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BY 4.0 license Open Access Published by De Gruyter February 5, 2024

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

  • Josef Diblík EMAIL logo and Miroslava Růžičková

Abstract

A singular nonlinear differential equation

z σ d w d z = a w + z w f ( z , w ) ,

where σ > 1 , is considered in a neighbourhood of the point z = 0 located either in the complex plane C 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 C { 0 } , and that the function f is analytic in a neighbourhood of the origin in C × C . 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 C or in the Riemann surface of either a rational or a logarithmic function. Within this domain, the property lim z 0 w ( 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.

MSC 2010: 34M35; 34M30; 34M10; 34A25

1 Introduction

A singular nonlinear differential equation

(1) z σ d w d z = a w + z w f ( z , w ) ,

where σ > 1 , is considered in a neighbourhood of the point z = 0 either in the complex plane C if σ N { 1 , 2 , } is a natural number, in a Riemann surface of a rational function if σ Q N is a rational number, or in the Riemann surface of logarithmic function if σ I R Q is an irrational number. In (1), z is an independent variable, w = w ( z ) , and a C { 0 } . The function f : D C is assumed to be analytic in a neighbourhood D of the point ( 0 , 0 ) C × C having the form:

(2) D = { ( z , w ) C × C : z < ρ , w < e k } ,

where ρ ( 0 , ) and k ( , ) are the fixed constants such that there exists a finite number M > 0 satisfying

M sup ( z , w ) D f ( z , w ) .

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 C if σ N , in a Riemann surface of a rational function if σ Q N , or in the Riemann surface of the logarithmic function if σ I . 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

(3) z w = h ( z , w ) and w ( 0 ) = 0

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 ) = 0 can be constructed as a power series convergent in a neighbourhood of z = 0 . In [31], assuming h ( z , 0 ) = o ( z 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 w ( 0 , 0 ) is singular, systems

(4) z σ y = h 1 ( z , y , w ) ,

(5) z w = h 2 ( z , y , w ) ,

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 [1317,1923,2527,30] 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 N , equation (1) is considered in the complex plane C , while, in the case of σ = m 1 m 2 Q N , where m 1 > m 2 > 1 and m 1 , m 2 N are relatively prime, we use the Riemann surface of the function w = z 1 m 2 and, if σ I , 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 C or in the aforementioned Riemann surfaces).

In the right-hand side of (1), a “perturbation” z w f ( z , w ) of a linear equation z σ w = a w 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 ) z w f ( 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 14) are formulated. Their proofs are given in Section 6. Since a major part of the proofs of Theorems 13 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 15 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.

2 Preliminaries

Consider an initial problem

(6) w = F ( z , w ) , w ( z 0 ) = w 0 ,

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 0 , w 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 C 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 C , respectively. Let 1 2 and 1 2 . If f 1 f 2 in 1 2 , f 2 is called an analytic continuation of f 1 to 2 , and vice versa. The analytic continuation is unique.

3 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.

3.1 On the coefficient a in (1)

Let the complex coefficient a in (1) be given in its exponential form:

a = a e i θ , θ [ 0 , 2 π ) .

Then, a substitution, geometrically expressing a rotation,

(7) z = v e i θ ( σ 1 ) , v C { 0 } ,

where v is a new independent variable, changes equation (1) into one of a similar form:

v σ d w d v = a w + v w f ( v e i θ ( σ 1 ) , w ) e i θ ( σ 2 ) ( σ 1 ) ,

with a positive coefficient a instead of the previous complex coefficient a . In the following investigation, we will assume that z w f ( z , w ) in (1) 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:

v w f ( v e i θ ( σ 1 ) , w ) e i θ ( σ 2 ) ( σ 1 ) = v w f ( v e i θ ( σ 1 ) , 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, whenever computations in this article depend on a , we assume that it is a positive number.

3.2 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:

(8) z ( φ ) = cos ( ν 0 ( σ 1 ) φ ) ( σ 1 ) c 1 ( σ 1 ) e i φ ,

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

(9) cos ( ν 0 ( σ 1 ) φ ) > 0 ,

and, moreover, as we need z ( φ ) < ρ , that

cos ( ν 0 ( σ 1 ) φ ) ( σ 1 ) c 1 ( σ 1 ) < ρ .

