# Lax Representation and Darboux Solutions of the Classical Painlevé Second Equation

Authors: Irfan Mahmood; Muhammad Waseem
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this article, we present Darboux solutions of the classical Painlevé second equation. We reexpress the classical Painlevé second Lax pair in new setting introducing gauge transformations to yield its Darboux expression in additive form. The new linear system of that equation carries similar structure as other integrable systems possess in the AKNS scheme. Finally, we generalize the Darboux transformation of the classical Painlevé second equation to theN-th form in terms of Wranskian.


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## Body

## 1. Introduction

The six classical Painlevé equations first were introduced by Paul Painlevé and his colleagues while classifying nonlinear second-order ordinary differential equations with respect to their solutions [1]. In the beginning, these were well entertained in mathematics because of their connection to nonlinear partial differential equation, where they appear as ordinary differential equation reduction of higher dimensional integrable that further pursued to establish the Painlevé test for these partial differential equations [2, 3]. For example, the Painlevé second (P-II) equation arises as ODE reduction of the Korteweg-de Vries (KdV) equation [4, 5]. Although various properties of these equations have been studied from the mathematical point of such as their integrability through the Painlevé test, their zero-curvature representation is a compatibility condition of associated set of linear systems [6–8] as well as in the framework of hamiltonian formalism [9]. These equation got considerable attention from the physical point of view because the number of problems in applied mathematics and in physics is made to be integrable in terms of their solutions, Painlevé transcendents. The Painlevé second equation is one of these six equation
(1)vxx=v3+xv−αapplied as a model to study electric field in semiconductor [10], where vxx=d2v/dx2 and α is parameter, whose rational solutions have been constructed in [11] by means of Yablonski-Vorobév polynomials and further, the determinantal generalization of its rational solutions is presented by [12]. Moreover, the classical Painlevé II equations can also be represented by the Noumi-Yamada systems [13], these systems are discovered by Noumi and Yamada while studying symmetry of Painlevé equations, and these systems also possess the affine Weyl group symmetry of type Al1. In the Noumi-Yamada setting, the symmetric form of the Painlevé II equation (1) is given by
(2)v0′=v0v2+v2u0+α0v1′=−v1v2−v2u1+α1v2′=v1−v0,where vi′=dvi/dz and α0, α1, are constant parameters. This can be shown that on eliminating v1 and v0, one can obtain Painlevé II equation (1) for v2. In proposition 1 below, and we introduce the Lax representation of the Painlevé II symmetric form (2). We also present the Darboux solutions of the Painlevé II equation with the help of Darboux transformations incorporating its gaued Lax pair. The Darboux transformation significantly has got considerable attention in theory of integrable systems to find their exact solutions. The successful implementation of these transformations can be found in [14–16], where the solitonic solutions have been generated to few integrable systems with their computational presentations.Proposition 1.
The Painlevé II symmetric 2 can be represented by Lax operatorsLz=P,L, where L and M are Lax operators.Proof.
For this purpose, consider a linear system(3)Lzψ=λψ,and the evolution of ψ with respect to z is given by
(4)ψz=Pψ.
The compatibility condition of this systems yields the Lax equation(5)Lz=P,L.
Here,λ is a spectral parameter, and λz=0. Now, we consider the Lax pair L,P in the following form
(6)L=L1OOOL2OOOL3,P=P1OOOP2OOOP3,where
(7)L1=10−v0−1,L2=10−v1−1,L3=−10−v21,and the elements of the matrix P are given by
(8)P1=ρ100−ρ1,L2=−ρ200ρ2,L3=−1012σ1,where
(9)ρ1=−v2−12α0v0−1,ρ2=−v2+12α1v1−1,σ=u0−v1+2v2O=0000.
When the above Lax matricesL and P are subjected to the Lax equation (5), we obtain the Painlevé II symmetric form (2), and one may expand determinant I+μLii in terms of the eigenvalues of L similar as done for the quasideterminants in [17].
The Painlevé second equation (1) arises as the compatibility condition of linear systems
(10)Ψλ=Ax;λΨ,Ψx=Bx;λΨwith the Lax pair
(11)A=4λv+αλ−1σ1−2v′σ2−4iλ2+ix+2iv2σ3B=vσ1−iλσ3that yields the Painlevé second equation (1), where σ1, σ2, and σ3 are Pauli spin matrices.Remark 2.
Under the standard Darboux transformation [18, 19] on components of the column vector Ψ=X/Y as
(12)X→X1=λY−λ1Y1X1XY→Y1=λX−λ1X1Y1Y,one can construct the 1-fold Darboux transformation to the solution of equation (1) by using its associated linear system (10) in the following form v1=vYX−12 that yields trivial Darboux solutions to equation (1) taking v=0 as a seed solution at α=0, which are meaningless. In the next section, we will present nontrivial Darboux solutions of that equation introducing gauge transformation that brings its Lax pair (11) to a new form as similar as we have for many integrable systems in the AKNS scheme. Further, by iteration, the N-fold Darboux transformation will be generalized to determinantal form in terms of Wronskian.


