Rational solutions of Painlev´e-II equation as Gram determinant

Xiaoen Zhang(张晓恩) and Bing-Ying Lu(陆冰滢)

1College of Mathematics and Systems Science,Shandong University of Science and Technology,Qingdao 266590,China

2SISSA,Via Bonomea 265,Trieste,Italy

Keywords: Painlev´e-II equation,Darboux transformation,rational solutions

1.Introduction

The six Painlev´e equations(PI-PIV)are a class of nonlinear ordinary differential equations which have long been studied.It originated from when the Painlev´e school tried to answer Picard’s question in Ref.[1]in the late 19th century.Because the general solutions cannot be reduced to elementary or known functions,today it is often regarded as a nonlinear type of special function.Due to transcendency,the solutions to the Painlev´e equations are often called the Painlev´e transcendents.In the late 1970s, Painlev´e equations garnered new attention when they were found to be “integrable”, thus not only extending the tools to analyze their solutions,but also expanding the theory of integrability.Although these Painlev´e equations were first derived from strictly mathematical considerations,these Painlev´e equations have appeared in many physical applications.Perhaps Ref.[2]is the first example of these applications.Subsequently,Painlev´e equations were widely used in many aspects of physics.For example,in Ref.[3],the authors showed that the density matrix of a one-dimensional system of impenetrable Bose gases at zero temperature can be precisely represented by the solution of a nonlinear differential equation: the Painlev´e transcendence and its generalization.In Ref.[4], the authors gave a detailed derivation by using the Painlev´e equation to describe the dynamics of the electrons and holes in a semiconductor.In Ref.[5], by using an auxiliary nonlinear Schr¨odinger equation, the authors found that the second Painlev´e transcendent solution can be used to study the propagation of the optical pulses in single mode dispersive fibers.In addition, the Painlev´e equation also has a good application in the light-matter interaction in nematic liquid crystals,[6]the high-energy physics[7]and the planar fluid flow.[8]

We are particularly interested in certain pattern formation related to rogue waves in integrable nonlinear waves and their connection to the Painlev´e transcendents.Very recently, the locations of the rogue wave patterns in nonlinear Schr¨odinger equation and the lump patterns in Kadomtsev-Petviashvili equation are found to be determined by the root structure of the Yablonskii-Vorob’ev polynomials hierarchy.[9-11]It has long been observed in Ref.[12] that the Painlev´e-II rational solutions can be represented by the Yablonskii-Vorob’ev polynomials, so the root structure is exactly the pole structure for Painlev´e-II rational solutions.This directly inspired us to add to the ample existing studies of Painlev´e-II rational solutions from a different perspective.

The general formula of Painlev´e-II equation is given by

As mentioned,the general solutions of Painlev´e equations are transcendental.However, Painlev´e-II equation is known to have rational solutions for some special parameters.Indeed,in Ref.[12] Yablonskii first found an iteration formula for the rational solutions withm ∈Z in Eq.(1), and Airault[13]showed that the parameter choice is both necessary and sufficient.Like soliton solutions in integrable systems, the rational solutions can be given by B¨acklund transformations.The B¨acklund transformation for Pailev´e-II equation is given by Lukashevich in 1971.[14]In 1985,[15]Murata derived the rational solutions of the second and the fourth Painlev´e equation with different integer parameters,and in particular showed the necessary and sufficient conditions for rational solutions to exist.There are numerous studies of rational solutions[13,16-21]through the B¨acklund transformation.

Painlev´e equations are considered integrable in the sense that they admit Lax representation.Indeed, Painlev´e-II equation admits the following Flaschka-Newell Lax representation

yields the Painlev´e-II equation (1).Due to integrability, one powerful tool for analysis of the Painlev´e equations is the Riemann-Hilbert problems.The book[22]is a particularly comprehensive and systematic documentation of this modern subject.The Riemann-Hilbert problem is also used to study the rogue wave patterns that we are interested in.Specifically in literature,[23]the authors found that in semiclassical focusing nonlinear Schr¨odinger equation, near a gradient catastrophe point, universal “rogue wave”-like patterns form can be mapped to the poles of the Painlev´e-I tritronqu´ee solution.In Ref.[24],the authors analyzed the asymptotics of the infinite order rogue waves,and the asymptotics in the transitional region was related to the Tritronqu´ee solutions of Painlev´e-II equation.

