Painlevé Analysis for (2 + 1) Dimensional Non-Linear Schrödinger Equation

ABSTRACT

This paper investigates a real version of a (2 + 1) dimensional nonlinear
Schr?dinger equation through adoption of Painlevé test by means of which
the (2 + 1) dimensional nonlinear Schr?dinger equation is studied according
to the Weiss *et al*. method and Kruskal’s simplification algorithms. According
to Painlevé test, it is found that the number of arbitrary functions required for
explaining the Cauchy-Kovalevskaya theorem exist. Finally, the associated
B?cklund transformation and bilinear form is directly obtained from the
Painlevé test.

KEYWORDS

(2 + 1) Dimensional Nonlinear Schrödinger Equation, Painlevé Analysis, Bäcklund Transformation, Bilinear Form

(2 + 1) Dimensional Nonlinear Schrödinger Equation, Painlevé Analysis, Bäcklund Transformation, Bilinear Form

1. Introduction

One of the most powerful nonlinear equations that has been discussed and used in different field of physics is the Non-linear Schrödinger equation (NLSE) in (2 + 1) dimension. The structure and integrability of (2 + 1) dimensional equations have received considerable attention in the last few years [1] [2] . Non-linear Schrödinger type equations are a particular case of interest. These equations were discovered by Calogero [3] and then discussed by Zakharov [4] . The geometrical properties of these equations have been studied [5] .

Historically, more and more problems involved nonlinearity; much attention had been paid to the integrability of nonlinear models. Close relationships were observed between Painlevé test and the integrability of non-linear Partial Differential Equations (PDEs). Therefore it carries significant importance to investigate whether the non-linear PDEs have the Painlevé characteristics. In order to verify this proposition various methods have been historically used by researchers like Ablowitz-Ramani-Segur (ARS) method [6] , the Weiss-Tabor-Carnevale (WTC) method [7] , the Kruskal simplification method [8] , Conte invariant method [9] and Pickering approach method [10] .

Various methods have verified Painlevé property but the Kruskal’s simplification and the Weiss-Tabor-Carnevale (WTC) methods are frequently employed for this verification. WTC method is significantly helpful in Painlevé test where non-linear PDEs are involved as tests of symmetry, Hirota’s method of bilinear forms and special solutions. However, this is less significant when applied to the coupled equation system or equation where high resonance is observable. This study proposes another prospective approach i.e. to apply WTC method with Kruskal’s simplification for verification of Painlevé test.

By observing chaotic behavior of certain systems and their non-integrability, significant advancement had been observed with respect to research on the integrability of non-linear PDEs thus leading to analysis of manifold classes of equations employing Painlevé test as a base [11] [12] [13] . Painlevé test had being used in various situations, although was neither observed as a necessity nor a condition for integrability of PDEs. The current study proposes a Painleve test analysis using (2 + 1) dimensions, for testing the integrability of Non-linear Schrödinger equation (NLSE). It was observed that such approach could satisfy requirements of Cauchy-Kovalevskaya theorem while providing satisfactory results.

The (2 + 1) dimensional NLSE under taken in this study is:

${p}_{z}-{\mu}^{2}{p}_{xx}-{p}_{yy}-pS-\delta {p}^{2}q=0,$

${q}_{z}+{\mu}^{2}{q}_{xx}+{q}_{yy}+qS+\delta {q}^{2}p=0,$

${S}_{xx}-{\mu}^{2}{S}_{yy}+2\delta {\left(pq\right)}_{xx}=0.$ (1)

In Equation (1), S is called a forcing term which shows the real quantity and ${\mu}^{2}=1$ or ${\mu}^{2}=-1$ prescribes the elliptic or hyperbolic nature respectively.