## 2. Gauge Transformation and Nontrivial Darboux Solutions

Proposition 3.
Let us consider a matrixG=1−i1i that transforms the old Lax pair (11) A,B to a new set as A→A~=GAG−1,B→B~=GBG−1, where the new Lax pair A~,B~ reproduces Painlevé second equation (1) as subjected to the zero-curvature condition A~x−B~λ=B~,A~.Proof.
Under above gauge transformations, the new Lax will take the following form(13)A~=−2v′σ1−4iλ2+ix+2iv2σ2+4λv+αλ−1σ3B~=vσ3−iλσ2,with the linear system
(14)Ψλ=A~x;λΨ,Ψx=B~x;λΨ,whose compatibility condition gives zero-curvature form in terms of Lax matrices (A~, B~) as A~x−B~λ=B~,A~. Now, having values for A~x, B~λ, and for commutator B~,A~ from gauged Lax pair (13) and substituting in the corresponding zero-curvature condition, some computations system reproduces the classical Painlevé second equation (1).Proposition 4.
The linear system (14) under the Darboux transformation (12) connects the next new solution and says v1 of (1) to its old one v as
(15)v1=−v+ddxlnX1Y1,where X1 and Y1 are particular solutions of the linear system (14) at λ=λ1.Proof.
Let us start with standard Darboux transformations (12) on components of the column vector and consider a column matrix Ψ=XY with the linear system (14), and we have the following transformed equation
(16)X′1=v1X1−λY1,and now from (12)
(17)X′1=λoYo′−λ1Y1′X1Xo+λY1X12X1′Xo−λ1Y1X1Xo′.
Now, after combining (16) and (17) and then comparing the coefficients of Xo and Yo, at λo=λ1=λXo, we obtain
(18)vY1X1−Y12X12λ=−v1Y1X1λ−λ.
The above expression yields after simplification and the 1-fold Darboux transformation as follows:(19)v1=−v+Y1X1−X1Y1λ1,(20)v1=−v+Y1X1−1−X1Y1−1λ1,where in the above expression (20), v is the old solution of the classical Painlevé II equation that generates its new solution v1 having X1 and Y1 as particular solutions of the system (14) at λ=λ1. Now, this can be shown that after substituting the values of λ1X1 and λ1X1 in (20) from the linear system (14), the 1-fold Darboux transformation can be expressed as
(21)v1=−v+ddxlnX1Y1.
Above result shows that the classical Painlevé second Darboux transformation in the additive structure where as if we pursue with the old Lax pair, the Darboux expression will be in product form which mostly happened for noncommutative case not recommended in the classical framework.


## 3.N-Fold Darboux Solutions in Terms of Wronskians

Here, we generalize the Darboux transformation in terms of Wronskian. The iterated Darbous solutions can be describe in a more compact way as Nth wronskian of the components of the eigenfunctions.


### 3.1. 1-Fold Darboux Solution

With the following settings,(22)Yi1=λiYi,Xi1=λiXi.The 1-fold Darboux transformationX1, Y1, and v1 can be written as below
(23)X1=W1X1,Y1,Xo,Yo2W1X1,Y11,Y1=W2X1,Y1,Xo,Yo2W2X1,Y11,(24)v1=−v+∂xlnW1X1,Y11W2X1,Y11,where W1 and W2 are Wronskians defined as
(25)W12=X1XoY11Yo1,W22=Y1YoX11Xo1,W11=X1,W21=Y1.


### 3.2. Wronskians for 2-Fold Darboux Transformation

The 2-fold Darboux transformation for theX component can be written as
(26)X2=λoYo1−λ1Y11X11Xo1,or
(27)X2=Yo11−Y111X111Xo1.And now, in terms of Wronskian, it can be expressed as(28)X2=W1Xk,Yk,Xo,Yo3W1Xk,Yk2.Similarly, for theY component, we have
(29)Y2=λoXo1−λ1X11Y11Yo1,which equivalently can be expressed as
(30)Y2=Xo11−X111Y11Yo1,or
(31)Y2=W2Xk,Yk,Xo,Yo3W2Xk,Yk2,where k=1,2. Now, the second iteration on (21) yields the next new solution to equation (1)
(32)v2=−v+∂xlnW1X1,Y12W2X1,Y12,where the Wronskians are defined as follows:
(33)W13=Y2Y1YoX21X11Xo1Y22Y12Yo2,W23=X2X1XoY21Y11Yo1X22X12Xo2,W12=X2X1Y21Y11,W22=Y2Y1X21X11.