Inequality (9) will hold if and only if

(10) 1 σ 1 ν 0 π 2 + 2 s π < φ < 1 σ 1 ν 0 + π 2 + 2 s π , s = 0 , ± 1 ,

and, obviously,

(11) z ( φ k ) = 0 , k = 0 , ± 1 ,

for

(12) φ k 1 σ 1 ν 0 + π 2 + π k , k = 0 , ± 1 , .

The closure of each loop of the curve (8) separated by an inequality (10) is a closed curve passing through the origin. If σ N , then there are σ 1 different curve arcs (8) lying in the complex plane C , defined, e.g., by: s = 0 , , σ 2 in (10). If σ = m 1 m 2 Q N , where m 1 > m 2 > 1 and m 1 , m 2 N are relatively prime, then there are m 2 1 different curve arcs (8) on the Riemann surface of the function w = z 1 m 2 defined, e.g., by s = 0 , , m 2 2 in (10). If σ I , 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.

Figure 1 shows different loops of (8) in the z plane as defined by angles φ satisfying (10), where s = 0 , 1 , 2 , ν 0 = 3 π 2 , and related to the values c = 1 (green loops), c = 2 (brown loops), and c = 3 (blue loops).

Figure 1 
                  Loops of (8) specified by 
                        
                           
                           
                              
                                 
                                    ν
                                 
                                 
                                    0
                                 
                              
                              =
                              3
                              π
                              ∕
                              2
                           
                           {\nu }_{0}=3\pi /2
                        
                     , 
                        
                           
                           
                              σ
                              =
                              4
                           
                           \sigma =4
                        
                     , 
                        
                           
                           
                              s
                              =
                              0
                              ,
                              1
                              ,
                              2
                              ,
                           
                           s=0,1,2,
                        
                      and 
                        
                           
                           
                              c
                              =
                              3
                              ,
                              2
                              ,
                              1
                           
                           c=3,2,1
                        
                     .
Figure 1

Loops of (8) specified by ν 0 = 3 π 2 , σ = 4 , s = 0 , 1 , 2 , and c = 3 , 2 , 1 .

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

(13) d w ( z ( φ ) ) d φ = d w ( z ( φ ) ) d z ( φ ) d z ( φ ) d φ = e i ν 0 z σ ( φ ) w [ a e i ν 0 + z ( φ ) e i ν 0 f ( z ( φ ) , w ) ] d z ( φ ) d φ ,

where

(14) d z ( φ ) d φ = 1 ( σ 1 ) ( ( σ 1 ) c ) 1 ( σ 1 ) ( cos ( ν 0 ( σ 1 ) φ ) ) ( 1 ( σ 1 ) ) 1 × ( sin ( ν 0 ( σ 1 ) φ ) ) ( σ 1 ) e i φ + cos ( ν 0 ( σ 1 ) φ ) ( σ 1 ) c 1 ( σ 1 ) i e i φ = z ( φ ) sin ( ν 0 ( σ 1 ) φ ) cos ( ν 0 ( σ 1 ) φ ) + i = i z ( φ ) cos ( ν 0 ( σ 1 ) φ ) e i ( ν 0 ( σ 1 ) φ ) .

Finally, using (8), (13), and (14),

(15) d w ( z ( φ ) ) d φ = e i ν 0 z σ ( φ ) w [ a e i ν 0 + z ( φ ) e i ν 0 f ( z ( φ ) , w ) ] i z ( φ ) cos ( ν 0 ( σ 1 ) φ ) e i ν 0 ( e i φ ) σ 1 = w [ a e i ν 0 + z ( φ ) e i ν 0 f ( z ( φ ) , w ) ] i cos ( ν 0 ( σ 1 ) φ ) ( z ( φ ) e i φ ) 1 σ = w [ a e i ν 0 + z ( φ ) e i ν 0 f ( z ( φ ) , w ) ] i cos ( ν 0 ( σ 1 ) φ ) cos ( ν 0 ( σ 1 ) φ ) ( σ 1 ) c 1 = i ( σ 1 ) c w [ a e i ν 0 + z ( φ ) e i ν 0 f ( z ( φ ) , w ) ] cos 2 ( ν 0 ( σ 1 ) φ ) .

