## Abstract

We illustrate how one can obtain hyperasymptotic expansions for solutions of nonlinear ordinary differential equations. The example is a Riccati equation. The main tools that we need are transseries expansions and the Riemann sheet structure of the Borel transform of the divergent asymptotic expansions. Hyperasymptotic expansions determine the solutions uniquely. A numerical illustration is included.

## 1. Introduction and summary

This is the first paper that discusses hyperasymptotic expansions for solutions of nonlinear ordinary differential equations (ODEs). The case of single rank linear ODEs is well understood (e.g. Murphy & Wood 1997; Olde Daalhuis 1998*a*). Utilising a first-order Riccati equation, we try to give a gentle introduction to this subject. In a sequel, we discuss the hyperasymptotic expansions of the first Painlevé equation and a more complicated second-order Riccati equation.

In hyperasymptotic expansions, we truncate a divergent asymptotic expansion and re-expand the remainder in expansions that involve the first hyperterminants. These re-expansions are again divergent, and we repeat the process of truncation and re-expansion. In this way, we incorporate the exponentially small terms and the Stokes phenomenon.

Riccati equations are the simplest nonlinear ODEs, but the hyperasymptotic expansions of the solutions of these equations already incorporate many of the important features of hyperasymptotic expansions of nonlinear ODEs. In the case of linear ODEs the general solutions are just linear combinations of, from an asymptotics point of view, special solutions of the linear ODEs. In the case of nonlinear ODEs, these linear combinations are replaced by so-called transseries, which are convergent series of the form , where *C* is a free constant and are formal (divergent) series. In the case of Riccati equations, these transseries are very natural objects.

The Stokes phenomenon plays a crucial role in asymptotic analysis and is well understood in the case of linear ODEs (e.g. Paris & Wood 1995). Since it plays a crucial role, we also have to be able to compute the so-called Stokes multipliers. Again, this is no problem in the case of linear ODEs (see Jurkat *et al*. 1976*a*,*b*; Immink 1990; Loday-Richaud 1990; Braaksma 1991; Olde Daalhuis & Olver 1995, 1998; Lutz & Schäfke 1997; Olde Daalhuis 1999).

For nonlinear ODEs with a rank of one singularity at infinity, the Stokes phenomenon is the change of the constant *C* in a transseries to *C*+*K*, when Stokes lines are crossed. Here, *K* is a Stokes multiplier. In contrast to linear ODEs, the Stokes phenomenon will not influence the dominant asymptotic behaviour of the solutions, and exponential asymptotics is needed to make the Stokes phenomenon visible. However, the computation of the Stokes multipliers is not any more difficult. On the boundary of the regions of validity of the asymptotic expansions, the solutions of the nonlinear ODEs have singularities, and the location of these singularities depends on the constant *C*. Hence, the Stokes phenomenon is also visible when one studies the location of singularities on these boundaries (e.g. Costin & Costin 2001).

In general, it is not possible to determine solutions of nonlinear ODEs uniquely via their Poincaré asymptotic expansions. In cases where the coefficients are real, one would like to be able to determine the solutions that are real-valued on the positive real axis. Medianization or balanced averaging are used in the literature (see Costin 1998) to prove that special transseries expansions (see (9.4)) converge to real-valued solutions. We will illustrate how hyperasymptotics can determine these solutions uniquely.

One way of obtaining the results for linear ODEs is to introduce the so-called Borel transform, which transforms the divergent asymptotic expansions into convergent expansions. In the case of linear ODEs the Borel transforms live in a so-called Borel plane and this plane contains only finitely many singularities. The complete Riemann sheet structure of the Borel plane is known. This information is used in Olde Daalhuis (1998*a*) to compute the complete hyperasymptotic expansion, and in Olde Daalhuis (1999) to compute the Stokes multipliers. In the case of nonlinear ODEs, the Borel plane contains infinitely many singularities. We will discuss the Riemann sheet structure. It has several interesting features that are special to nonlinear ODEs. At each level in hyperasymptotics, only a finite number of the singularities play a role, and the results of the linear paper Olde Daalhuis (1998*a*) can be copied to nonlinear ODEs.