2. Painlevé Analysis

Painlevé test is frequently used to determine integrability of non-linear equations. A PDE which lacks any movable dimension (Algebraic or Logarithmic) is called Painlevé-type (P-type) [14] . An ordinary differential equation (ODE) carries movable singularities without having movable branch points which place the same ODE into P-type equations. P-test to analyze the singularities structure for

ODE was explained by Ablowitz et al. [15] , further extended by Weiss-Tabor- Carnevale (WTC) to PDE [16] the method provides a great basis for investigation of integrability of various non-linear equations. A brief procedure for WTC is explained below.

Firstly, perform the leading order analysis, secondly, identify resonance position of arbitrary functions as per Laurent series and finally, existence of a significant number of arbitrary functions are verified without the introduction of movable critical manifold.

3. (2 + 1) Dimensional Non-Linear Schrodinger Equation

In this paper the Painlevé test is performed for (2 + 1) dimensional non-linear Schrödinger equation Equation (1) as described in three steps constituting the Kruskal’s simplification algorithms such as determination of leading order, determination of resonance position and verification of the resonance conditions.

3.1. Leading Order Analysis

For estimation of leading singularity let’s suppose that:

$p~{p}_{0}{\xi}^{{\alpha}_{1}},\text{\hspace{0.17em}}q~{q}_{0}{\xi}^{{\alpha}_{2}},\text{\hspace{0.17em}}s~{s}_{0}{\xi}^{{\alpha}_{3}}$ (2)

whereas ${\alpha}_{j}\left(j=1,2,3\right)$ will be all negative integers, $\xi =x-g\left(y,z\right)$ and $\left({p}_{0},{q}_{0},{s}_{0}\right)$ all are functions of $\left(y,z\right)$ . The same assumption were used by Jimbo et al. [17] and Goldstein & Infeld [18] the same were previously proposed in private communication by M. Kruskal [19] . As per original explanation of Weiss et al., [17] $\xi $ is a function of $\left(x,y,z\right)$ . Substituting Equation (2) in Equation (1) followed by balancing the leading term of $\xi $ , and considering that these exponents will be all negative integers, get the only possible values i.e. ${\alpha}_{1}={\alpha}_{2}=-1$ and ${\alpha}_{3}=-2$ .

According to the condition

$\delta {p}_{0}{q}_{0}=2\left({\mu}^{2}{\xi}_{x}^{2}-{\xi}_{y}^{2}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{0}=-4{\mu}^{2}{\xi}_{x}^{2}.$ (3)

3.2. Determine the Resonance Position

As per WTC method, the system of Equation (1) which could satisfy the Painlevé test with the general solution is presented as:

$P={\displaystyle {\sum}_{i=0}^{\infty}{p}_{i}{\xi}^{i+{\alpha}_{1}}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}q={\displaystyle {\sum}_{i=0}^{\infty}{q}_{i}{\xi}^{i+{\alpha}_{2}}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}S={\displaystyle {\sum}_{i=0}^{\infty}{s}_{i}{\xi}^{i+{\alpha}_{3}}}.$ (4)

This equation contains six arbitrary functions with ${p}_{i},{q}_{i},{s}_{i}$ in addition to the increasing function $\xi $ , where the final solution is a single value regarding the arbitrary movable non-characteristic singular manifold as shown in the Laurent series.

Now substituting Equation (2), Equation (3) and Equation (4) into Equation (1) and equating the coefficients of $\left({\xi}^{i-3},{\xi}^{i-3},{\xi}^{i-4}\right)$ we get

$\left[\begin{array}{ccc}\delta {q}_{0}^{2}& G-4{\xi}_{y}^{2}& {q}_{0}\\ -G+4{\xi}_{y}^{2}& -\delta {p}_{0}^{2}& -{p}_{0}\\ 2\delta {q}_{0}I{\xi}_{x}^{2}& 2\delta {p}_{0}I{\xi}_{x}^{2}& I\left({\xi}_{x}^{2}-{\mu}^{2}{\xi}_{y}^{2}\right)\end{array}\right]\left[\begin{array}{c}{p}_{i}\\ {q}_{i}\\ {s}_{i}\end{array}\right]=0$ , (5)

where

$G=\left(i-1\right)\left(i-2\right)\left({\mu}^{2}{\xi}_{x}^{2}+{\xi}_{y}^{2}\right)$