Let w = w ( z ( φ ) ) in (15) be represented by its algebraic form, i.e. w ( z ( φ ) ) = y 1 ( φ ) + i y 2 ( φ ) with y 1 ( φ ) and y 2 ( φ ) being the real and imaginary parts of w ( z ( φ ) ) , respectively. Then,

d w ( z ( φ ) ) d φ = d y 1 ( φ ) d φ + i d y 2 ( φ ) d φ = i ( σ 1 ) c w [ a e i ν 0 + z ( φ ) e i ν 0 f ( z ( φ ) , w ) ] cos 2 ( ν 0 ( σ 1 ) φ ) = i ( σ 1 ) c ( y 1 ( φ ) + i y 2 ( φ ) ) a ( cos ν 0 i sin ν 0 ) cos 2 ( ν 0 ( σ 1 ) φ ) + i ( σ 1 ) c ( y 1 ( φ ) + i y 2 ( φ ) ) [ Re ( z ( φ ) e i ν 0 f ( z ( φ ) , w ) ) + i Im ( z ( φ ) e i ν 0 f ( z ( φ ) , w ) ) ] cos 2 ( ν 0 ( σ 1 ) φ ) .

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):

(16) y 1 = ( σ 1 ) c cos 2 ( ν 0 ( σ 1 ) φ ) ( ( a sin ν 0 ) y 1 ( a cos ν 0 ) y 2 ) + ( σ 1 ) c cos 2 ( ν 0 ( σ 1 ) φ ) ( y 2 Re ( z e i ν 0 f ( z , w ) ) y 1 Im ( z e i ν 0 f ( z , w ) ) ) ,

(17) y 2 = ( σ 1 ) c cos 2 ( ν 0 ( σ 1 ) φ ) ( ( a cos ν 0 ) y 1 + ( a sin ν 0 ) y 2 ) + ( σ 1 ) c cos 2 ( ν 0 ( σ 1 ) φ ) ( y 1 Re ( z e i ν 0 f ( z , w ) ) y 2 Im ( z e i ν 0 f ( z , w ) ) ) .

3.3 Real system equivalent to (1) on a system of rays

Consider rays running from the origin

(18) z ( t ) = t e i ν , 0 < t < ρ , ν = const ,

and transform equation (1) along these into a system of two real equations. Since

d w d z = d w d t d t d z = d w d t e i ν ,

(1) can be written as:

t σ d w d t = e i ( σ 1 ) ν w ( a + t e i ν f ( t e i ν , w ) ) .

Assuming w ( z ( t ) ) by its algebraic form w ( z ( t ) ) = x 1 ( t ) + i x 2 ( t ) , we derive

t σ d w d t = t σ d x 1 ( t ) d t + i d x 2 ( t ) d t = e i ( σ 1 ) ν w ( z ( t ) ) ( a + t e i ν f ( t e i ν , w ( z ( t ) ) ) ) = e i ( σ 1 ) ν ( x 1 ( t ) + i x 2 ( t ) ) ( a + t e i ν f ( t e i ν , w ( z ( t ) ) ) ) = ( cos ( σ 1 ) ν i sin ( σ 1 ) ν ) ( x 1 ( t ) + i x 2 ( t ) ) ( a + Re ( t e i ν f ( t e i ν , w ( z ( t ) ) ) + i Im ( t e i ν f ( t e i ν , w ( z ( t ) ) ) ) ) ) .

Functions x 1 and x 2 satisfy the following system of ordinary differential equations:

(19) t σ x 1 = a ( cos ( σ 1 ) ν ) x 1 + a ( sin ( σ 1 ) ν ) x 2 + ( cos ( σ 1 ) ν ) ( x 1 Re ( t e i ν f ) x 2 Im ( t e i ν f ) ) + ( sin ( σ 1 ) ν ) ( x 1 Im ( t e i ν f ) + x 2 Re ( t e i ν f ) ) ,

(20) t σ x 2 = a ( sin ( σ 1 ) ν ) x 1 + a ( cos ( σ 1 ) ν ) x 2 + ( cos ( σ 1 ) ν ) ( x 1 Im ( t e i ν f ) + x 2 Re ( t e i ν f ) ) + ( sin ( σ 1 ) ν ) ( x 1 Re ( t e i ν f ) + x 2 Im ( t e i ν f ) ) .