There are many ways to derive Painlev´e-II rational solutions.One standard method is the B¨acklund transformation,given nearly 50 years ago.However,to the best of our knowledge, few has studied the rational solution of Painlev´e equations with the Darboux transformation[25,26]directly.The Darboux transformation for Painlev´e-II rational solutions is still an open problem up to now.Observe that the∂y-equation in the Lax pair (2) is similar to the spectral problem in the AKNS system.It is also shown that the Painlev´e-II equation is related the modified Korteweg-de Vries equation via a self-similar transformation.[27-30]Thus it is natural to borrow the idea from the Darboux transformation of AKNS system to construct the solution of the Painlev´e-II equation.

In the present work,we modify the Darboux transformation to construct the rational Painlev´e-II solutions.Under the same framework, we also derive the rational solution with a Gram determinant representation.With our determinantal formula, we get the asymptotics of largeyby expansion.The rational solutions of Painlev´e-II equation are often compared to the soliton solutions in integrable nonlinear wave equations,since they can be represented by determinantal formula, and correspond to fully discrete spectral data etc.[29]However,the Flaschka-Newell determinantal formula is different from our Gram determinant formula.Indeed, in our Darboux transformation, all of the spectral data are at the same point so we have to employ limiting technique.The process is much more like deriving the rogue wave solutions in integrable nonlinear waves.The Gram determinant representation is also comparable to rogue waves.This also in a way attests to the connection between the roots of Yablonskii-Vorob’ev polynomial(poles of Painlev´e-II rational solutions) and rogue wave patterns found in Refs.[10,11].However, the transformation for Painlev´e-II has even one more difference, the generalized Darboux-B¨acklund transformation we used is not the so called auto-transformation in comparison, which we believe also makes our work even more interesting.

Ultimately we believe to study the rogue wave pattern,Riemann-Hilbert method can give us a better analytical handle, therefore we always have the Riemann-Hilbert problem in mind.In Ref.[24], the authors studied the asymptotics of infinite order rogue waves via the Riemann-Hilbert method,which was first constructed by using the Darboux transformation.Based on this idea, we are able to solve the Painlev´e-II Riemann-Hilbert problem given in Ref.[31].

The innovation of this paper consists of the following two points: (i) We derive the rational solutions of Painlev´e-II equation via two types of generalized Darboux transformation.In the first case,from the“seed solution”p=0,m=0,we give the iterated Darboux transformation at the same spectrumλ1=0 and derive the corresponding B¨acklund transformation.Compared to some other rational solutions in integrable systems, the derivation of Painlev´e-II rational solution is more difficult.To achieve it, we would need to use special limiting techniques: the Darboux transformation is iterated at the same spectrum and the spectrumλ1must be chosen at a special point.In the second case, we rewrite the generalized Darboux transformation by another equivalent formula and derive the rational solutions as a Gram determinant.In particular,we can analyze the asymptotics of largeyby this formula.(ii) Our generalized Darboux transformation can be used to construct the solution of Riemann-Hilbert problem given in Ref.[31], which provides a new way for solving the corresponding Riemann-Hilbert problem for the Painlev´e-II equation.

The outline of this paper is as follows.In Section 2, we give detailed derivation of rational solutions via the generalized Darboux transformation.Then, we also give the corresponding B¨acklund transformation.In Section 3,the generalized Darboux transformation is rewritten as another equivalent formula, which gives the solution in Gram determinant form,and can be used to analyze the asymptotics for largey.In Section 4, we give a brief introduction to the Riemann-Hilbert problem given by Ref.[31] and solve it with the aid of the Darboux transformation.The final part is the conclusion.