The structure of this paper is as follows. In §2, we introduce our first-order Riccati equation. Of course, the solutions can be written as quotients of solutions of a second-order linear ODE and we can obtain the Stokes multipliers from this linear ODE.

This quotient of general solutions of the linear ODE can be expanded in §3 as a series of the form . When in a sector in the complex *z*-plane each of the *v*_{n}(*z*) is replaced by its asymptotic expansion , one obtains the corresponding transseries. We will express *v*_{n}(*z*) in terms of the solutions of the linear ODE. Hence, these objects are well known. We also show how the Stokes phenomenon for the linear ODE causes the Stokes phenomenon for the solutions of the Riccati equation. From the special expressions for *v*_{n}(*z*), it will become obvious that each *v*_{n}(*z*) also has its own transseries expansion.

The original transseries expansion corresponds to the large parameter . It is surprising that the *v*_{n}(*z*), *n*≥2, also have transseries expansions that correspond to .

In §4, we show that the results in this paper are not special to Riccati equations. We show how to obtain the transseries without using this special representation as a quotient of solutions of the linear ODE; that is, we obtain the transseries directly from the nonlinear ODE. In this way, we also obtain the recurrence relations of the coefficients of the asymptotic expansions of each *v*_{n}(*z*).

All of these transseries expansions are needed to determine the complete Riemann sheet structure of the Borel transform in §5 and, once we have this structure, it is just a small step to obtain the hyperasymptotic expansion in §6. In §6, we give the first four levels of the hyperasymptotic expansion of one of the special solutions of the Riccati equation. We also illustrate the growth of the terms in these expansions, and the exponential improvements that can be obtained at each level.

The hyperasymptotic expansions are in terms of so-called hyperterminants. We give the definition of these functions in §7. They are the simplest functions that incorporate the Stokes phenomenon at each level. Hyperterminants are multi-valued functions of several variables, and in the case of nonlinear ODEs, we have to compute these hyperterminants on branch-cuts of one or more of the variables. Hence, special care has to be taken.

Since we obtain the complete Riemann sheet structure of the Borel transform in §5, we will use this information to compute the Stokes multipliers in §8. Hence, the link to the solutions of the linear ODE in §2 is not needed at all.

It is impossible to determine the solutions of the Riccati equation that are real on the positive real axis uniquely from the Poincaré asymptotic expansion. In §9, we show how the level-1 hyperasymptotic expansions determine these solutions uniquely. A numerical illustration is included.

## 2. The Riccati equation

The Riccati differential equation that we will study here is(2.1)

The corresponding linear equation is(2.2)

Thus, the solutions of equation (2.1) can be written as *v*(*z*)=*w*′(*z*)/*w*(*z*), where *w*(*z*) is a solution of equation (2.2). The four solutions of equation (2.2) that we will use in this paper are *w*_{1,±}(*z*), *w*_{2}(*z*) and *w*_{2,−}(*z*), and they are uniquely determined by(2.3)as (see ch. 7 in Olver 1974). The connection relations between these solutions are(2.4)where the constants *K*_{1}, *K*_{2} are the so-called Stokes multipliers. In fact, the solutions of equation (2.2) can be expressed in terms of Whittaker functions (for notation and properties of this function, see Olver 1974):(2.5)

Hence, the Stokes multipliers are known as(2.6)

We will restrict our discussion to the solutions of equation (2.1) that behave like *v*(*z*)∼−*z*^{−2} as in the sector . These solutions can be written as(2.7)where is a constant. The asymptotic relations (2.3) show us that *v*(*z*, *C*) has poles at *z*=*z*_{n}, where , as integer .

## 3. Transseries expansions: special representation approach

In this section, we construct transseries expansions for the solutions of the Riccati equation. We introduce the Stokes phenomenon for nonlinear ODEs in terms of transseries expansions. This information will be used in §5, where we discuss the singularity structure of the Borel plane.

We can expand the right-hand side of equation (2.7) as(3.1)where(3.2)*n*=1, 2, 3, …. In the derivation of *v*_{n,±}(*z*), we have used the Wronskian relation .