### 3.3.N-Fold Darboux Transformation

For theN-time iterated Darboux transformation, we have the following results for X and Y components, respectively
(34)XN=W1Xk,Yk,Xo,YoN+1W1Xk,YkN,YN=W2Xk,Yk,Xo,YoN+1W2Xk,YkN,where k=1,2,⋯N and Xk and Yk are particular solutions of the Lax pair at λ=λk.For odd values ofN, determinants W1 and W2 are
(35)W1N+1=XNXN−1⋯X1Xo⋮YNN−1YN−1N−1⋯Y1N−1YoN−1⋮XNNXN−1N⋯X1NXoN,W2N+1=YNYN−1⋯Y1Yo⋮XNN−1XN−1N−1⋯X1N−1XoN−1⋮YNNYN−1N⋯Y1NYoN.For even values ofN, we have the following determinants:
(36)W1N+1=YNYN−1⋯Y1Y0⋮XNN−1XN−1N−1⋯X1N−1XoN−1⋮YNNYN−1N⋯Y1NYoN,W2N+1=XNXN−1⋯X1X0⋮YNN−1YN−1N−1⋯Y1N−1oN−1⋮XNNXN−1N⋯X1NXoN.Finally, the all possible Darboux solutions of (1) can generalized to the N-th form in terms of Wronskian as below
(37)vN=−v+∂xlnW1Xk,YkNW2Xk,YkN,where v is the initial or “seed” solution that can be taken as trivial at α=0 in (1), and with this simplest choice, one can construct all possible nontrivial solutions to that equation.


## 3.1. 1-Fold Darboux Solution

With the following settings,(22)Yi1=λiYi,Xi1=λiXi.The 1-fold Darboux transformationX1, Y1, and v1 can be written as below
(23)X1=W1X1,Y1,Xo,Yo2W1X1,Y11,Y1=W2X1,Y1,Xo,Yo2W2X1,Y11,(24)v1=−v+∂xlnW1X1,Y11W2X1,Y11,where W1 and W2 are Wronskians defined as
(25)W12=X1XoY11Yo1,W22=Y1YoX11Xo1,W11=X1,W21=Y1.


## 3.2. Wronskians for 2-Fold Darboux Transformation

The 2-fold Darboux transformation for theX component can be written as
(26)X2=λoYo1−λ1Y11X11Xo1,or
(27)X2=Yo11−Y111X111Xo1.And now, in terms of Wronskian, it can be expressed as(28)X2=W1Xk,Yk,Xo,Yo3W1Xk,Yk2.Similarly, for theY component, we have
(29)Y2=λoXo1−λ1X11Y11Yo1,which equivalently can be expressed as
(30)Y2=Xo11−X111Y11Yo1,or
(31)Y2=W2Xk,Yk,Xo,Yo3W2Xk,Yk2,where k=1,2. Now, the second iteration on (21) yields the next new solution to equation (1)
(32)v2=−v+∂xlnW1X1,Y12W2X1,Y12,where the Wronskians are defined as follows:
(33)W13=Y2Y1YoX21X11Xo1Y22Y12Yo2,W23=X2X1XoY21Y11Yo1X22X12Xo2,W12=X2X1Y21Y11,W22=Y2Y1X21X11.


## 3.3.N-Fold Darboux Transformation

For theN-time iterated Darboux transformation, we have the following results for X and Y components, respectively
(34)XN=W1Xk,Yk,Xo,YoN+1W1Xk,YkN,YN=W2Xk,Yk,Xo,YoN+1W2Xk,YkN,where k=1,2,⋯N and Xk and Yk are particular solutions of the Lax pair at λ=λk.For odd values ofN, determinants W1 and W2 are
(35)W1N+1=XNXN−1⋯X1Xo⋮YNN−1YN−1N−1⋯Y1N−1YoN−1⋮XNNXN−1N⋯X1NXoN,W2N+1=YNYN−1⋯Y1Yo⋮XNN−1XN−1N−1⋯X1N−1XoN−1⋮YNNYN−1N⋯Y1NYoN.For even values ofN, we have the following determinants:
(36)W1N+1=YNYN−1⋯Y1Y0⋮XNN−1XN−1N−1⋯X1N−1XoN−1⋮YNNYN−1N⋯Y1NYoN,W2N+1=XNXN−1⋯X1X0⋮YNN−1YN−1N−1⋯Y1N−1oN−1⋮XNNXN−1N⋯X1NXoN.Finally, the all possible Darboux solutions of (1) can generalized to the N-th form in terms of Wronskian as below
(37)vN=−v+∂xlnW1Xk,YkNW2Xk,YkN,where v is the initial or “seed” solution that can be taken as trivial at α=0 in (1), and with this simplest choice, one can construct all possible nontrivial solutions to that equation.


## 4. Conclusion

Here, we have presented other equivalent Lax representation classical Painlevé second equation through the gauge-like transformations. This has been shown that the Lax pair in the new setting remained helpful to construct the Darboux transformation in additive structure as other integrable systems possess. That transformations may yield all possible nontrivial Darboux solutions of the classical Painlevé second equation taking seed solution as zero in that transformation. Finally,N-fold Darboux transformations have been presented in terms of Wronskians. Still, work is in progress to construct the exact solution through its Darboux transformation involving the associated Riccati forms of classical Painlevé second.



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