2.The Darboux transformation of Painelve´-II equation

In this section, we would like to construct the generalized Darboux transformation of Painlev´e-II equation and derive the rational solutions.The rational solutions of Painlev´e-II equation play an analogous role as the solitons of the integrable partial differential equations.However, to the best of our knowledge, the rational solutions of Painlev´e-II equation have never been constructed by the well known Darboux transformation.From the Lax pair of Eq.(2), we can see that the∂y-part of Lax pair is the same as the∂x-equation, what we usually call the Lax equation,in the Lax pair for the defocusing NLS equation,[22]while the∂λ-part is different.It has two types of singularity.One is atλ=∞and the other one is atλ=0.What is more, the∂λ-equation contains a constantm, which leads to a new difficulty in the Darboux transformation.To overcome these difficulties, we use two kinds of limiting skills following the original idea of the so-called generalized Darboux transformation.[33,34]We want to emphasize here that there is a difference between the rational solutions of Painlev´e-II equation and the higher order rogue wave of integrable equations.The former ones are meromorphic functions with simple poles in the complex plane but the latter ones are global solutions in the (x,t)-plane.The detailed derivation is shown as follows.

To construct the rational solutions,we start with the“seed solution”p=0,m=0.Since the∂y-part of Lax pair is the spectral problem of AKNS system, we can set the Darboux transformation matrix as

Equipped with the Darboux transformation defined above, we know that the original Lax pair (2) is converted to a new one by replacingp,mwithp[1],m1.In the transformed Lax pair we use the superscript[1]inp[1]to mean the first order rational solution,and the subscript inm1is the constant in correspondence withp[1].

In Ref.[34], the authors showed that the∂y-equation in the new Lax pair is valid,i.e.,it has the same form except with newpunder this transformation.Therefore we only need to check whether the Darboux transformation also satisfies the∂λ-part of Lax pair.Plugging the Darboux matrix Eq.(3)and the original seed solutionp=0,m=0 into the new Lax pair,we have

which should be a constant independent ofy.Therefore, the spectral parameterλ1is chosen to be zero and the entries ofφ1should satisfy|φ1,1|=|φ1,2|.To achieve this, we can set the constant vectorcasc=(1,-1)T, thenφ1,1φ∗1,2=φ∗1,1φ1,2and|φ1,1|=|φ1,2|.Note that now both the numerator and the denominator are zero.Expand for infinitesimalλ1,we getm1=1,and the first order rational solutionp[1]is

Summarizing the above calculation, we obtain rational solutions of all orders.In Ref.[13], Airault pointed out thatm ∈Z is the necessary and sufficient condition for Painlev´e-II solution to be rational.By uniqueness, we conclude all general rational solutions are given by our generalized Darboux transform in Theorem 1.

Theorem 1 Settingλ1=-λ∗1=0, the generalized Darboux transformation matrices for the Painlev´e-II equation(1)are given by

wherepis the seed solutionp=0 in Eq.(3).

We remark on the specificity of Painlev´e-II rational solutions in the transformation.

Remark 1 It is clear that each Painlev´e-II equation (1)involves one constantm, and that from our transformation in each iteration, we take a different value ofmk.Therefore,the B¨acklund transformation between two rational solutions is not the auto-B¨acklund transformation.For each order of the rational solutions,the constantsmkdiffer.This makes our B¨acklund transformation different than what usually appears in integrable partial differential equations.

We derived the B¨acklund transformation (31) through generalized Darboux transformation.In fact, apart from this method, we can also give a B¨acklund transformation forp[k]from Eq.(1)itself.Before discussing it,we first give a proposition about the Schwarzian derivative.

In the following section, we will prove in Theorem 2 that the constantmkcorresponding to thekth order rational solutionp[k]ismk= (-1)k-1k.If we set (-1)k-2p[k-1]=˜p, (-1)k-2p[k]=p, then the two B¨acklund transformations Eqs.(31)and(46)are equivalent.

3.The asymptotics for large y

In the last section,through generalized Darboux transformations(32),we obtain the rational solutions in Eq.(34).The B¨acklund transformation(31)is also directly given from iteration.In this section,we will continue to discuss some properties of the rational solutions of Painlev´e-II equation.So far,the rational solutions in Eq.(34) are in abstract form, which does not lend itself to easy analysis of their properties.In this section, following the idea in Ref.[37], we convert the generalized Darboux transformation (32) into another equivalent formulation and rewrite the rational solutions as a Gram determinant.A great benefit of this type of formula is that it can be easily used to analyze the asymptotics for largey.The detailed calculation is shown as follows in Corollary 1.