The expansion (3.1) converges in the sector . The functions *v*_{n,±}(*z*) have asymptotic expansion(3.3)as . In §5, we will show that in the sector −*π*<ph *z*<0 the Borel–Laplace transform of is *v*_{n,−}(*z*), and in the sector 0<ph *z*<*π* it is *v*_{n,+}(*z*). Hence, in the sector , it is more natural to have an expansion for *v*(*z*, *C*) in terms *v*_{n,+}(*z*). We use the first connection relation in (2.4) in (2.7) and obtain(3.4)Thus, the expansion in terms of *v*_{n,+}(*z*) is(3.5)It follows from equation (3.3) that the transseries expansions of function *v*(*z*,*C*) are(3.6)The change from constant *C* to constant *C*+*K*_{1} is the Stokes phenomenon for nonlinear ODEs. In contrast to linear ODEs, the extra terms that are switched on when the Stokes line (the positive real *z*-axis) is crossed are exponentially small everywhere in the sector of validity. Hence, in that region, exponential asymptotics is needed to make the Stokes phenomenon visible.

Note that in the upper half-plane, the function *v*(*z*,*C*) has poles at *z*=*z*_{n}, where , as integer . Thus, the Stokes phenomenon causes a shift in the location of the poles. (For more details on the location of singularities on the boundary of the region of validity of asymptotic expansions, see Costin & Costin (2001).)

We can also construct transseries that correspond to *v*_{n,±}(*z*). Let us introduce(3.7)

The Stokes phenomenon for this expansion is again the change of constant from *C* to *C*+*K*_{1} as the positive real axis is crossed.

The function *v*(*z*,*C*) does not have a corresponding transseries expansion in the sector . Hence, it is surprising that in that sector there are transseries expansions that correspond to the functions *v*_{n,−}(*z*), *n*≥2. Let(3.8)

The Stokes phenomenon for this expansion is the change of constant from to as the negative real axis is crossed.

Since, by definition, , it has transseries expansion(3.9)as .

## 4. Transseries expansions: ODE approach

In the analysis above, we obtained the transseries expansions via the special features of Riccati equations. We can also obtain the transseries via a direct substitution of a transseries(4.1)into the nonlinear ODE (2.1), where the are defined in (3.3). We collect the terms that are of exponential order e^{−nz} and denote the Borel–Laplace transform of as .1 The result is that *v*_{0}(*z*) is, of course, a solution of the original nonlinear ODE (2.1), and all of the other satisfy the linear inhomogeneous ODE(4.2)For the coefficients in (3.3), we obtain via (2.1) and (4.2)(4.3)(4.4)and for *n*≥2 we have , and(4.5)

The only freedom that we have is the choice of *a*_{01}. We set it to minus unity and put that freedom of choice in the constant *C*.

The functions *v*_{n}(*z*) themselves also have transseries expansions. They are(4.6)and can be obtained via direct substitution of (4.6) into (4.2) and through the use of induction. The *C* in (4.6) is the same as in (4.1). Hence, there is only one free variable.

The transseries in (4.1) and (4.6) are for in the sector and in the sector . In §3 we noted that for *n*≥2 we also have the transseries expansions(4.7)and these are for in the sector and in the sector .

Here we note that, in contrast to transseries (4.6), it seems not to be possible to obtain the coefficients of transseries (4.7) from direct substitution into (4.2). For example, when we substitute the transseries expansion into (4.2), then the coefficient *c*_{1} cannot be determined since is a formal solutions of the homogenous differential equation (4.2), with *n*=1. In this paper we can compute the exact coefficients via the special representations in §3. In general it is unlikely that the exact values of the *c*_{j} can be determined. These constants can be seen as Stokes multipliers and, with the method given in Olde Daalhuis (1999), it is possible to determine them numerically to any precision (see also §8).

## 5. The Borel transforms

We can define the Borel transform as(5.1)This function is analytic in the sector , and on the boundary of this sector it has two types of singularities: logarithmic singularities at negative even integers, and square-root singularities at negative odd integers. It is slightly more convenient to have only one type of singularity in the complex *t*-plane. For that reason we will define the Borel transform as follows.

Let *α* be a constant such that . We define(5.2)If we take , we then obtain via (2.1) the differential equation(5.3)The inverse Laplace transform (see (5.5)) of this differential equation is(5.4)Via equation (4.3), we can show that the Borel transform *y*_{0}(*t*) defined in (5.2) is a solution of this equation.