And

$I=\left(i-2\right)\left(i-3\right)$

That value at which the determinant of the system matrix vanishes is called resonance position. It is establish that

$\mathrm{det}\left[\text{.}\right]=i\left(i+1\right)\left(i-2\right){\left(i-3\right)}^{2}\left(i-4\right)$

Therefore we get the resonances at

$i=-1,0,2,3,3,4.$ (6)

3.3. Coefficient of Expansion at the Resonance Position

Resonance at first position i = −1 is observed to relate the arbitrary singularity manifold $\xi $ . For the Equation (1) to pass the Painlevé test, it is required that second resonance position at i = 0, third at i = 2, fourth & fifth at i = 3, 3 and sixth at i = 4 should be identically satisfied. We utilize the Kruskal’s simplification for verification of resonance conditions, such as

$\xi =x+g\left(y,z\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{p}_{i}={p}_{i}\left(y,z\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{q}_{i}={q}_{i}\left(y,z\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{i}={s}_{i}\left(y,z\right).$ (7)

With the resolution of Equation (7), Equation (3) is decreased to

$\delta {p}_{0}{q}_{0}=2\left({\mu}^{2}-{g}_{y}^{2}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{s}_{0}=-4{\mu}^{2}$ (8)

Case a: For i = −1.

This point to the arbitrariness of

$\xi =x+g\left(y,z\right)$ (9)

Case b: For i = 0.

This point to the arbitrariness of ${p}_{0}$ or ${q}_{0}$ (See Equation (3))

Case c: For i = 1,

Equating the coefficients of $\left({\xi}^{-2},{\xi}^{-2},{\xi}^{-3}\right)$ we get,

${p}_{1}=\frac{\left({\mu}^{2}\left(3{g}_{yy}-{g}_{z}\right){g}_{y}^{2}+{g}_{z}+{g}_{yy}\right){q}_{0}+2{q}_{0y}{g}_{y}\left(1-{\mu}^{2}{g}_{y}^{2}\right)}{\delta {q}_{0}^{2}\left(1+{\mu}^{2}{g}_{y}^{2}\right)}$ (10)

${q}_{1}=\frac{{\mu}^{2}\left({q}_{0}{g}_{yy}-{q}_{0}{g}_{z}-2{q}_{0y}{g}_{y}\right)}{\left(1+{\mu}^{2}{g}_{y}^{2}\right)}$ (11)

${s}_{1}=0$ (12)

Case d: For i = 2,

Eliminating the coefficients of $\left({\xi}^{-1},{\xi}^{-1},{\xi}^{-2}\right)$ we get,

${p}_{0z}+4{g}_{y}^{2}{p}_{2}-\delta \left({q}_{2}{p}_{0}^{2}+{q}_{0}{p}_{1}^{2}\right)-{p}_{0}{s}_{2}-{p}_{0yy}-\left({s}_{1}+2\delta {q}_{1}{p}_{0}\right){p}_{1}=0$ (13)

${q}_{0z}-4{g}_{y}^{2}{q}_{2}+\delta \left({p}_{2}{q}_{0}^{2}+{p}_{0}{q}_{1}^{2}\right)+{q}_{0}{s}_{2}+{q}_{0yy}+\left({s}_{1}+2\delta {p}_{1}{q}_{0}\right){q}_{1}=0$ (14)

${s}_{0yy}-2{s}_{1y}{g}_{y}-{s}_{1}{g}_{yy}=0$ (15)

From Equation (8) it is observed that Equation (15) is identically satisfied viva Equation (8) and (12).

And hence there are only two Equations (13) and (14) for three unknown variables ${p}_{2}$ , ${q}_{2}$ and ${s}_{2}$ which clarifies that one of them is arbitrary. Therefore the resonance condition at i = 2 is satisfied.

Case e: For i = 3 (double resonance).