Corollary 1 The generalized Darboux transformation matrices(32)can be rewritten as the following equivalent formula:

Directly substitute the above to the Darboux matrix (52) and take the limitεi →0,i=2,3,...,k, then we can get the formula(48).Correspondingly,the rational solution of Painlev´e-II equation can be given with the following formula:

which equals to Eq.(51),it completes the proof.

Apparently the rational solutions in Eqs.(34) and (51)have different forms, each of which has its own advantage.The former can be used to derive the B¨acklund transformation,to which the latter does not have an obvious connection.However, the latter writes all rational Painlev´e-II solution in a compact formula.Moreover, the formula is made up of determinants, which renders analytical properties for largeyasymptotics very accessible.Using the new representation,we find the constantmkin Eq.(1)for thekth order rational solutionp[k], thus also completing the B¨acklund transformation Eq.(31).

Theorem 2 The constantmkin the Painlev´e-II equation Eq.(1), corresponding to thekth order rational solutionsp[k]in Eq.(51)is equal to(-1)k-1k.

Proof Following the idea in Ref.[38], we first expandp[k]Eq.(51)at the neighborhood ofy=∞and substitute this series into the Painlev´e-II equation(1).Then the constantmkcan be calculated by comparing the coefficients ofypolynomials.Clearly,the rational solution Eq.(51)is the quotient of two Gram determinants,thus we can look at the denominator and the numerator separately.Firstly, we discuss the asymptotics of the denominator det(M).From the definition ofMijin Eq.(50),we know that whileMijdoes not generally admit simple formula, the leading behavior for largeyis quite easy to compute:

Based on this asymptotical expression in Eq.(55),whenyis large andkis even, the denominator det(M)can be expanded in the following formula:

Similarly,whenyis large,the numerator det(G)can be expanded as

Therefore,

The upper left block ofHeis a Hilbert matrix,while the lower right block is the negative of a Hilbert matrix.Employing basic property of Hilbert matrices, we can derive the following identity:

A similar asymptotic formula as Eq.(60) forp[k]with oddkcan also be obtained,withHereplaced byHo.The subscript o indicates the odd case.In the odd case,Hois instead given by

Consequently, the asymptotics ofp[k]becomesp[k]=k/y+O(y-2).Substituting this expression into Eq.(1),then we obtain that the constantmkis equal tok.

Example Next,we compute here the first several rational solutions derived from the generalized Darboux transformation.Whenk=1, the Darboux transformationT1(λ;y) and the first order solutionp[1]are

Whenk=2, the Darboux transformationT[2](λ;y) and the second order rational solutionp[2]are

Obviously, the first order rational solutionp[1]satisfies the Painlev´e-II equation (1) withm1=1 and the second order rational solutionp[2]satisfies it withm2=-2.Similar to the rational solutions of KdV equation,[39]both the rational solutionsp[1]andp[2]are meromorphic functions with simple poles.Indeed these solutions have different analytic properties from the rogue waves, even though in Ref.[11] the authors showed the close connection of the two,which is an important motivation for our study.In Fig.1, we show the location of zeros and poles of higher order rational solutions.

Fig.1.The poles (green) and the zeros (blue) of the rational solutions of Painlev´e-II equation.The corresponding parameters are m9 =9 (a),m14=-14(b)and m18=-18(c),respectively.

4.Riemann-Hilbert representation about the rational solutions of Painleve´-II equation

In the previous two sections, we constructed two types of Darboux transformations about the rational solutions and derived the corresponding B¨acklund transformation.Both of them are studied under the Flaschka-Newell Lax representation Eq.(2).It is well known that the Painlev´e-II equation has three different Lax representations,i.e.,the Flaschka-Newell Lax representation, the Jimbo-Miwa Lax representation and Bortola-Bothner representation.In Ref.[31],the authors gave a detailed introduction of the three representations and constructed three different Riemann-Hilbert problems accordingly.In this section,we will give the solution of the first(Flaschka-Newell) Riemann-Hilbert problem for Painlev´e-II equation.