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 0 m π , m Z , i.e.

(21) sin ν 0 0 .

In the following, we consider system (16), (17) in the domain

Ω cr { ( φ , y 1 , y 2 ) R 3 : 0 < z ( φ ) < ρ , y 1 2 + y 2 2 < e 2 k } ,

where k and ρ are the same as in the definition of the domain D 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 cos ( ν 0 ( σ 1 ) φ k ) = 0 . 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

(22) ε ( ν 0 ) ε 0 sin ν 0 ,

where ε 0 is a fixed number satisfying

(23) 0 < ε 0 < min ρ , a 2 M .

Define cylinders C ( λ ) as sets

(24) C ( λ ) { ( φ , y 1 , y 2 ) Ω cr : y 1 2 + y 2 2 = e 2 λ } , λ = const , λ ( , k ) .

Lemma 1

Assume that φ varies in a fixed domain defined by (10). Let c and ν 0 , satisfying (21), be fixed such that

(25) 0 < z ( φ ) ε ( ν 0 ) .

Then, any integral curve ( φ , y 1 ( φ ) , y 2 ( φ ) ) of system (16), (17) intersecting at a value φ = φ * a cylinder C ( λ ) , i.e. if ( φ * , y 1 ( φ * ) , y 2 ( φ * ) ) satisfy

y 1 2 ( φ * ) + y 2 2 ( φ * ) = e 2 λ ,

behaves as follows. The integral curve ( φ , y 1 ( φ ) , y 2 ( φ ) ) , as φ increases, is passing

  1. from domain

    (26) y 1 2 + y 2 2 > e 2 λ

    into domain

    (27) y 1 2 + y 2 2 < e 2 λ

    if sin ν 0 < 0 and inequality

    (28) w ( z ( φ ) ) 2 = y 1 2 ( φ ) + y 2 2 ( φ ) < e 2 λ < e 2 k

    holds for every admissible φ > φ * .

  2. from domain (27) into domain (26) if sin ν 0 > 0 and inequality (28) holds for every admissible φ < φ * .

Proof

Consider the behaviour of integral curves of system (16), (17) intersecting cylinders C ( λ ) . To do this, compute the scalar product ( N , T ) at an arbitrary point of cylinder C ( 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

N = ( 0 , y 1 , y 2 ) and T = 1 , d y 1 d φ , d y 2 d φ ,

we have

(29) ( N , T ) = y 1 d y 1 d φ + y 2 d y 2 d φ = ( σ 1 ) c e 2 λ ( a sin ν 0 Im ( z e i ν 0 f ( w , z ( φ ) ) ) ) cos 2 ( ν 0 ( σ 1 ) φ ) .

We will show that (29) implies

(30) sgn ( N , T ) = sgn sin ν 0

whenever (25) holds. Indeed, for z = Re z + i Im z and

e i ν 0 f ( z , w ) = Re ( e i ν 0 f ( z , w ) ) + i Im ( e i ν 0 f ( z , w ) ) ,

we have

Im ( z e i ν 0 f ( z , w ) ) = Re z Im ( e i ν 0 f ( z , w ) ) + Im z Re ( e i ν 0 f ( z , w ) ) .

Therefore, from (22), (23), and (25), it follows

Im ( z e i ν 0 f ( z , w ) ) z M + z M = 2 z M 2 ε ( ν 0 ) M 2 ε 0 sin ν 0 M < a sin ν 0 .

Then, taking into account that

( σ 1 ) c e 2 λ cos 2 ( ν 0 ( σ 1 ) φ ) > 0 ,

from (29), we derive

sgn ( N , T ) = sgn ( a sin ν 0 Im ( z e i ν 0 f ( z , w ) ) ) = sgn sin ν 0

and (30) holds. Finally, we remark that (22), (23), and (25) imply z ( φ ) < ρ , i.e. the values of z used are within the domain D defined by (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.