By collecting the coefficients of $\left({\xi}^{0},{\xi}^{0},{\xi}^{-1}\right)$ and according to Equation (8) we get,

$\begin{array}{l}2\left({\mu}^{2}-{g}_{y}^{2}\right){p}_{3}+\left({g}_{yy}+2\delta {q}_{1}{p}_{0}+2\delta {q}_{0}{p}_{1}-{g}_{z}+{s}_{1}\right){p}_{2}+2{p}_{2y}{g}_{y}+{p}_{1yy}\\ +\left(\delta {q}_{3}{p}_{0}+{s}_{3}\right){p}_{0}+\left(\delta {q}_{1}{p}_{1}+{s}_{2}+2\delta {q}_{2}{p}_{0}\right){p}_{1}-{p}_{1z}=0\end{array}$ (16)

$\begin{array}{l}2\left({\mu}^{2}-{g}_{y}^{2}\right){q}_{3}+\left({g}_{yy}+2\delta {q}_{0}{p}_{1}+2\delta {q}_{1}{p}_{0}+{g}_{z}+{s}_{1}\right){q}_{2}+2{q}_{2y}{g}_{y}+{q}_{1yy}\\ +\left(\delta {p}_{3}{q}_{0}+{s}_{3}\right){q}_{0}+\left(\delta {p}_{1}{q}_{1}+{s}_{2}+2\delta {p}_{2}{q}_{0}\right){q}_{1}-{q}_{1z}=0\end{array}$ (17)

${s}_{1yy}=0$ (18)

Obviously due to Equation (12), Equation (18) is identically satisfied. Now we solve ${s}_{3}$ from Equation (17) we get,

$\begin{array}{c}{s}_{3}={p}_{0}^{-1}[{p}_{1z}-2\left({\mu}^{2}-{g}_{y}^{2}\right){p}_{3}-\left({g}_{yy}+2\delta {q}_{1}{p}_{0}+2\delta {q}_{0}{p}_{1}-{g}_{t}+{s}_{1}\right){p}_{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\delta {q}_{1}{p}_{1}+{s}_{2}+2\delta {q}_{2}{p}_{0}\right){p}_{1}-{p}_{1yy}-2{p}_{2y}{g}_{y}-\delta {q}_{3}{p}_{0}^{2}]\end{array}$ (19)

Now putting Equation (19) into Equation (17) and using Equations (10)-(14), we find that Equation (17) is satisfied identically, which shows that ${p}_{3}$ and ${q}_{3}$ must be arbitrary if ${s}_{3}$ is determined by Equation (19). Hence the resonance condition at i = 3 is satisfied.

Case f: for i = 4.

Vanishing the coefficients of $\left(\xi ,\xi ,{\xi}^{0}\right)$ and according to Equation (8) and Equation (11) we get,

$\begin{array}{l}\left(3{\mu}^{2}+{g}_{y}^{2}\right){p}_{4}+\delta {p}_{0}^{2}{q}_{4}+{p}_{0}{s}_{4}+\left(2\delta {q}_{1}{p}_{0}2{g}_{z}+2{g}_{yy}\right){p}_{3}+4{p}_{3y}{g}_{y}-{p}_{2z}+{p}_{2yy}\\ +\left(2\delta {q}_{3}{p}_{0}+{s}_{3}+{q}_{2}{p}_{1}\right){p}_{1}+\left(\delta {q}_{0}{p}_{2}+{s}_{2}+2\delta {p}_{1}{q}_{1}+2\delta {q}_{2}{p}_{0}\right){p}_{2}=0\end{array}$ (20)

$\begin{array}{l}\left(3{\mu}^{2}+{g}_{y}^{2}\right){q}_{4}+\delta {q}_{0}^{2}{p}_{4}+{q}_{0}{s}_{4}+\left(2\delta {q}_{0}{p}_{1}-2{g}_{z}+2{g}_{yy}\right){q}_{3}+4{q}_{3y}{g}_{y}+{q}_{2t}+{q}_{2yy}\\ +\left(2\delta {p}_{3}{q}_{0}+{s}_{3}+{q}_{1}{p}_{2}\right){q}_{1}+\left(\delta {p}_{0}{q}_{2}+{s}_{2}+2\delta {p}_{1}{q}_{1}+2\delta {q}_{0}{p}_{2}\right){q}_{2}=0\end{array}$ (21)