The factor *t*+1 in (5.4) shows us that the nearest possible singularity in the complex *t*-plane to the origin is a branch point at *t*=−1. The convolution in (5.4) will produce ‘copies’ of this branch point at *t*=−2,−3,−4, …. Thus, in the complex *t*-plane, *y*_{0}(*t*) has singularities only at *t*=0,−1,−2,−3, …, and (5.2) converges for |*t*|<1. It can also be shown that the function *y*_{0}(*t*) does not grow faster than an exponential function as *t*→∞; that is, there exists a *β*>0 such that is bounded as *t*→∞.

For *n*=0,1,2,3, …, let *γ*_{n,±} be a contour that starts at , encircles *t*=−*n* once in the positive direction and terminates at . Then, the Laplace transform of the Borel transform *y*_{0}(*t*) is the function(5.5)The function *v*_{0,−}(*z*) is the unique solution of (2.1) with asymptotic behaviour (3.3), with *n*=0, as *z*→∞ in the sector . Hence, the functions *v*_{0,±}(*z*) defined in this section are the same as the functions *v*_{0,±}(*z*) defined in (3.2).

We also define(5.6)and(5.7)

As with the case of *y*_{0}(*t*), it can be shown (see next paragraph or for proofs Costin (1998)) that in the complex *t*-plane, the Borel transform *y*_{n}(*t*) has singularities only at the points *t*=−1,−2,−3, …, and that for *n*=1,2,3, …, the Borel–Laplace transform *v*_{n,−} (*t*) has asymptotic behaviour (3.3), as *z*→∞ in the sector . In fact, the functions *v*_{n,±}(*z*), defined in this section, are the same as the functions *v*_{n,±}(*z*) defined in (3.2), but we are not going to use this identification in the remainder of this paper.

The Borel transform *y*_{n}(*t*), *n*=1,2,3, …, satisfies the equation(5.8)which can be used to prove the statements made in the previous paragraph.

Let us now identify the singularities in the Borel plane. For the function *v*_{0,−}(*z*) defined in (5.5), the expansion in terms of *v*_{0,−}(*z*) (compare (3.1)) is just one term, and if the Stokes multiplier *K*_{1} is not zero, then its expansion in terms of *v*_{n,+}(*z*) (compare (3.5)) is an infinite series. This infinite series can be obtained from integral representation (5.5) by first collapsing contour *γ*_{0,−} onto the straight line that starts at the origin and ends at −i∞, and then rotating this contour clockwise over the singularities of *y*_{0}(*t*) on the negative real axis (see figure 1). The contribution of the singularity at *t*=−*p* should give us . It follows from (5.7) that for *p*=1,2,3, …, the singular behaviour of *y*_{0}(*t*) at is .

Hence, for *n*=1,2,3, …, we obtain from the connection relation(5.9)that the singular behaviour of *y*_{n}(*t*) at *t*=*p* e^{−πi}, *p*>*n*, is(5.10)

Similarly, for *n*=0,1,2, …, we obtain the connection relation(5.11)that the singular behaviour of *y*_{n}(*t*) at *t*=*p* e^{−πi}, *p*>*n*, is(5.12)

Finally, for *n*=2,3,4, …, we have the connection relation(5.13)where is the function *v*_{n,+}(*z*) defined in (5.7), but with . From this connection relation we obtain that the singular behaviour of *y*_{n}(*t*) at *t*=*p* e^{−πi}, 0<*p*<*n*, is(5.14)This singularity structure is illustrated in figure 2.

## 6. The hyperasymptotic expansions

The only results that we need from the previous sections are the asymptotic expansions of *v*_{n,±}(*z*) and the connection relations. However, it will also be very convenient to refer to the singularity structure of the Borel plane. For example, since the nearest singularity to the origin is at distance 1, the optimal number of terms in the asymptotic expansion of *v*_{0,−}(*z*) is . In this section, *N* will be an integer such that as *z*→∞.

To obtain a hyperasymptotic expansion for *v*_{0,−}(*z*) that is correct up to a relative error of order , we have to include, in our analysis, all the singularities of *y*_{0}(*t*) in the Borel *t*-plane that are at ‘distance’ <*R* from the original. The analysis in Olde Daalhuis (1998*a*) does not use the fact that the Borel transform in that paper originates from a linear ODE. It only uses the knowledge of which singularities are at ‘distance’ <*R* in the Borel plane. Hence, we can and will refer to that paper for the hyperasymptotic expansions, the optimal number of terms at each level and for remainder estimates.