Figure 2 
                  Loops of (8) specified by 
                        
                           
                           
                              
                                 
                                    ν
                                 
                                 
                                    0
                                 
                              
                              =
                              π
                              ∕
                              2
                           
                           {\nu }_{0}=\pi /2
                        
                      and 
                        
                           
                           
                              σ
                              =
                              3
                           
                           \sigma =3
                        
                      if 
                        
                           
                           
                              ε
                              
                                 (
                                 
                                    
                                       
                                          ν
                                       
                                       
                                          0
                                       
                                    
                                 
                                 )
                              
                              =
                              1
                           
                           \varepsilon \left({\nu }_{0})=1
                        
                     , 
                        
                           
                           
                              c
                              =
                              3
                           
                           c=3
                        
                     , 2, 1, 0.5, and 0.3.
Figure 2

Loops of (8) specified by ν 0 = π 2 and σ = 3 if ε ( ν 0 ) = 1 , c = 3 , 2, 1, 0.5, and 0.3.

4.2 Behaviour of solutions along system of rays (18)

Assume, in the definition of rays (18),

ν 1 σ 1 π 2 + m π , m = 0 , ± 1 , ,

i.e.

(31) cos ( σ 1 ) ν 0 .

In this part, we will examine the behaviour of solutions to Systems (19) and (20) in the domain:

Ω r s { ( t , x 1 , x 2 ) R 3 : 0 < t < ρ , x 1 2 + x 2 2 < e 2 k } ,

where k and ρ are the same as in the definition of the domain D 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 r s . Define cones K ( δ ) as sets

(32) K ( δ ) ( t , x 1 , x 2 ) Ω r s : x 1 2 + x 2 2 = δ 2 exp 2 a cos ( σ 1 ) ν t p 1 ,

where δ and p are the fixed parameters satisfying δ > 0 and p ( 1 , σ ) . Put

(33) t ν , p * ε p * cos ( σ 1 ) ν 1 μ ,

where μ satisfies 0 < μ < min { 1 , σ p } and ε p * is a positive number satisfying

(34) 0 < ε p * < min ρ , a 4 M ρ 1 μ + ( p 1 ) a ρ σ p μ 1 μ .

If an integral curve ( t , x 1 ( t ) , x 2 ( t ) ) of the system (19), (20) intersects a fixed cone K ( δ ) at a point t = t * ( 0 , t ν , p * ] , then

w ( z ( t * ) ) 2 = x 1 2 ( t * ) + x 2 2 ( t * ) = δ 2 exp 2 a cos ( σ 1 ) ν ( t * ) p 1 ,

where δ satisfies

(35) δ < exp k + a cos ( σ 1 ) ν ( t * ) p 1

by the definitions of K ( δ ) and Ω r s .

Lemma 2

Let ν satisfying (31), p ( 1 , σ ) , δ satisfying (35), and μ ( 0 , min { 1 , σ p } ) be fixed. Then, either (i) or (ii) in the following holds.

  1. If cos ( σ 1 ) ν > 0 , then an arbitrary integral curve ( t , x 1 ( t ) , x 2 ( t ) ) of Systems (19) and (20) intersecting the cone K ( δ ) at a point t = t * ( 0 , t ν , p * ] satisfies the inequality:

    (36) w ( z ( t ) ) 2 = x 1 2 ( t ) + x 2 2 ( t ) < δ 2 exp 2 a cos ( σ 1 ) ν t p 1 < e 2 k , if t ( 0 , t * ) .

    Moreover,

    (37) lim t 0 + w ( z ( t ) ) 2 = 0 .

  2. If cos ( σ 1 ) ν < 0 and ω ( 0 , t ν , p * ) , then an arbitrary integral curve ( t , x 1 ( t ) , x 2 ( t ) ) of the system (19), (20) intersecting the cone K ( δ ) at a point t = t * [ ω , t ν , p * ] satisfies the inequality:

    (38) w ( z ( t ) ) 2 = x 1 2 ( t ) + x 2 2 ( t ) δ 2 exp 2 a cos ( σ 1 ) ν t p 1 < e 2 k , if t [ t * , t ν , p * ]

    and

    (39) w ( z ) 2 = x 1 2 ( t ) + x 2 2 ( t ) > δ 2 exp 2 a cos ( σ 1 ) ν t p 1

    if t * ε < t < t * , where ε is a small positive number.