$4\delta {\mu}^{2}\left({q}_{0}{p}_{4}+{p}_{0}{q}_{4}+{p}_{3}{q}_{1}+{p}_{2}{q}_{2}+{p}_{1}{q}_{3}\right)+2\left({\mu}^{2}-{f}_{y}^{2}\right){s}_{4}-2{s}_{3y}{f}_{y}-{s}_{3}{f}_{yy}-{s}_{2yy}=0$ (22)

Firstly we simplify Equations (20) and (21) for ${p}_{4}$ and ${q}_{4}$ and then putting into Equation (22) according to Equations (8) (10) (11) (13) (14), and (19) we can find that Equation (22) is identically satisfied.

So any one of ${p}_{4}$ , ${q}_{4}$ and ${s}_{4}$ is arbitrary. Therefore the resonance condition at i = 4 is also satisfied.

Hence we proved that the Equation (1) satisfies the Painlevé test.

4. Associated Bäcklund Transformation

To construct the Bäcklund transformation of Equation (1), let us truncate the Laurent series at the constant level term to give

$p=\frac{{p}_{0}}{\xi}+{p}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}q=\frac{{q}_{0}}{\xi}+{q}_{1},\text{\hspace{0.17em}}\text{\hspace{0.17em}}S=\frac{{s}_{0}}{{\xi}^{2}}+\frac{{s}_{1}}{\xi}+{s}_{2}.$ (23)

Here ${p}_{1}={p}_{1}\left(x,y,z\right)$ , ${q}_{1}={q}_{1}\left(x,y,z\right)$ and ${s}_{1}={s}_{1}\left(x,y,z\right)$ , ${s}_{2}={s}_{2}\left(x,y,z\right)$ .

where the pair of functions $\left(p,{p}_{1}\right)$ ; $\left(q,{q}_{1}\right)$ and $\left(s,{s}_{2}\right)$ satisfies Equation (1), and hence Equation (23) may be treated as the associated Bäcklund transformation of Equation (1).

5. Bilinear Form

In order to derive the Hirota’s bilinear form we consider the vacuum solutions ${p}_{1}={q}_{1}={s}_{2}=0$ in Equation (23) then we have

$p=\frac{{p}_{0}}{\xi},\text{\hspace{0.17em}}\text{\hspace{0.17em}}q=\frac{{q}_{0}}{\xi},\text{\hspace{0.17em}}\text{\hspace{0.17em}}S=\frac{{s}_{0}}{{\xi}^{2}}+\frac{{s}_{1}}{\xi}.$ (24)

This suggest that we take the Hirota’s bilinear transformation in the form

$P=\frac{V}{U},\text{\hspace{0.17em}}\text{\hspace{0.17em}}q=\frac{W}{U},\text{\hspace{0.17em}}\text{\hspace{0.17em}}S=4{\mu}^{2}{\left(\mathrm{ln}U\right)}_{xx}.$ (25)

In Equation (25) U, V and W are functions of variables x, y and z where U is a real function and V, W are complex functions. Using Equation (25) and the Hirota’s bilinear operator Equation (24), Equation (1) can be transforming into bilinear forms as,

$\left({D}_{z}-{\mu}^{2}{D}_{x}^{2}-{D}_{y}^{2}\right)W\cdot U=0,$ (26)

$\left({D}_{z}+{\mu}^{2}{D}_{x}^{2}+{D}_{y}^{2}\right)V\cdot U=0,$ (27)

$\left({\mu}^{2}{D}_{x}^{2}-{D}_{y}^{2}\right)U\cdot U+\delta V\cdot W=0.$ (28)

Here D is called the bilinear operator defined by [20] .

