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This paper investigates a real version of a (2 + 1) dimensional nonlinear Schr
ödinger equation through adoption of Painlevé test by means of which the (2 + 1) dimensional nonlinear Schr
ödinger 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
äcklund transformation and bilinear form is directly obtained from the Painlevé test.

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 [

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 [

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 [

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

p z − μ 2 p x x − p y y − p S − δ p 2 q = 0 ,

q z + μ 2 q x x + q y y + q S + δ q 2 p = 0 ,

S x x − μ 2 S y y + 2 δ ( p q ) x x = 0. (1)

In Equation (1), S is called a forcing term which shows the real quantity and μ 2 = 1 or μ 2 = − 1 prescribes the elliptic or hyperbolic nature respectively.

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) [

ODE was explained by Ablowitz et al. [

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.

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.

For estimation of leading singularity let’s suppose that:

p ~ p 0 ξ α 1 , q ~ q 0 ξ α 2 , s ~ s 0 ξ α 3 (2)

whereas α j ( j = 1 , 2 , 3 ) will be all negative integers, ξ = x − g ( y , z ) and ( p 0 , q 0 , s 0 ) all are functions of ( y , z ) . The same assumption were used by Jimbo et al. [

According to the condition

δ p 0 q 0 = 2 ( μ 2 ξ x 2 − ξ y 2 ) , s 0 = − 4 μ 2 ξ x 2 . (3)

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

P = ∑ i = 0 ∞ p i ξ i + α 1 , q = ∑ i = 0 ∞ q i ξ i + α 2 , S = ∑ i = 0 ∞ s i ξ i + α 3 . (4)

This equation contains six arbitrary functions with p i , q i , s i in addition to the increasing function ξ , 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 ( ξ i − 3 , ξ i − 3 , ξ i − 4 ) we get

[ δ q 0 2 G − 4 ξ y 2 q 0 − G + 4 ξ y 2 − δ p 0 2 − p 0 2 δ q 0 I ξ x 2 2 δ p 0 I ξ x 2 I ( ξ x 2 − μ 2 ξ y 2 ) ] [ p i q i s i ] = 0 , (5)

where

G = ( i − 1 ) ( i − 2 ) ( μ 2 ξ x 2 + ξ y 2 )

And

I = ( i − 2 ) ( i − 3 )

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

det [ . ] = i ( i + 1 ) ( i − 2 ) ( i − 3 ) 2 ( i − 4 )

Therefore we get the resonances at

i = − 1 , 0 , 2 , 3 , 3 , 4. (6)

Resonance at first position i = −1 is observed to relate the arbitrary singularity manifold ξ . 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

ξ = x + g ( y , z ) , p i = p i ( y , z ) , q i = q i ( y , z ) , s i = s i ( y , z ) . (7)

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

δ p 0 q 0 = 2 ( μ 2 − g y 2 ) , s 0 = − 4 μ 2 (8)

Case a: For i = −1.

This point to the arbitrariness of

ξ = x + g ( y , z ) (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 ( ξ − 2 , ξ − 2 , ξ − 3 ) we get,

p 1 = ( μ 2 ( 3 g y y − g z ) g y 2 + g z + g y y ) q 0 + 2 q 0 y g y ( 1 − μ 2 g y 2 ) δ q 0 2 ( 1 + μ 2 g y 2 ) (10)

q 1 = μ 2 ( q 0 g y y − q 0 g z − 2 q 0 y g y ) ( 1 + μ 2 g y 2 ) (11)

s 1 = 0 (12)

Case d: For i = 2,

Eliminating the coefficients of ( ξ − 1 , ξ − 1 , ξ − 2 ) we get,

p 0 z + 4 g y 2 p 2 − δ ( q 2 p 0 2 + q 0 p 1 2 ) − p 0 s 2 − p 0 y y − ( s 1 + 2 δ q 1 p 0 ) p 1 = 0 (13)

q 0 z − 4 g y 2 q 2 + δ ( p 2 q 0 2 + p 0 q 1 2 ) + q 0 s 2 + q 0 y y + ( s 1 + 2 δ p 1 q 0 ) q 1 = 0 (14)

s 0 y y − 2 s 1 y g y − s 1 g y y = 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 ( ξ 0 , ξ 0 , ξ − 1 ) and according to Equation (8) we get,