First we give a brief review of the Riemann-Hilbert problem corresponding to the Flaschka-Newell Lax representation from Ref.[31].Firstly, the authors constructed two types of fundamental solution matricesV∞(λ;y) andV0(λ;y) of the Flaschka-Newell Lax pair Eq.(2).The former solutionV∞(λ;y)has a convergent expansion when|λ|is large and the latter solution is given in the neighborhood ofλ=0.Then there exists a so-called monodromy matrixGm(λ;y)such that

whereθ(λ;y) =λy+2λ3.This immediately gives a Riemann-Hilbert problem forMm(λ;y).

Riemann-Hilbert problem 1[31]Letm ∈Z andy ∈C,seekMm(λ;y)satisfying the following conditions.

(i) AnalyticityMm(λ;y) is analytic for|λ|/= 1, andMm,+(λ;y),Mm,-(λ;y)are the continuous boundary values from the interior and the exterior.

(ii) Jump condition When|λ|=1,Mm(λ;y) satisfies the following jump condition:

To solve this Riemann-Hilbert problem,in Sections 2 and 3,we constructed two types of generalized Darboux transformation and derived the general rational solutions of Painlev´e-II equation.The solution of the Riemann-Hilbert problem 1 can be directly given by the Darboux transformation.

Theorem 3 The solution of Riemann-Hilbert problem 1 can be given by

whereT[m](λ;y) is the generalized Darboux transformation defined in Eq.(48).

Proof Based on the basic property of Darboux transformation, we know thatT[m](λ;y)e-iθσ3can solve the Lax pair Eq.(6) withptaken as themth order rational solutionp[m]and the constant as (-1)m-1m.From the symmetry in the∂y-equation of the Lax pair,we haveσ1BFN(-λ;y)σ1=BFN(λ;y).Therefore,the Darboux transformation matrix also have symmetry

In addition, the Darboux transformation matrix has a convergent series expansion forλ →∞.Notice thatT[m]almost coincides with the fundamental solutionV∞(λ;y) defined in Ref.[31], except for a sign difference for evenm.With the discrete symmetry of the Painlev´e-II equation(p(y),m)→(-p(y),-m), we can take a simple transformation toT[m](λ;y),and now the formula is valid for allm ∈Z.Thus,we have derived the connection between the generalized Darboux transformation and the Riemann-Hilbert problem 1 when|λ|>1.For|λ|<1,the solution of the Riemann Hilbert problem 1 can also be given by the jump condition.Therefore,the solution of the Riemann-Hilbert problem 1 is given in the whole complex plane C,that is,

5.Discussion and conclusion

In this paper, we construct the general rational solutions to Painlev´e-II equation by the generalized Darboux transformation.We are able to compactly write the rational solutions as Gram determinant.In the generalized Darboux transformation,the spectral parameterλ1is chosen as a special valueλ1=0.Under this condition, the fundamental solutionφ1becomes a polynomial.In particular,the spectrum in the Darboux transformation is a removable singularity,which is similar to the study of rogue wave.In Section 2,we use two methods to derive the B¨acklund transformation of the rational solutions.One is directly from the Painlev´e-II equation itself,and the other one is derived from the iteration steps in the Darboux transformation.The two transformations are shown to be equivalent by a simple transformation.In addition, from the exact form of rational solutions simply represented by Gram determinants,we can also analyze the asymptotics of largey.In Section 4, we prove that our Darboux transformation can solve the Riemann-Hilbert problem in Ref.[31], which has never been reported before.

It is known that the generalized Darboux transformation can be used to derive the rogue wave of NLS equation and the rational solutions of Painlev´e-II equation,both written as a Gram determinant.In view of the result in Ref.[10],we conjecture that there exist certain connections between these two kinds of solutions.Our goal is to continue the study of this topic via the Riemann-Hilbert problem in the future.

Acknowledgments

The authors sincerely thank Professor Liming Ling for his guidance and help.Project supported by the National Natural Science Foundation of China(Grant No.12101246).