Proof

Let us investigate the behaviour of the integral curves intersecting cones K ( δ ) . Compute the scalar product ( N , t σ T ) at an arbitrary point of a fixed cone K ( 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

N = ( p 1 ) a ( cos ( σ 1 ) ν ) δ 2 t p exp 2 a cos ( σ 1 ) ν t p 1 , x 1 , x 2 , T = 1 , d x 1 d t , d x 2 d t ,

we have

(40) ( N , t σ T ) = ( p 1 ) a ( cos ( σ 1 ) ν ) δ 2 t σ p exp 2 a cos ( σ 1 ) ν t p 1 + x 1 t σ d x 1 d t + x 2 t σ d x 2 d t = δ 2 exp 2 a cos ( σ 1 ) ν t p 1 ( a cos ( σ 1 ) ν + V ) ,

where

V ( sin ( σ 1 ) ν ) Im ( t e i ν f ) + ( cos ( σ 1 ) ν ) ( Re ( t e i ν f ) + ( 1 p ) a t σ p ) .

Now, (40) will be used to determine sign ( N , T ) , which is obviously equal to sign ( N , t σ T ) . In the following, we show that

(41) sign ( N , T ) = sign cos ( σ 1 ) ν .

For V , we obtain

(42) V Im ( t e i ν f ) + Re ( t e i ν f ) + ( 1 p ) a t σ p = t μ ( Im ( t 1 μ e i ν f ) + Re ( t 1 μ e i ν f ) + ( p 1 ) a t σ p μ ) t μ ( ρ 1 μ Im ( e i ν f ) + ρ 1 μ Re ( e i ν f ) + ( p 1 ) a ρ σ p μ ) = t μ ( ρ 1 μ ( cos ν ) Im f + ( sin ν ) Re f + ρ 1 μ ( cos ν ) Re f ( sin ν ) Im f + ( p 1 ) a ρ σ p μ ) t μ ( 2 ρ 1 μ Im f + 2 ρ 1 μ Re f + ( p 1 ) a ρ σ p μ ) t μ ( 4 M ρ 1 μ + ( p 1 ) a ρ σ p μ ) .

Then, by (33), (34), and (42),

V t μ ( 4 M ρ 1 μ + ( p 1 ) a ρ σ p μ ) ( t ν , p * ) μ ( 4 M ρ 1 μ + ( p 1 ) a ρ σ p μ ) = ( ε p * ) μ cos ( σ 1 ) ν ( 4 M ρ 1 μ + ( p 1 ) a ρ σ p μ ) < a cos ( σ 1 ) ν ,

and formula (41) holds. Therefore,

(43) sign ( N , T ) = sign cos ( σ 1 ) ν = + 1 , if 0 < t t ν , p * and cos ( σ 1 ) ν > 0

and

(44) sign ( N , T ) = sign cos ( σ 1 ) ν = 1 , if 0 < t t ν , p * and cos ( σ 1 ) ν < 0 .

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

lim t 0 + δ 2 exp 2 a cos ( σ 1 ) ν t p 1 = ,

and this contradicts ( t , x 1 , x 2 ) Ω r s . Since δ in (32) satisfies inequality (35), if t varies within interval [ t * , t ν , p * ] , we have

δ 2 exp 2 a cos ( σ 1 ) ν t p 1 < exp 2 k + 2 a cos ( σ 1 ) ν ( t * ) p 1 exp 2 a cos ( σ 1 ) ν t p 1 e 2 k .

Therefore, any solution w = w ( z ( t ) ) = x 1 ( t ) + i x 2 ( t ) of (1) different from the zero solution and such that ( t , x 1 ( t ) , x 2 ( t ) ) intersects the cone (32) at t = t * [ ω , t ν , p * ] , satisfies (38) if t * t t ν , p * , and (39) if t satisfies t * ε < t < t * , 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 neighbourhood 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 neighbourhood 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

w ( t e i ν ) 2 > δ 2 exp 2 a cos ( σ 1 ) ν t p 1 ,

if t * ε < t < t * , where ε is a small positive number. However, if t 0 + , then the modulus w ( t e i ν ) is exponentially growing and lim t 0 + w ( t e i ν ) = .

5 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

O C { z C : 0 < z < ρ } ,

if the complex plane C is considered, or

O { z : 0 < z < ρ } ,

if a Riemann surface is considered. With the notation Arg z 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 C or on a Riemann surface .

