h 0, 0 0. The authors prove that a solution ( ) = w w z such that ( ) = w 0 0can be constructed as a power series convergent in a neighbourhood of = z 0. In [31], assuming ( ) m , the authors use functional-analytical methods to show that (3) admits an analytic solution expressed by a power series with an initial power z m .As a particular case of system (3), arising when the Jacobian matrix ( ) ′ h 0, 0 w is singular, systems and their modifications are investigated where > σ 1 is an integer, z is an independent complex variable, ( ) = y y z , ( ) = w w z , h i , = i 1, 2, are holomorphic vector-functions in a neighbourhood of ( ) 0, 0, 0 vanishing there.We refer to [13][14][15][16][17][19][20][21][22][23][25][26][27]30] and to references therein.As the principal method of investigation is often used a construction of formal solutions in the form of special series (that are not necessarily exact power series) convergent in a subset of a neighbourhood of = z 0. A substantial restriction used in the above-mentioned results is the assumption that σ in (4) is an integer as the methods used by their authors are not applicable in cases of σ being rational or irrational.In this study, we suggest a "geometrical" method connected with the properties of the function z σ and allowing us to omit this restriction in the case of scalar equation (1).If ∈ σ , equation (1) is considered in the complex plane , while, , where > > m m 1 are relatively prime, we use the Riemann surface of the function = ∕ w z m 1 2 and, if ∈ σ , the Riemann surface of the logarithmic function.Moreover, the properties of solutions to systems (4) and ( 5) have only been studied in a subset of the origin (in a sector with its vertex at the point = z 0).Our investigation of the asymptotic properties of solutions to equation (1) cover, in a sense, the whole neighbourhood of the singular point = z 0 (in or in the aforemen- tioned Riemann surfaces).
In the right-hand side of (1), a "perturbation" ( ) zwf z w , of a linear equation ′ = z w aw σ is considered.Such a form of nonlinearity is quite natural because, if we use, for example, the term ( ) f z w * , instead, the assumptions of our results reduce such a general form to the form used in (1), i.e. to ( ) ( ) ≔ f z w zwf z w * , , .In some formulas throughout this article, if no ambiguity can arise, a simplified notation is used of the dependent variables not indicating their dependence on independent variables.The geometrical method of investigation suggested in this study is quite different from the methods used previously and can be used for analysing other classes of equations in the complex domain.
For the reader's convenience, in Section 2, we recall some auxiliary notions and concepts well known from the theory of functions of complex variable.Transformations applied to equation (1) with focus on systems equivalent on given curves and rays to (1) are discussed in Section 3, while in Section 4, the behaviour of solutions is studied of the systems derived.In Section 5, the results of this article (Theorems 1-4) are formulated.Their proofs are given in Section 6.Since a major part of the proofs of Theorems 1-3 is identical for an arbitrary value of σ , we consider only one variant of the proof where the differences depending on natural, rational, or irrational values of σ are emphasized.The proof of Theorem 4 is a consequence of the common part of the proof.Each of Examples 1-5 accompanies the constructions performed in Section 6.Nevertheless, in Section 7, a more complex example is considered.Concluding remarks and open problems formulated are given in Section 8.A close connection of the findings of this article with the unlimited growth of moduli of solutions near the singular point = z 0 (the so-called blow-up phenomenon) is mentioned and discussed as well.

Preliminaries
Consider an initial problem where z and w are the complex variables and F is a complex-valued function.By a special case of the well known Cauchy-Kovalevskaya theorem if the function F is analytic in a neighbourhood of the point ( ) z w , 0 0 , problem (6) has a unique analytic solution ( ) = w w z in a neighbourhood of the point z 0 (we refer, e.g., to [8]).Recall also that an analytic function is a function that can be expressed by a convergent power series.A holomorphic function is a function that is differentiable in each neighbourhood of the point of its domain.For complex functions, the notions of an analytic and a holomorphic function are equivalent.
Let ⊂ be a path-connected domain.A curve lying in is said to be simple if it does not cross itself.A domain is simply connected if any simple closed curve in can be continuously shrunk into a point while remaining in .A domain that is not simply connected is called a multiply-connected domain.The symbol ∂ denotes the boundary of , and stands for the closure of (i.e. for the set ∪ ∂ ).The concept of analytic continuation (extension) is used in this article as well.It means the following.Let f 1 and f 2 be two analytic functions in open domains 1 and 2 of the complex plane , respectively.Let ∩ ≠ ∅ and , f 2 is called an analytic continuation of f 1 to 2 , and vice versa.The analytic continuation is unique.