In this section we will also use the notation for the hyperterminants. The reader is referred to §7 for the notation.

This level is the optimal truncated version of the asymptotic expansion (3.3). We have(6.1)where(6.2)as *z*→∞ in the sector .

Now we re-expand the remainder. Since *y*_{0}(*t*) has singularities at all negative integers (or since there are infinitely many terms in connection relation (5.9) with *n*=0), infinitely many terms should appear in the re-expansion:(6.3)

However, according to Olde Daalhuis (1998*a*), the optimal number of terms at level 1 is , and *N*_{k}=0 for *k*=2, 3, 4, …. Hence, only one of the re-expansions contributes to the level-1 hyperasymptotic expansion, and the optimal version is(6.4)where(6.5)as *z*→∞ in the sector .

At this level we re-expand the re-expansion of level 1. Formally, this is an infinite series of infinite series. As we saw at the previous level, only several expansions contribute when we take an optimal number of terms. The optimal result is(6.6a)(6.6b)where(6.7)

as *z*→∞ in the sector . The two sums in (6.6*a*) are part of the first (level 1) re-expansion, and only the first of these is re-expanded into the sum in (6.6*b*). Hence (6.6) contains two level-1 re-expansions and one level-2 re-expansion.

In figure 3, we illustrate the growth of the terms in this level-2 hyperasymptotic expansion of *v*_{0,−}(*z*), with *z*=10.0−3.1i. The figure clearly shows that the first sum in (6.6*a*) is the dominant re-expansion of the original Poincaré asymptotic expansion, and that the sum in (6.6*b*) is the dominant re-expansion of the first sum in (6.6*a*).

The optimal result at this level is(6.8)as *z*→∞ in the sector −*π*<ph *z*<0, where the terms that were switched on in the first re-expansion are(6.9)The level-2 re-expansion of the first two terms on the right-hand side of (6.9) is(6.10a)(6.10b)where the two sums in (6.10*a*) are the re-expansions of the first sum in (6.9), which corresponds to *y*_{1} seeing singularities 2*K*_{1}*y*_{2} and , and the two sums in (6.10*b*) are the re-expansions of the second sum in (6.9), and this corresponds to *y*_{2} seeing singularities 3*K*_{1}*y*_{3} and *K*_{2}*y*_{1}. The level-3 re-expansion is (6.11)

These are the re-expansions of the first sum in (6.10*a*), and again they correspond to *y*_{2} seeing singularities 3*K*_{1}*y*_{3} and *K*_{2}*y*_{1}.

Hence, with each level, we improve the accuracy by exp(−|*z*|). We could continue this process. However, the number of sums increases exponentially with each level. We illustrate the improvements in table 1, where we use the first four levels of hyperasymptotic expansions to approximate *v*_{0,−}(*z*) for *z*=10.0−3.1i.

## 7. Hyperterminants and nonlinear ODEs

In Olde Daalhuis (1998*b*), the hyperterminants are defined and a method is given to compute these functions to any given accuracy. The definition is(7.1)where we use the notation *θ*_{j}=ph *σ*_{j} and . From integral representation (7.1), it is obvious that the hyperterminants are multi-valued functions with respect to *z*, and also with respect to *σ*_{j}. Connection relations with respect to all these variables are given in Olde Daalhuis (1998*b*).

In Olde Daalhuis (1998*b*) the hyperterminants are expanded into convergent series involving the hypergeometric functions(7.2)

These hypergeometric functions have the branch-cut at 1+(*σ*_{1}/*σ*_{0})>1, that is, for the cases where *σ*_{0} and *σ*_{1} have the same phase. As can be seen in §6, this is very common in hyperasymptotic expansions of nonlinear ODEs. The reason for this is that the singularities in the Borel plane are collinear. In applications, special care has to be taken in choosing the correct branches of these functions. We did that in the numerical calculations in §6, but have omitted the details here.