Define

ψ ( n ) ( 2 n + 1 ) π σ 1 , n = 0 , ± 1 , ± 2 , ,

(45) ν 0 + ( n , r ) ( σ 1 ) ( ψ ( n ) + r ) = ( 2 n + 1 ) π + ( σ 1 ) r , r R ,

(46) ν 0 ( n , r ) ( σ 1 ) ( ψ ( n ) r ) = ( 2 n + 1 ) π ( σ 1 ) r , r R ,

and

(47) ν + ( n , r ) ψ ( n ) + r + π 2 ( σ 1 ) = 1 σ 1 ν 0 + ( n , r ) + π 2 ( σ 1 ) , r R ,

(48) ν ( n , r ) ψ ( n ) r π 2 ( σ 1 ) = 1 σ 1 ν 0 ( n , r ) π 2 ( σ 1 ) , r R .

Let r 1 be a sufficiently small fixed positive number satisfying the inequality:

r 1 < π 2 ( σ 1 ) .

Before formulating Theorems 13, the following comments may be useful. Theorem 1 uses domains O 1 n , n = 0 , , σ 2 if σ N . These domains, as we show in the proof, are located on the complex plane C in sectors between the angles ν ( n , r 1 ) and ν + ( n , r 1 ) 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

ν = ν 0 ( n , r 1 ) , ν ( n , r 1 ) < φ ψ ( n ) ,

while the second one by ν and φ such that

ν = ν 0 + ( n , r 1 ) , ψ ( n ) φ < ν + ( n , r 1 ) ,

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

lim φ ν ( n , r 1 ) z ( φ ) = 0 , lim φ ν + ( n , r 1 ) z ( φ ) = 0 ,

respectively.

Theorem 2 uses domains O 2 n , n = 0 , , m 1 m 2 1 if σ = m 1 m 2 Q N and m 1 , m 2 N are relatively prime. They have the same properties as domains O 1 n in Theorem 1 except for their location; these are located on the Riemann surface of the function w = z 1 m 2 .

Similarly, domains O 3 n , n = n 0 , , n 0 + N , where n 0 is an integer and N N { 0 } if σ I in Theorem 3, have the same properties as well located, however, on the Riemann surface of the logarithmic function.

All the above-mentioned domains O 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 .

Theorem 1

Let σ N . Then, there exist infinitely many analytic solutions w = w ( z ) to (1) defined on the complex plane C in a multiply-connected domain:

O 1 O C n = 0 σ 2 O 1 n ,

where O 1 n O C , n = 0 , , σ 2 are simply connected open domains such that

{ 0 } O 1 n , O ¯ 1 n 1 O ¯ 1 n 2 = { 0 } , s 1 , s 2 = 0 , , σ 2 , n 1 n 2 ,

and inequality w ( z ) < e k holds if z O 1 .

Theorem 2

Let σ = m 1 m 2 Q N where m 1 , m 2 N 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 1 m 2 in a multiply-connected domain

O 2 O n = 0 m 1 m 2 1 O 2 n ,

where O 2 n O , n = 0 , , m 1 m 2 1 are simply connected open domains such that

{ 0 } O 2 n , O ¯ 2 n 1 O ¯ 2 n 2 = { 0 } , n 1 , n 2 = 0 , , m 1 m 2 1 , n 1 n 2 ,

and inequality w ( z ) < e k holds if z O 2 .

Theorem 3

Let σ I . Then, for any integer n 0 and any natural number N N { 0 } , there exist infinitely many analytic solutions w = w ( z ) to (1) on a part ( n 0 , N ) of the Riemann surface of the logarithmic function defined by inequalities:

2 n 0 π σ 1 < Arg z < 2 ( n 0 + N + 1 ) π σ 1

in a multiply-connected domain

O 3 ( O ( n 0 , N ) ) n = n 0 n 0 + N O 3 n ,

where O 3 n ( O ( n 0 , N ) ) , n = n 0 , , n 0 + N are simply connected open domains such that

{ 0 } O 3 n , O ¯ 3 n 1 O ¯ 3 n 2 = { 0 } , n 1 , n 2 = n 0 , , n 0 + N , n 1 n 2 ,

and inequality w ( z ) < e k holds if z O 3 .