Transformations of equation (1)
Several auxiliary transformations of equation ( 1) are necessary for its analysis.In Section 3.1, we show why the coefficient a in (1) can be assumed to be real and positive.Then, two types of real two-dimensional systems equivalent to (1) are considered.In Section 3.2, we derive an equivalent two-dimensional real system along given curves starting and ending at the point = z 0, while in Section 3.3, an equivalent real two-dimensional real system along given rays leading off the point = z 0 is obtained.

On the coefficient a in (1)
Let the complex coefficient a in (1) be given in its exponential form: iθ Then, a substitution, geometrically expressing a rotation, with a positive coefficient | | a instead of the previous complex coefficient a.In the following investigation, we will assume that | is sufficiently small.This is true if either | | z or | | w is small enough.Then, due to (7), this property remains in force also for the expression: vwf ve w e vwf ve w , , .
Therefore, the coefficient a in (1) can be assumed, without loss of generality, real and positive, i.e. a can be replaced by its modulus | | a .We implicitly use this property in the investigations in the following and, when- ever computations in this article depend on a, we assume that it is a positive number.

Real system equivalent to (1) on the loops of a given curve
The behaviour of solutions to (1) in a small neighbourhood of the point = z 0 will be studied along the loops of a curve defined as: , where ν 0 is a fixed real number, > c 0 is a real parameter, and φ is a real independent variable.If c and ν 0 are fixed, we will assume that φ varies in such a way that and, moreover, as we need

−
and, obviously, The closure of each loop of the curve (8) separated by an inequality ( 10) is a closed curve passing through the origin.If ∈ σ , then there are − σ 1 different curve arcs (8) lying in the complex plane , defined, e.g., by: , where > > m m 1 in (10).If ∈ σ , then there is a countable set of disjunct loops of curve (8) defined by = ± s 0, 1,… in (10).We consider them on the Riemann surface of the logarithmic function.

0
, and related to the values = c 1 (green loops), = c 2 (brown loops), and = c 3 (blue loops).We will transform equation (1) into a system of two ordinary differential equations on parts of the closures of the loops of curve (8) with independent variable φ satisfying inequalities (10).Then, ( ) ( ( )) = w z w z φ .To reduce the computations, we do not always write the argument ( ) z φ of w or the argument φ of z (similarly, we proceed when new dependent variables α and β in the following are used).By the chain rule, we derive where Finally, using (8), (13), and ( 14), Equalling the real and imaginary parts, we see that y 1 and y 2 satisfy the following system of ordinary differential equations equivalent to equation ( 1) on the given loop segment of the curve (8): iν iν ) iν iν