## 8. The computation of the Stokes multipliers

In the case of the special example discussed in this paper, we have simple expressions (2.6) for the Stokes multipliers *K*_{1} and *K*_{2}. In general, we have to use hyperasymptotics to compute the Stokes multipliers. The general situation is discussed in Olde Daalhuis (1999). Here we simply mention how we could have computed *K*_{1} and *K*_{2} via hyperasymptotics.

Note that the function *y*_{0}(*t*) is defined via (5.2), that its nearest singularity is at *t*=−1, and that the local expansion at *t*=−1 is *K*_{1}*y*_{1}(*t*). This can be used to obtain the asymptotics of the late coefficients in (5.2),(8.1)as *p*→∞. Since we can compute all the coefficients *a*_{sj} via the recurrence relations given in §4, we can use (8.1) to compute the Stokes multiplier *K*_{1}. The optimal number of terms in the divergent series in (8.1) is approximately *p*/2. For example, if we take *p*=100 then(8.2)All the digits are correct in this approximation.

To obtain an approximation that involves *K*_{2}, we recall that *K*_{2} plays a role in the singularity structure of the function *y*_{2}(*t*). This function is defined via (5.6), with *n*=2, and its nearest singularities are at *t*=−3, with local expansion 3*K*_{1}*y*_{3}(*t*) and at *t*=−1, with local expansion *K*_{2}*y*_{1}(*t*). (See (5.10) and (5.14).) Hence, the late coefficients grow as follows:(8.3)as *p*→∞. In the two divergent series, the optimal number of terms is again *p*/2. Because we can compute *K*_{1} via (8.1), we can use (8.3) to compute *K*_{2}. If we take *p*=100 we obtain the approximation(8.4)All the digits are correct in this approximation.

## 9. Real solutions on the real line

So far, the main solutions of the Riccati equation (2.1) are *v*_{0,±}(*z*) From an asymptotics point of view, these solutions are very special, since they are uniquely determined by their asymptotic behaviour in a large sector:(9.1)All the coefficients in the Poincaré asymptotic expansions (3.3), with *n*=0, are real. However, these two functions are not real on the positive real axis. This is a direct consequence of the Stokes phenomenon, which takes place when crossing the positive real axis. To obtain the imaginary part of *v*_{0,−}(*z*) on the positive real axis we use the level-1 hyperasymptotic expansion (6.4), and the fact that for *z*>0 we have the identity(9.2)Thus,(9.3)as *z*→∞ along the positive real axis. The Stokes multiplier *K*_{1}, defined in (2.6), is purely imaginary, and thus the factor *K*_{1}/i in (9.3) is real.

To obtain a solution that is real on the real axis, we have to choose a special value for *C* in expansion (3.1). The value of *C* that cancels the imaginary part of is, of course, , where *A* is any real number. Thus, the function(9.4)is real on the positive real axis.

We note that if we want to determine on the positive real axis the solutions of the Riccati equation (2.1) uniquely via their asymptotic expansions, and require that they are real on the positive real axis, then hyperasymptotics is needed. These solutions are uniquely determined by the constant *A*. The level-1 hyperasymptotic expansion is(9.5)where and(9.6)

Since in the right-hand side of (9.5) the term containing the factor is of larger order than order estimate (9.6), it follows that is uniquely determined by this level-1 hyperasymptotic expansion.

For the final illustration we take *A*=0, *z*=15.5, *N*=16 and calculatewhich is one of the real-valued solutions of (2.1),which are the first two terms on the right-hand side of (9.5)(and is also the level-1 hyperasymptotic expansion of *v*(*z*, 0); note that the imaginary part is not zero),which cancels the imaginary part in the previous result, and

## Acknowledgments

This work was supported by EPSRC grant GR/R18642/01.

## Footnotes

↵In this section, we are mainly interested in determining the ODEs (4.2), the recurrence relations (4.3)–(4.5) and other (formal) transseries solutions of (4.2). All of these results are independent of the choice of the

*z*-sector in which*v*_{n}(*z*) is regarded to be the Borel–Laplace transform of . The two obvious choices are 0<ph*z*<*π*and −*π*<ph*z*<0. We will specify the*z*-sectors in later sections.- Received July 16, 2004.
- Accepted February 1, 2005.

- © 2005 The Royal Society