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 O 1 n , n = 0 , , σ 2 . Geometrically, the domain O 1 is a neighbourhood of the origin with σ 1 blow-up holes O 1 n , n = 0 , , σ 2 . In the case of Theorem 2, the point z = 0 is a unique point common to the closures of domains O 2 n , n = 0 , , m 1 m 2 1 . The domain O 2 is a neighbourhood of the origin on the Riemann surface of the function w = z 1 m 2 with m 1 m 2 blow-up holes O 2 n , n = 0 , , m 1 m 2 1 . If Theorem 3 can be applied, then the point z = 0 is a unique point common to the intersections of closures of domains O 3 n , n = n 0 , , n 0 + N . The domain O 3 is a subset of the neighbourhood of the origin on the Riemann surface associated with the logarithmic function with N + 1 blow-up holes O 3 n , n = n 0 , , n 0 + N .

Theorem 4

Let w = w ( z ) be an arbitrary solution mentioned in Theorems 13. Then,

lim z 0 , z O i w ( z ) = 0 ,

where i = 1 , 2 or i = 3 if Theorems 1, 2, or 3can be applied, respectively. Moreover, for any fixed p , 1 < p < σ , and for every z O i , i = 1 , 2 , 3 , we have

(49) w ( z ) min e k , δ exp a sin ( σ 1 ) r 1 z p 1 ,

where δ < exp ( k + ( a sin ( σ 1 ) r 1 ) ( t * ) p 1 ) , t * ( 0 , ε p * sin ( σ 1 ) r 1 ] and ε p * satisfies (34).

6 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 given loops and rays in a domain P n ω , while in Section 6.2, its analytic continuation is constructed to certain domains Ω n + and Ω n lying on different sides of P n ω . Finally, in Section 6.3, the analytic continuation is constructed to a domain P ( n ) . Each construction is demonstrated by an example. Sections 6.46.6 explain how some solutions can be analytically continued to domains P ( n ) P ( n + 1 ) , P ( n ) P ( n + 1 ) P ( n + 2 ) , or P ( n ) P ( n + N ) , respectively. This proves Theorems 13 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.

6.1 Analytic solutions on a domain P n ω

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 n ω .

6.1.1 Basic constructions using curve arcs and rays

Let, in (18),

(50) ν = ψ ( n ) ,

where n { 0 , ± 1 , ± 2 , } be fixed. Then,

cos ( σ 1 ) ν = cos ( σ 1 ) ψ ( n ) = cos π = 1 < 0 ,

and (31) holds. Let

(51) z 0 = z ω n ω exp ( i ψ ( n ) )

be fixed, where ω is an arbitrarily small fixed number such that (we refer to (33) and (34))

(52) 0 < ω < t ψ ( n ) , p * = ε p * cos ( σ 1 ) ψ ( n ) 1 μ = ε p * < ρ ,

p is fixed and satisfies p ( 1 , σ ) . By the Cauchy-Kovalevskaya theorem, the initial condition

(53) w n 0 ( z 0 ) = w n 0 ,

where w n 0 < e k , defines a unique analytic solution w = w n 0 ( z ) 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:

(54) w n 0 ( z ) < e k ,

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.

Figure 3 
                     Visualization – to Example 1.
Figure 3

Visualization – to Example 1.

By Lemma 2, part (ii), with t = t * = ω , the analytic continuation of solution w n 0 ( z ) for all z = t exp ( i ψ ( n ) ) , where t satisfies

(55) 0 < ω z = t ε p * ,

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 n 0 ( z ) does not preserve the property (54), lim t 0 + w n 0 ( t exp ( i ψ ( n ) ) ) = , and the blow-up growing to infinity is of an exponential type.

Consider segments of two loops of the curve (8), where either ν 0 = ν 0 + ( n , r ) or ν 0 = ν 0 ( n , r ) (we refer to (45) and (46)), r is a fixed number satisfying

(56) 0 < r 1 r r 2 < π 2 ( σ 1 )

and the angle φ varies within the limits given by the value s = 0 in (10), i.e.

(57) 1 σ 1 ν 0 ± ( n , r ) π 2 < φ < 1 σ 1 ν 0 ± ( n , r )