Real system equivalent to (1) on a system of rays
Consider rays running from the origin and transform equation ( 1) along these into a system of two real equations.Since (1) can be written as: w z t by its algebraic form ( ( )) ( ) , we derive Functions x 1 and x 2 satisfy the following system of ordinary differential equations: )) 4 Auxiliary results on the behaviour of solutions In this section, we prove two lemmas used in proving the results of this article.In Section 4.1, the behaviour of solutions to (1) is considered along loop segments of the curve (8), while in Section 4.2, the behaviour of solutions to (1) is considered along rays (18) defined in Section 3.2.
4.1 Behaviour of solutions along segments of curve (8) Assume, in the definition of curves (8), ≠ ν mπ In the following, we consider system ( 16), (17) in the domain , , : 0 , , where k and ρ are the same as in the definition of the domain given by ( 2), ( ) z φ is defined by (8), and assume that (9) holds as well.The values φ k , defined by (12), have property (11), i.e. these values define singular points = φ φ k of system ( 16), (17) because . It is clear that the conditions are met at every point of Ω cr of the well-known theorems on the existence and uniqueness of solutions to initial Cauchy problem as well as on the continuous dependence of solutions on the initial data.
Let ( ) ε ν 0 be a positive number such that where ε 0 is a fixed number satisfying Define cylinders ( ) λ as sets Lemma 1. Assume that φ varies in a fixed domain defined by (10).Let c and ν 0 , satisfying (21), be fixed such that Then, any integral curve ( ( ) ( )) φ y φ y φ , , Proof.Consider the behaviour of integral curves of system ( 16), (17) intersecting cylinders ( ) λ .To do this, compute the scalar product ( ) → → N T , at an arbitrary point of cylinder ( ) λ with fixed λ, where → N is its normal vector directed outwards and → T is a vector of the vector field defined by system ( 16), (17).As ( ) we have We will show that (29) implies Therefore, from ( 22), (23), and (25), it follows  (2) and that formula (30) is independent of the value of λ.The geometrical meaning of equation ( 30) is as given in parts (i) and (ii) of the lemma.□ Remark 1.Note that the φ-axis itself (i.e. the set of points ( ) φ, 0, 0 , where φ satisfies (10)) is also an integral curve of ( 16) and (17); therefore, no other integral curve intersects the φ-axis.For a sufficiently large c, the considered loops of (8), as it follows from (25), are completely contained in the ( ) ε ν 0 -neighbourhood of the point = z 0. If c is sufficiently small, then loops of (8) are not connected in the ( ) ε ν 0 -neighbourhood of the point = z 0 with this ( ) ε ν 0 -neighbourhood containing their parts (Figure 2).In most of the following figures, the value ( ) ε ν 0 can be seen on the plots scaled down to fit into a single figure.
4.2 Behaviour of solutions along system of rays (18) Assume, in the definition of rays (18), In this part, we will examine the behaviour of solutions to Systems (19) and (20) in the domain: , , : 0 , , where k and ρ are the same as in the definition of the domain given by ( 2) and ( ) z t is defined by (18), i.e. | ( )| = z t t.For system (19), (20), the hypotheses of the well-known theorems on the existence and uniqueness of solutions to initial Cauchy problem as well as on the continuous dependence of solutions on the initial data are true at every point of Ω rs .Define cones ( ) δ as sets where δ and p are the fixed parameters satisfying > δ 0 and where and ε* p is a positive number satisfying ( ) where δ satisfies by the definitions of ( ) δ and Ω rs .
of the system (19), ( 20) , satisfies the inequality: and where ε is a small positive number.
Proof.Let us investigate the behaviour of the integral curves intersecting cones ( ) , σ at an arbitrary point of a fixed cone ( ) δ , where → N is a normal vector to (32) directed outwards and → T is a vector of the vector field defined by the system ( 19), (20).Since we have where Now, (40) will be used to determine Then, by (33), (34), and (42), and formula (41) holds.Therefore, and The geometric meaning of ( 43) is expressed by Inequality (36) in (i), which implies (37).Now, concerning conclusion (ii).In the case (44), we have Therefore, any solution of (1) different from the zero solution and such that ( ( ) ( )) intersects the cone (32 , , and (39 , where ε is a small positive number.□ Remark 2. In this article, the existence of analytic solutions ( ) = w w z to (1) defined in a subset of a neighbour- hood of the point = z 0 and satisfying | ( )| < w z e k is established.Nevertheless, the behaviour of the solutions ( ) = w w z in the domain, which is defined as the complement of the above-mentioned subset of a neighbour- hood of the point = z 0, can be quite different and its modulus can tend to infinity as → z 0 and z belongs to this complement.Such a blow-up behaviour can be proved by Lemma 2, part (ii).Indeed, it is visible from (39) that , where ε is a small positive number.However, if → + t 0 , then the modulus | ( )| w te iν is expo- nentially growing and

Main results
This chapter is concerned with the existence of analytic solutions to (1) defined on subsets of neighbourhoods of the point = z 0 and with the size of their moduli.We will consider a neighbourhood if a Riemann surface is considered.With the notation z Arg applied, it will denote the principal value of the argument of a complex number z on the relevant domain, i.e. either on the complex plane or on a Riemann surface .