The power-series can be further used with the bilinear form obtained in this paper to construct a soliton solution by expanding the dependent variables.

6. Conclusions

A demonstration on the system which passes the Painlevé test is presented in this paper and further explained that the system has multi-linear variable solutions having arbitrary function, also the associated Bäcklund transformation and bilinear form are obtained directly from the Painlevé test.

The analysis points that our equation having six positions of resonance at i = −1, 0, 2, 3, 3, 4, there are six arbitrary functions at these resonance positions. Therefore our equation conform the Painlevé criterion of integrability as explained in Cauchy-Kovalevskaya theorem. However, there is one further proposition of this paper that Painlevé analysis cannot be used to derive the Lax pair because in the case of coupled system there is still no specialized method for derivation of the Lax pair for coupled non-linear equations.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (grant No. 11371361), the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology (2014), and the Key Discipline Construction by China University of Mining and Technology (Grant No. XZD 201602). The authors thank to the reviewer’s for their helpful advices.

Cite this paper

Iqbal, M. and Zhang, Y. (2017) Painlevé Analysis for (2 + 1) Dimensional Non-Linear Schrödinger Equation.*Applied Mathematics*, **8**, 1539-1545. doi: 10.4236/am.2017.811112.

Iqbal, M. and Zhang, Y. (2017) Painlevé Analysis for (2 + 1) Dimensional Non-Linear Schrödinger Equation.

References

[1] Ablowitz, M.J. and Clarkson, P.A. (1992) Solitons, Non-Linear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.

[2] Konopelchenko, B.J. (1993) Solitons in Multi-Dimension. World Scientific, Singapore. https://doi.org/10.1142/1982

[3] Calogero, F. (1975) Exactly Solvable One-Dimensional Many-Body Problems. Lettere al Nuovo Cimento, 14, 43.

[4] Zakharov, V.E. (1980) Solitons. Springer, Berlin.

[5] Mason, L.J. and Sparling, G.A.J. (1992) Twister Correspondences for the Soliton Hierarchies. Journal of Geometry and Physics, 8, 243.

[6] Ablowitz, M.J. and Clarkson, P.A. (1999) Solitons, Non-Linear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.

[7] Weiss, J., Tabor, M. and Carnevale, G. (1983) The Painlevé Property of Partial Differential Equations. Journal of Mathematical Physics, 24, 522-526. https://doi.org/10.1063/1.525721

[8] Jimbo, M., Kruskal, M.D. and Miwa, T. (1982) The Painlevé Test for the Self-Dual Yang-Mills Equations. Physics Letters A, 92, 59. https://doi.org/10.1016/0375-9601(82)90291-2

[9] Conte, R. (1989) Invariant Painlevé Analysis of Partial Differential Equations. Physics Letters A, 140, 383-389.

[10] Conte, R., Fordy, A.P. and Pickering, A. (1993) A Perturbation Painlevé Approach to Non-Linear Differential Equations. Journal of Physics D, 69, 33-58.

[11] Weiss, J. (1983) The Painlevé Property for Partial Differential Equations. II: Bäcklund Transformation, Lax Pairs, and the Schwarzian Derivative. Journal of Mathematical Physics, 24, 1405. https://doi.org/10.1063/1.525875

[12] Weiss, J. (1984) Bäcklund Transformation and Linearization’s of the Henon-Heiles System. Physics Letters, 102A, 329.

[13] Weiss, J. (1984) Bäcklund Transformation and the Henon-Heiles System. Physics Letters, 105A, 387.

[14] Ince, E.L. (1956) Ordinary Differential Equations. Dover, New York.

[15] Ablowitz, M.J., Ramani, A. and Segur, H.A. (1980) Connection between Nonlinear Evolution Equations and Ordinary Differential Equation of P Type. Journal of Mathematical Physics, 21, 715-721.

[16] Weiss, J., Tabor, M. and Carnevale, G. (1983) The Painlevé Property for Partial Differential Equations. Journal of Mathematical Physics, 24, 522-526. https://doi.org/10.1063/1.525721

[17] Jimbo, M., Kruskal, M.D. and Miwa, T. (1982) Painlevé Test for the Self-Dual Yang-Mills Equation. Physics Letters, 92A, 59.

[18] Goldstein, P. and Infeld, E. (1984) The Zakharov Equations: A non-Painlevé System with Exact N Soliton Solutions. Physics Letters, 103A, 8.

[19] Kruskal, M.D. (1982) Private Communications.

[20] Ablowitz, M.J. and Segur, H. (1981) Solitons and the Inverse Scattering Transform. SIAM, Philadelphia. https://doi.org/10.1137/1.9781611970883

[1] Ablowitz, M.J. and Clarkson, P.A. (1992) Solitons, Non-Linear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.

[2] Konopelchenko, B.J. (1993) Solitons in Multi-Dimension. World Scientific, Singapore. https://doi.org/10.1142/1982

[3] Calogero, F. (1975) Exactly Solvable One-Dimensional Many-Body Problems. Lettere al Nuovo Cimento, 14, 43.

[4] Zakharov, V.E. (1980) Solitons. Springer, Berlin.

[5] Mason, L.J. and Sparling, G.A.J. (1992) Twister Correspondences for the Soliton Hierarchies. Journal of Geometry and Physics, 8, 243.

[6] Ablowitz, M.J. and Clarkson, P.A. (1999) Solitons, Non-Linear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge.

[7] Weiss, J., Tabor, M. and Carnevale, G. (1983) The Painlevé Property of Partial Differential Equations. Journal of Mathematical Physics, 24, 522-526. https://doi.org/10.1063/1.525721

[8] Jimbo, M., Kruskal, M.D. and Miwa, T. (1982) The Painlevé Test for the Self-Dual Yang-Mills Equations. Physics Letters A, 92, 59. https://doi.org/10.1016/0375-9601(82)90291-2

[9] Conte, R. (1989) Invariant Painlevé Analysis of Partial Differential Equations. Physics Letters A, 140, 383-389.

[10] Conte, R., Fordy, A.P. and Pickering, A. (1993) A Perturbation Painlevé Approach to Non-Linear Differential Equations. Journal of Physics D, 69, 33-58.

[11] Weiss, J. (1983) The Painlevé Property for Partial Differential Equations. II: Bäcklund Transformation, Lax Pairs, and the Schwarzian Derivative. Journal of Mathematical Physics, 24, 1405. https://doi.org/10.1063/1.525875

[12] Weiss, J. (1984) Bäcklund Transformation and Linearization’s of the Henon-Heiles System. Physics Letters, 102A, 329.

[13] Weiss, J. (1984) Bäcklund Transformation and the Henon-Heiles System. Physics Letters, 105A, 387.

[14] Ince, E.L. (1956) Ordinary Differential Equations. Dover, New York.

[15] Ablowitz, M.J., Ramani, A. and Segur, H.A. (1980) Connection between Nonlinear Evolution Equations and Ordinary Differential Equation of P Type. Journal of Mathematical Physics, 21, 715-721.

[16] Weiss, J., Tabor, M. and Carnevale, G. (1983) The Painlevé Property for Partial Differential Equations. Journal of Mathematical Physics, 24, 522-526. https://doi.org/10.1063/1.525721

[17] Jimbo, M., Kruskal, M.D. and Miwa, T. (1982) Painlevé Test for the Self-Dual Yang-Mills Equation. Physics Letters, 92A, 59.

[18] Goldstein, P. and Infeld, E. (1984) The Zakharov Equations: A non-Painlevé System with Exact N Soliton Solutions. Physics Letters, 103A, 8.

[19] Kruskal, M.D. (1982) Private Communications.

[20] Ablowitz, M.J. and Segur, H. (1981) Solitons and the Inverse Scattering Transform. SIAM, Philadelphia. https://doi.org/10.1137/1.9781611970883