2 ( μ 2 − g y 2 ) p 3 + ( g y y + 2 δ q 1 p 0 + 2 δ q 0 p 1 − g z + s 1 ) p 2 + 2 p 2 y g y + p 1 y y + ( δ q 3 p 0 + s 3 ) p 0 + ( δ q 1 p 1 + s 2 + 2 δ q 2 p 0 ) p 1 − p 1 z = 0 (16)

2 ( μ 2 − g y 2 ) q 3 + ( g y y + 2 δ q 0 p 1 + 2 δ q 1 p 0 + g z + s 1 ) q 2 + 2 q 2 y g y + q 1 y y + ( δ p 3 q 0 + s 3 ) q 0 + ( δ p 1 q 1 + s 2 + 2 δ p 2 q 0 ) q 1 − q 1 z = 0 (17)

s 1 y y = 0 (18)

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

s 3 = p 0 − 1 [ p 1 z − 2 ( μ 2 − g y 2 ) p 3 − ( g y y + 2 δ q 1 p 0 + 2 δ q 0 p 1 − g t + s 1 ) p 2 − ( δ q 1 p 1 + s 2 + 2 δ q 2 p 0 ) p 1 − p 1 y y − 2 p 2 y g y − δ q 3 p 0 2 ] (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 ( ξ , ξ , ξ 0 ) and according to Equation (8) and Equation (11) we get,

( 3 μ 2 + g y 2 ) p 4 + δ p 0 2 q 4 + p 0 s 4 + ( 2 δ q 1 p 0 2 g z + 2 g y y ) p 3 + 4 p 3 y g y − p 2 z + p 2 y y + ( 2 δ q 3 p 0 + s 3 + q 2 p 1 ) p 1 + ( δ q 0 p 2 + s 2 + 2 δ p 1 q 1 + 2 δ q 2 p 0 ) p 2 = 0 (20)

( 3 μ 2 + g y 2 ) q 4 + δ q 0 2 p 4 + q 0 s 4 + ( 2 δ q 0 p 1 − 2 g z + 2 g y y ) q 3 + 4 q 3 y g y + q 2 t + q 2 y y + ( 2 δ p 3 q 0 + s 3 + q 1 p 2 ) q 1 + ( δ p 0 q 2 + s 2 + 2 δ p 1 q 1 + 2 δ q 0 p 2 ) q 2 = 0 (21)

4 δ μ 2 ( q 0 p 4 + p 0 q 4 + p 3 q 1 + p 2 q 2 + p 1 q 3 ) + 2 ( μ 2 − f y 2 ) s 4 − 2 s 3 y f y − s 3 f y y − s 2 y y = 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.

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

p = p 0 ξ + p 1 , q = q 0 ξ + q 1 , S = s 0 ξ 2 + s 1 ξ + s 2 . (23)

Here p 1 = p 1 ( x , y , z ) , q 1 = q 1 ( x , y , z ) and s 1 = s 1 ( x , y , z ) , s 2 = s 2 ( x , y , z ) .

where the pair of functions ( p , p 1 ) ; ( q , q 1 ) and ( s , s 2 ) satisfies Equation (1), and hence Equation (23) may be treated as the associated Bäcklund transformation of Equation (1).

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 = p 0 ξ , q = q 0 ξ , S = s 0 ξ 2 + s 1 ξ . (24)

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

P = V U , q = W U , S = 4 μ 2 ( ln U ) x x . (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,

( D z − μ 2 D x 2 − D y 2 ) W ⋅ U = 0 , (26)

( D z + μ 2 D x 2 + D y 2 ) V ⋅ U = 0 , (27)

( μ 2 D x 2 − D y 2 ) U ⋅ U + δ V ⋅ W = 0. (28)

Here D is called the bilinear operator defined by [

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.

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.

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.

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