Define
and Let r 1 be a sufficiently small fixed positive number satisfying the inequality: 1 . 1 Before formulating Theorems 1-3, the following comments may be useful.Theorem These domains, as we show in the proof, are located on the complex plane in sectors between the angles ( ) and their boundaries are generated by segments of two loops of the curve ( 8), the first one being specified by ν and φ such that while the second one by ν and φ such that provided that the parameter c in (8), being the same for both arcs, is sufficiently large.Such specifications of ν and φ imply the properties Theorem 2 uses domains are relatively prime.They have the same properties as domains n 1 in Theorem 1 except for their location; these are located on the Riemann surface of the function , where n 0 is an integer and { } ∈ ∪ N 0 if ∈ σ in Theorem 3, have the same properties as well located, however, on the Riemann surface of the logarithmic function.
All the above-mentioned domains in , = i 1, 2, 3 are excluded from formulation of results below because the solutions ( ) w z become there, as it will follow from the proofs, unbounded by their moduli (with blow-up effect arising).In what follows, we will call these domains "blow-up holes." For more details, we refer to Section 6.We refer as well to Figure 11 where the case = σ 2 is visualized and to Figure 15 illustrating the case = σ 3.
Then, there exist infinitely many analytic solutions ( ) = w w z to (1) defined on the complex plane in a multiply-connected domain: where ∈ m m , 1 2 are relatively prime.Then, there exist infinitely many analytic solutions ( ) = w w z to (1) defined on the Riemann surface of the function = ∕ w z m 1 2 in a multiplyconnected domain where are simply connected open domains such that Theorem 3. Let ∈ σ .Then, for any integer n 0 and any natural number { } ∈ ∪ N 0 , there exist infinitely many analytic solutions of the Riemann surface of the logarithmic function defined by inequalities: are simply connected open domains such that Remark 3. From the formulation of Theorem 1, it follows that the point = z 0 is a unique point common to the closures of all domains n 1 , = − n σ 0,…, 2. Geometrically, the domain 1 is a neighbourhood of the origin with − σ 1 blow-up holes n 1 , = − n σ 0,…, 2. In the case of Theorem 2, the point = z 0 is a unique point common to the closures of domains n . The domain 2 is a neighbourhood of the origin on the Riemann surface of the function . If Theorem 3 can be applied, then the point = z 0 is a unique point common to the intersections of closures of domains n . The domain 3 is a subset of the neighbourhood of the origin on the Riemann surface associated with the logarithmic function with . Theorem 4. Let ( ) = w w z be an arbitrary solution mentioned in Theorems 1-3.Then, , or 3 can be applied, respectively.Moreover, for any fixed p, < < p σ 1 , and for every ∈

Proofs
The proofs are divided into several sections and are based on the results given in Sections 4.1 and 4.2.In Section 6.1, we show how the solution of an initial Cauchy problem to (1) can be analytically continued using Vanishing and blow-up solutions  13 given loops and rays in a domain P nω , while in Section 6.2, its analytic continuation is constructed to certain domains + Ω n and − Ω n lying on different sides of P nω .Finally, in Section 6.3, the analytic continuation is con- structed to a domain ( ) P n .Each construction is demonstrated by an example.Sections 6.4-6.6 explain how some solutions can be analytically continued to domains ( ) ( ) Pn N , respectively.This proves Theorems 1-3 and Theorem 4 as well because property (49), as pointed out in 6.2.1 and in 6.2.2, is a consequence of the results of Sections 4.1 and 4.2.

Analytic solutions on a domain P nω
In Section 6.1.1,we show how the solution of a Cauchy problem can be analytically continued using some of the previously defined curve loops and segments of rays, while in Section 6.1.3,using some properties of loop segments given in Section 6.1.2,this solution is analytically continued in a domain P nω .

Basic constructions using curve arcs and rays
Let, in (18), where Let be fixed, where ω is an arbitrarily small fixed number such that (we refer to (33) and (34)) p is fixed and satisfies ( ) ∈ p σ 1, .By the Cauchy-Kovalevskaya theorem, the initial condition where , defines a unique analytic solution ( ) = w w z n 0 to equation (1) in a neighbourhood of z 0 .Our aim is to enlarge the domain of the existence of such a solution in a subset of the neighbourhood of the point = z 0 (in the complex plane or on a Riemann surface), preserving the property: by the method of analytic continuation using the previously defined curves and rays.In Example 1, the following constructions are used as shown in Figure 3.
By Lemma 2, part (ii), with = = t t ω * , the analytic continuation of solution ( ) preserves the property (54).We refer to Remark 2, where an explanation is given of the behaviour of solutions as → + t 0 .In such a case, the solution ( ) w z n 0 does not preserve the property (54), , and the blow-up growing to infinity is of an exponential type.
Consider segments of two loops of the curve (8), where either (we refer to (45) and ( 46)), r is a fixed number satisfying and the angle φ varies within the limits given by the value = s 0 in (10), i.e. (57) In (56), the number r 2 will be specified later, and the number r 1 is defined as: for an arbitrary but fixed ( ) Since the construction of r 2 in the following does not depend on r 1 , such a definition of r 1 is correct.For the "+" and " − " cases, inequality (57) has the forms: and respectively.Assumption ( 21) is satisfied since Therefore, for ν 0 defined by (45) or by (46), we have, by (22), . Obviously, by (8), Vanishing and blow-up solutions  15 Assume that (63) holds.Then, both the above-mentioned segments intersect the ray (18), where ν is given by (50) and t satisfies (55), at the point , provided that (deducting from (55)) where .
If the parameter c is fixed (being sufficiently large) and if ω is sufficiently small, then, simultaneously, the set of values | | z * n satisfying (64) will be non-empty and (63) will hold as well.This is shown in the following.Let ω be fixed and sufficiently small.Define the number as the solution of the equation: This maximum will be achieved for . Then, from (65), we derive From formulas (67) and (68) we deduce the following.If, for an increasing c and decreasing ω, the product − cω σ 1 remains the same, then the values r 2 , r 1 will be fixed as well.Inequality (63) will be satisfied if with (63) still holding for all [ ] ∈ r r r , 1 2 .Obviously, it is possible to apply the above-mentioned reasoning even after replacing ω with an < ω ω  , and (63) holds.These loops are defined by the formula: , where φ varies within domains (57).Moreover, it is easy to show that the arcs (70) are symmetric with respect to the given ray.
Example 1. Assume that, in equation ( 1), we have = σ 2, = a 1, and = M 2. Let = ρ 1 and = k 0. The following constructions are visualized by Figure 3.In accordance with (23), put and, by (22), we can set (in the formula, the dependence on r is emphasized and notation ( ) 0.02 sin 0.02 sin .
r 0 0 0 In accordance with (56), define a range for r by numbers r 1 and r 2 satisfying = ∕ r π 3 2 and = ∕ r π 6 1 Although we have defined the number r 2 "ad hoc" (not using its definition (66)) to better illustrate all computations, this choice is in accordance with (66) as shown in the following.Let us take a number ( ) and the solution of equation ( 66) gives the value = ∕ r π 3 .Inequality (63) holds as well since The curve arcs (70) are defined by the angles φ within domains (71) and (72), i.e.
that are parts of domains ( 60) and (61), i.e. , is analytically continued.Note that, for different values of ≠ n 0, we obtain identical constructions.

Auxiliary lemma
The following Lemma 3 and Remark 5 on the mutual positions of different segments of the loops of curve (8) will be used in further constructions.Instead of r 1 and r 2 in their formulations, arbitrary values r* 1 and r* 2 satisfying ( ) can be used.For an illustration, we refer to Example 2 and Figure 4.
Lemma 3. Let numbers r 1 and r 2 be fixed such that Consider two curve arcs ( ) of (8) defined by the following formulas: where c 1 and c 2 are the positive constants, ( )   (85) Now, on the interval ( ) ( ) ≤ < + ψ n φ ν n r , 1 , we will investigate the properties of the function: By (85), Due to (79), we have . Then, and, analysing (87), we have < c c

and, consequently, ( ( )) ′ = > ψ n
A 0. Computation of the second-order derivative leads to Therefore, ( ) φ satisfies the following Cauchy initial problem for a differential second-order equation: The general solution of ( 88) is where, for the specification of arbitrary constants C 1 and C 2 by the initial conditions, we use the equations The solution of the problem (88) is The angle φ varies within the interval indicated in (84), which implies 1 For such values of φ, we have ( ) − < σ φ sin 1 0; therefore, for ( ) φ expressed by ( 86) and ( From this inequality, we deduce that ( ) > φ 0 also holds in the same interval, where .
This inequality is equivalent with (84).□ Remark 5. A property similar to that formulated in Lemma 3, i.e. the inequality can be proved in much the same way, if the following segments of loops are considered instead of (80) and (81): Example 2. We use some constructions from Example 1 to illustrate Lemma 3 and Remark 5. We refer to Figure 4 where the two red and two green loop segments can be seen passing through the point = z te iπ , where = ∕ ≐ t 2 10,000 0.00014.Put = c 2,500 6 where while the outer green one is defined by the formula: where The inner red segment is defined by the formula: where while the outer red one is defined by the formula: where .
This is in accordance with the assertions of Lemma 3 and Remark 5.
Remark 6.Let us point out another property of the mutual position of different segments of loops of the curve (8), which is obvious.If two segments , where The same property holds if ( ) . This property is visualized in Figure 5, where = r r 2 with details described in Example 3.
Example 3. In the following, some constructions from Example 1 are used to illustrate the assertions of Remark 6.We refer to Figure 5 where the two red and two green loop segments are visualized passing through the points = z ε e ** p iπ and = z ωe iπ .The inner green loop segment is defined by the formula: while the outer green one is defined by the formula: The inner red loop segment is defined by the formula: while the outer red one is defined by the formula: 2 We see that ig og 0 0 and Vanishing and blow-up solutions  23 This is in accordance with the assertions of Remark 6. , defined by initial problem (53) with the initial point ( ) w z 0 , where = z z ωn 0 is determined by (51), on a domain P nω , defined as a domain covered by all the above-mentioned loop segments of the curve (8).The boundaries for t and r are the following.The value t varies within the interval (69), where ω is sufficiently small and ε** p is defined in such a way that the segments of loops (70) of curve ( 8) are defined by angles φ satisfying (71) and ( 72).The value r varies as indicated in (56) with r 2 defined by (66) and with r 1 defined by (58).Then, among others, the inequality holds.From the method of construction and ( 56), ( 71), (72), and (73), it follows that the domain P nω is simply connected and lies in the sector ( ) φ n , centred at the point = z 0, defined as: The sector ( ) φ n is located either in the complex plane or in a Riemann surface and does not contain the origin.Applying (56), the length of the interval for φ defining ( ) 2 Assume now that n is not fixed.Tracing carefully computations in Parts 6.1.1 and 6.1.2,we see that these do not depend on any of the values n.So, for every n, the above-mentioned considerations are correct.This means that two domains P n ω 1 and P n ω 2 , where ≠ n n 1 2 , are geometrically identical (having identical forms).These domains can differ only by their location in the complex plane or in a Riemann surface and, in such a case, they do not intersect.The following obvious statement is true about the number of domains P nω with different locations (two domains P n ω for ≠ n n Now, we describe in detail the construction of the domain P nω based on the properties of the curve segments given in Lemma 3, Remark 5, and Remark 6.For a visualization, we refer to Figure 6, which illustrates the relevant constructions of Example 4. The domain P nω contains all points between the following two curves including all boundary points except for the point = z 0. The boundary of P nω is formed by two closed, simple, and continuous curves "embedded" in each other (below called the inner and outer boundaries) with a unique common point = z 0. The inner boundary is formed by two segments of loops (70) specified by (for the definitions of ± ν 0 and ± ν , we refer to formulas (45)-( 48)): and passing through the point , determined by (51), while the outer boundary is defined by two segments of loops (70) specified by: (93) and passing through the point For the definition of ε** p , we refer to the explanation accompanying inequalities (69).This construction is correct since, by Lemma 3 and Remark 5, arcs defined by (90) and (92) have no intersection for ( ) ( ) < < + ψ n φ ν n r , 1 and arcs defined by (91) and (93) have no intersection for ( ) ( ) Example 4. Using several constructions of Example 1 again, we will construct the domain P ω 0 .The inner boundary of P ω 0 consists of two segments of loops (70) passing through the point = z ωe iπ with = r r 1 and = c 5,000 3 .The red segment in Figure 6 is defined as: 5,000 3 , while the green one in Figure 6, is defined as: where while the green one in Figure 6 is defined as: e π φ e cos 0, cos 4 3 5,000 3 , where The solution ( ) w z n 0 can be analytically continued from the domain P nω to certain domains + Ω nω and − Ω nω intersecting P nω and lying on opposite sides of P nω .Let us give their description.Domains + Ω nω and − Ω nω are bounded by curve arcs of type (8) as well as by some parts of segments of rays ( 18) that correspond to certain values of the parameters described in the following.For = σ 2, + Ω ω 0 , and − Ω ω 0 , they are shown in Figures 7 and 8. Regarding the corresponding constructions, we refer to Example 5.

Construction of domain + +
Ω nω A domain + Ω nω is constructed using (i) the loop segments of curve (8) with ( , with suitable parameters c and with the domain for φ (ii) parts of the rays (18), where the angle ν varies within the same domain, i.e.
Now, let us describe the construction in detail.We will generate the boundary of + Ω nω using segments ( ) of two curve loops (8) and one segment ( ) of a ray (18).Their construction is shown in the following.Consider a segment of loop (70) passing through the point = z ε e ** p iπ and specified by (92), i.e.
where ( ) ( ) ≤ < + ψ n φ ν n r , 2 and the parameter c ** ε p is the solution of the equation: Next, define the value ( ( )) ** The first segment ( ) is defined as a segment of the loop passing through the point ( ( ))  Vanishing and blow-up solutions  where the parameter c I is determined from the equation ( ** The second segment ( ) is defined as a segment of the loop passing through the point ( ( )) having the domain shown in the following, i.e.
where the value of parameter c II is determined from the equation ( The segment ( ) of a ray (18) defined as: is a part of the boundary as well.By construction, | ( ( . Obviously, the intersection ∩ + P Ω nω nω is nonempty containing an open set.For the analytical continuation, it is sufficient to use the loop segments, given by the equations: for each ≥ c c I and , for each ≥ c c II .These segments cover all the domain + Ω nω .To show this, Remark 6 can be modified easily to the case considered. Here are some properties of segments of the loops of curve ( 8) with ν 0 defined by (95) and segments of rays lying in + Ω nω that form the analytical continuation of the solution ( ) w z n 0 from the domain P nω to the domain + Ω nω .The closures of segments of the curve loops and rays with the domains (96) and (97) have two intersection points.The first one is at = z 0 since The second one, provided that ( ) Without loss of generality, we can assume that > c 0 is so large that < t ρ * .The analytic continuation along segments of the loop or ray has the following properties.Since is nonzero for every fixed r and ν and its minimum value, as it follows from (108), is positive and Thus, in Lemma 2, irrespective of the value ν within interval (97), we can set We conclude that, within the domain + Ω nω , the exponential estimate in (49) is a consequence of Inequality (110).

-
Similarly, the domain − Ω nω is generated using (i) segments of the curve loops (8) with suitable parameters c, with where ( ) , is defined by (48), [ ] ∈ r r r , 1 2 and with the domain for φ (ii) parts of the rays (106), where the angle ν varies within the same domain: and The following computations is much the same as those in Section 6.2.1.We omit the details defining the boundary of − Ω nω by segments of two loops and one segment of a ray as follows: and , , , where c I is determined from equation (102): , (116) .Obviously, the intersection ∩ − P Ω nω nω is nonempty containing an open set.For the analytical continuation, it is sufficient to use segments given by the equations: and , , , where ≥ c c I and and where ≥ c c II .These arcs cover all the domain − Ω nω .The following are the properties of segments of loops of the curve (8) with ν 0 defined by (111) and segments of rays lying in − Ω nω and forming the analytical continuation of the solution ( ) w z n 0 from the domain P nω to the domain − Ω nω .The closures of the segments of loops and rays with domains (96) and ( 97) have two points of intersection.The first one is the point = z 0 since The second one, provided that * and t* is defined by (107).The above-mentioned analytic continuation along the loops of the curves has the following properties.Since   The parameter c I is determined from equation (102), i.e.
( where Finally, by equation ( 106), the segment of the ray ( ) is defined by:  Finally, by equation ( 117), the segment of the ray ( ) − z t r10 is defined by: The domain − Ω ω 0 and its boundary can be seen in Figure 8.The union ∪ + −

P n P n 1
From the previous considerations, we conclude that the solution ( ) w z n 0 is analytically continued on the domain: The last property follows from (47), ( 48), ( 56), (96), and (112) because, in the opposite case, the inequality  (this intersection is highlighted in blue), the following will be true as well: . Note that, for = σ 2 or for a rational = ∕ σ m m In both cases, we assume that the parameter c is identical and sufficiently large.A suitable value of c can be determined, e.g., as the solution of equation: Assuming that φ varies within the domain described by the inequalities: where ( ) ∈ γ γ 0, 0 is fixed, consider the behaviour of solutions of integral curves given by (1) on ℓ ( ) n 1 and ℓ ( ) n 2 with respect to a fixed cylinder ( ) λ defined by (24), i.e. with respect to the cylinder:

Example
Let, in an equation (1), = σ 3, = a 1, and = M 2. Let = ρ 1 and = k 0. By (50), we have and there are two different values for = n 0, 1 in the complex plane , ( ) = ∕ ψ π 0 2 and ( ) = ∕ ψ π 1 3 2.The respective constructions can be seen in Figure 14.In accordance with (23), put and the solution of equation ( 66) gives the value = ∕ r π 6 .For ± ν 0 defined by formulas ( 45) and ( 46) with = n 0, 1, we have, by (50), For completeness, we compute the angles, defined by ( 47) and ( 48):   while the green one is defined as: The curve ( ) is defined as (we refer to formulas (101) and ( 102)) where the value of the parameter c I is given by the equations: where the value of parameter c II is given by the equation:

2 . 1 ,
Following the above-mentioned recommendation, if we replace ω with ω 1 and c with c 1 we can derive the same solution.Since, for = r r 1 ,

2 .
3 shows the segments of the curve loops defined by (70), i.e. 000 0.00014, and = z ωe iπ and the domains for φ are given by inequalities (78) with = = ∕ r For the red segment, the computations with ( ) while, for the green one, the values ( ) Along these segments, the solution of the initial problem (see (51) and (53)):

2 .1
The inner green segment is defined by the formula:Vanishing and blow-up solutions  21
Let n be fixed.Varying t and r in admissible boundaries, by the theorem on the existence of a unique solution to the initial problem, we obtain an analytic continuation of the solution ( ) w z n 0 the Riemann surface of the logarithmic function.

1
The outer boundary of P ω 0 consists of two segments of loops (70) passing through the point = z ε e ** p iπ with = r r 2 and = ∕ c 5,000 3 .The red segment in Figure 6 is defined as: 6: To Example 4domain P ω 0 .

0 1
having the domain shown in the following, i.e.
) of Lemma 1 is applicable.Therefore, on the given arcs,

2 .
By Formulas (110) and (118), the convergence to zero has an exponential character.Example 5. Let = n 0 and = σ We utilize Examples 1 and 4 to construct the domain + Ω ω 0 .We use the green loop segment, shown in Figure 6, of the outer boundary of P ω 0 , passing through the point = z ε e ** p iπ (for the exact value of ε** p , we refer to (77)) having equation (94), i.e.

1
must hold.Simplifying (122), we obtain < r 0, which contradicts (56), i.e. contains an open set and, by construction, is a simply connected domain.

2 , 2 ,
there exists only a single domain ( ) P n .All others have identical forms and locations.Since = σ A circle neighbourhood of the origin is drawn in Figure 11.Its radius r equals | ( )|

Remark 8 . 1 and ℓ ( ) 0 2 .
We use Example 5 again to visualize curves ℓ ( ) 0 Knowing that domains of the type ( ) P n have the same form, without loss of generality, we can use the previous constructions.Therefore, for clarity, we do not replace the value = n 0 with = n 1 (except for ( ) P 1 ).Both curves are visualized in Figures12 and 13(highlighted in blue and in red), where, by formula (124), ( ) formulas (119) and (120).Solving equation (130), where = n 0, leads to = c c II .The formulas for ℓ

o 8 1 2 . 2 1 2 .
The inner boundary of P ω 0 consists of two segments of loops (70) passing through the point = ∕ z ωe iπ 2 with = r r 1 .The red segment is defined as: The outer boundary of P ω 0 consists of two segments of loops (70) passing through the point = ∕ The red segment is defined as: The inner boundary of P ω 1 consists of two segments of loops (70) passing through the point = ∕ z ωe iπ 3 2 with = r r 1 .The red segment is defined as: The outer boundary of P ω 1 consists of two segments of loops (70) passing through the point = ∕ The red is defined as:

2 Figure 15
Figure 15 shows a circle neighbourhood of the origin in where there exist solutions with the properties indicated in Theorem 1.The radius of the circle | ( )| = = ⋅ + − r z 0 1.5 10 21 as (we refer to formulas (103) and (104)):

1 ,
we refer to Figure15.Inequality (49) (being formally the same as Inequality (123)) now has the form: