Bäcklund transformations for some non-linear evolution equations using Painlevè
analysis
M. F. El-Sabbagh, M. M. Hassan, E. Hamed
Mathematics Department, Faculty of Science, Minia University, Minia, Egypt.
Abstract
In this paper we present the Painlevè test for (3+1) -dimensional Yu, Toda, Sasa and
Fukuyama (YTSF) equation, the Ablowitz, Kaup, Newell and Segure (AKNS) equation,
the generalized shallow water wave equation and (3+1) -dimensional Jimbo-Miva
equation. The associated Bäcklund transformations are obtained directly from the
Painlevè test.
Keywords: (3+1)-dimensional YTSF equation, the AKNS equation, the generalized
shallow water wave equation, (3+1)-dimensional Jimbo-Miva equation and Painlevè
analysis.
1. Introduction and Painlevé analysis method
Painlevè property is a method of investigation for the integrability properties of
many NLEEs. If a PDE which has no points such as movable branch, algebraic and
logarithmic then is called P-type. An ordinary differential equation (ODE) might still
admit movable essential singularities without movable branch points. This method does
not identify essential singularities and therefore it provides only necessary conditions
for an ODE to be of P-type. Singularity structure analysis admitting the P-property
advocated by Ablowitz et al. for ODEs and extended to PDE by Weiss, Tabor and
Carnevale (WTC), plays a key role of investigating the integrability properties of many
NLEEs. The well-known procedure of WTC requires,
• The determination of leading orders Laurent series,
• The identification of powers at which the arbitrary functions can enter into
the Laurent series called resonances,
• Verifying that, at the resonance values, sufficient number of arbitrary
functions exist without introducing the movable critical manifold.
According to the WTC method, the general solution of PDE is in the form

u( x, t )    ( x, t ) u j ( x, t ) j ( x, t )
(1.1)
j 0
1
where α is negative,  ( x, t )  0 is the equation of singular manifold. The functions u j
(j=0,1,2,…) have to be determined by substitution of expansion (1.1) into the PDE, so
it becomes

E
j 0
j
(u 0 ,...u j ,  ) j  q ( x, t )  0 ,
(1.2)
where q is some negative constant. Ej depends on  only by the derivatives of  . The
successive practical steps of Painleve´ analysis are the following:
• Determine the possible leading orders p by balancing two or more terms of
the PDE and expressing that they dominate the other terms.
• Solve equation E0 = 0 for non-zero values of u0; this may lead to several
solutions, called branches.
• Find the resonances, i.e. the values of k for which uk cannot be determined
from equation Ej = 0. This last equation has generally the form
j>0
(1.3)
E j  ( j  1) P( j ) xjtn2 u j  Q(u0 ,...u j 1 ,  )  0 ,
where n is the order of the PDE, 0 ≤ j≤ n and P is a polynomial of degree n -1.
The values of the resonances are the zeros of P.
• Determine whether the resonances are ‘compatible’ or not. At resonance,
after substitution in (1.3) of the previously computed ui, i ≤ j -1, the
function Q is either zero or non-zero then in the case uj can arbitrarily be
chosen and the expansion (1.1) does not exist for arbitrary  , so the
resonance is called compatible.
2. The (3+1)-dimensional Yu, Toda, Sasa and Fukuyama equation
Consider the YTSF equation [15, 16]
 4u xt  u xxxz  4u x u xz  2u xx u z  3u yy  0.
(2.1)
We first present the Painlevè test of the potential-YTSF equation. The leading order of
solution of Eq. (2.1) is assumed as
u  u 0  .
(2.2)
Substituting Eq. (2.2) into (2.1) and equating the most dominant
terms, the following results are obtained
u 0  2 x . ,
  1
(2.3)
For finding the resonances, the full Laurent series.

u u 0    u j j 1 ,
1
(2.4)
j 1
is substituted into Eq. (2.1) and by equating the coefficients of  j 5 , the polynomial
equation in j is found as
j 4  10 j 3  23 j 2  10 j  24  0.
2
(2.5)
Using Eq. (2.5), the resonances are found to be
j = -1, 1, 4, 6.
(2.6)
As usual, the resonance at j= -1 corresponds to the arbitrariness of singular
manifold  ( x, y, z , t )  0 . In order to check the existence of sufficient number of
arbitrary functions at the other resonance values, the full Laurent expansion (2.4) is
substituted in Eq. (2.1). From the coefficient of  5 , the explicit value of u0 is obtained
as given in Eq. (2.3).
To construct a Bäcklund transformation of Eq. (2.1), let us truncate the Laurent
series at the constant level term to give
u  u 0 1  u1 .
Hence
u
2 x

(2.7)
 u1 ,
(2.8)
where the pair of function (u, u1) satisfy Eq. (2.1) and hence Eq. (2.8) is the associated
Bäcklund transformation of Eq. (2.1) relating a solution u with a known solution u1 of
the Eq. (2.1) which can be taken to be so.
We can also construct another Bäcklund transformation of Eq. (2.1) to be
u  u 0 1  u1  u 2 ,
(2.9)
where u, u1 satisfy Eq. (2.1) while u2 satisfying
u2 
1
2
2
2
2
[
4



8




8



16



12


xx
z
x
xxx
z
x
xxz
x
t
x
y
3
24 z x
 16u    8u   20 xxx xz  24 x xx xxz ],
2
1x x z
3
1z x
(2.10)
Hence Eq. (2.9) is a Bäcklund transformation too. Also let's truncate the Laurent series
again we get
u  u 0 1  u1  u 2  u 3 2 ,
(2.11)
where u, u1 satisfy Eq. (2.1) and u2 is given by Eq. (2.10) while u3 is given by
u3 
1
2
[
24
u

 xz  4 xx xxz  24u2 x z xx  6 x yy  8 x xxxz
2
x
3
24 z x
 2 xxxx z  3 xz xxx  8 xxt  12 xy y  16 x xt  8u1xz x2
 16u1x x xz  8u1x xx z  4u1xx x z  12u1z x xx ],
We can make more truncations to the Laurent series at the constant level term u 4 and it
will produce another Bäcklund transformation for Eq. (2.1).
3
3 .The Ablowitz, Kaup, Newell and Segure equation
Consider the AKNS equation [17]
4hxt  8hx hxz  4hx hxx  hxxxz  0,
(3.1)
We first present the Painlevè test of the AKNS equation. The leading order of solution
of Eq. (3.1) is assumed as

0
(3.2)
substituting Eq. (3.2) into (3.1) and equating the most dominant terms, the following
results are obtained
hh ,
h0   x .   1.
(3.3)
For finding the resonances, the full Laurent series.

h h 0    h j j 1 ,
1
(3.4)
j 1
j 5
is substituted into Eq. (3.1) and by equating the coefficients of  , the polynomial
equation in j is found as
j 4  10 j 3  23 j 2  10 j  24  0.
Using Eq. (3.5), the resonances are found to be
j = -1, 1, 4, 6
(3.5)
(3.6)
As usual, the resonance at j=-1 corresponds to the arbitrariness of singular manifold
 ( x, y, z , t )  0 . In order to check the existence of sufficient number of
arbitrary functions at the other resonance values, the full Laurent expansion (3.4) is
substituted in Eq. (3.1).
To construct the Bäcklund transformation of Eq. (3.1), let us truncate the Laurent
series at the constant level term to give
h  h0 1  h1 .
Hence
h
(3.7)
x
 h1 ,

(3.8)
where the pair of function (h, h1) satisfy Eq. (3.1) and hence Eq. (3.8) is the associated
Bäcklund transformation of Eq. (3.1) relating a solution u with a known solution h 1 of
the Eq. (3.1).
We can also construct the Bäcklund transformation of Eq. (3.1) to be
h  h0 1  h1  h2 ,
(3.9)
4
where h, h1 satisfy Eq. (3.1) and h2 is given by
h2 
1
[16h1x x2 z  4 x z xx  8h1z x3  4 x z xxx  4 x2 xx z
3
24 x  z
 2 z xx2  8 x2t ],
(3.10)
Eq. (3.5), and hence Eq. (3.9) is a Bäcklund transformation too.
Also let's truncate the Laurent series again we get
h  h0 1  h1  h2  h3 2 ,
(3.11)
where h, h1 satisfy Eq. (3.1) while h2 and h3 are given by Eqs. (3.10) and
h3 
1
[16h1x x xz  14 xx xxz  4h1xx x z  12h1z x xx   xxxx z  8h1xz x2  8 h1x z xx
3
24 x  z
 24h 2  xz x2  8 x xt  4 x xxxz  16h2 x z xx  8h2 x z xxx  4 xxt ],
We can make more truncations to the Laurent series at the constant level term u4 and it
will produce another Bäcklund transformation for Eq. (3.1).
4. The generalized shallow water wave equation
Consider the generalized shallow water wave equation [18, 19] in the form
u xxtt  u x u xt  ut u xx  u xt  u xx  0.
(4.1)
We first present the Painlevè test of the generalized shallow water wave equation. The
leading order of solution of Eq. (4.1) is assumed as
u  u 0  .
(4.2)
Substituting Eq. (4.2) into Eq. (4.1) and equating the most dominant terms, the
following results are obtained
u0 
12
.
  t ,
  1
(4.3)
For finding the resonances, the full Laurent series.
u u 0 
1

  u j
j 1
,
(4.4)
j 1
is substituted into Eq. (4.1) and by equating the coefficients of
equation in j is found as
j 4  10 j 3  23 j 2  10 j  24  0.
 j 5 , the polynomial
(4.5)
Using Eq. (4.5), the resonances are found to be j = -1, 1, 4, 6
As usual, the resonance at j= -1 corresponds to the arbitrariness of singular manifold
 ( x, y, z, t )  0 . In order to check the existence of sufficient number of arbitrary
functions at the other resonance values, the full Laurent expansion (4.4) is substituted in
5
Eq. (4.1). From the coefficient of  , the explicit value of u0 is obtained as given in
Eq. (4.3).
To construct Bäcklund transformations of Eq. (4.1), let us truncate the Laurent
series at the constant level term to give
5
u  u 0 1  u1 .
Hence
u
(4.6)
12t
 u1 ,
(   )
(4.7)
where the pair of function (u, u1) satisfy Eq. (4.1) and hence Eq. (4.12) is the associated
Bäcklund transformation of Eq. (4.1) relating a solution u with a known solution u 1 of
the Eq. (4.1).
We can also construct another Bäcklund transformation of Eq. (4.1) to be
u  u 0 1  u1  u 2 ,
(4.8)
where u, u1 satisfy Eq. (4.1) while u2 is given by
u2 
1
24 
2 2
x t
[2u0u1t x2  4u0 xt2  u0u0t xx  u02xt  8u0 x xtt
 4u0 xxtt  4u0 x xxt  2u0tt x2  u0 x u0t x  u0u0 xt x  2u0 xxt2
 u0 x u0 xt  2u0 x u0t x  2u0 xt  8u0 xt xt  2u0 xxtt  8u0t x xt
 2u0u1x xt  u0u0 xxt  2u0 x2  4u0 x xtt  4u0t xxt ],
(4.9 )
Therefore Eq. (4.13) is a Bäcklund transformation too. Also let 's truncate the Laurent
series again we get
u  u 0 1  u1  u 2  u 3 2 ,
(4.10)
where u, u1 satisfy Eq. (4.1) while u2 and u3 are given by Eqs. (4.9) and
u3 
1
24 
2 2
x t
[u0 xt  u0 xxtt  u0u1xxt  u0u1t xx  2u0u 2t x2  u 2u0 x xt
 2u0 x u1t x  2u0u 2 x xt  2u0u 2t xx  u0u 2 x xt  2u0u 2 x xt  u0 x u0 xt
 u0t u0 xx  2u0 x u 2 xt  u 2u0t x2  u0u1xt x  u0u 2t x2  u0 x u1xt  u0u1x xt
 u0t u1x x  2u0 x x  u0 xt  4u0 xt x ,t  2u0 x xtt  2u0 xtt x  2u0t xxt  u0tt xx
 u0 xxtt  u0t x  u0 xx  2u0 xxtt ] .
We can make more truncations to the Laurent series at the constant level term u 4 and it
will produce another Bäcklund transformation for Eq. (4.1).
6
5. The (3+1)-dimensional Jimbo-Miva equation
Consider the Jimbo-Miva equation [20, 21] in the form
u xxxy  3u yu xx  3u xu xy  2u yt  3u xz  0.
(5.1)
We first present the Painlevè test of the Jimbo-Miva equation. The leading order of
solution of Eq.(5.1) is assumed as
u  u 0  .
(5.2)
Substituting Eq. (5.2) into (5.1) and equating the most dominant terms, the following
results are obtained
u 0  2 x . ,   1
(5.3)
For finding the resonances, the full Laurent series.
u u 0 
1

  u j j 1 ,
(5.4)
j 1
is substituted into Eq. (5.1) and by equating the coefficients of  j 5 , the polynomial
equation in j is found as
j 4  10 j 3  23 j 2  10 j  24  0.
(5.5)
Using Eq. (5.5), the resonances are found to be j = -1, 1, 4, 6 .
As usual, the resonance at j=-1 corresponds to the arbitrariness of singular manifold
 ( x, y , z , t )  0 .
In order to check the existence of sufficient number of arbitrary
functions at the other resonance values, the full Laurent expansion (5.4) is substituted in
Eq. (5.1). From the coefficient of  5 , the explicit value of u0 is obtained as given in Eq.
(5.3).
To construct Bäcklund transformations of Eq. (5.1), let us truncate the Laurent
series at the constant level term to give
u  u 0 1  u1 .
(5.6)
Hence
2 x
u
 u1 ,
(5.7)

where the pair of function (u,u1) satisfy Eq. (5.1) and hence Eq. (5.7) is the associated
Bäcklund transformation of Eq. (5.1) relating a solution u with a known solution u 1 of
the Eq. (5.1).
We can also construct other Bäcklund transformations of Eq. (5.1) to be
u  u 0 1  u1  u 2 ,
(5.8)
where u, u1 satisfy Eq. (5.1) while u2 is given by
u2 
1
24 
3
x y
[12xxxxy  12x2xxy  8x yt  4x yxxx  12u1 yx3  12u1xx2 y ], (5.9)
7
and hence Eq. (5.8) is a Bäcklund transformation too. Also let 's truncate the Laurent
series again we get
u  u0 1  u1  u2  u3 2 ,
(5.10)
where u, u1 satisfy Eq. (5.1) and u2 satisfy Eq. (5.9) ) while u3 is given by
u3 
1
24 x3 y
[6u1xx x y  30u2 x y xx  2t  18u1 y x xx  12 x xz  6 xx z
 4 xy xxx  8 x xxxy  4 xt y  4 x yt  2 xxxx y  6u2 x x2 y  18u2 xy x2
 6u1xy x2  6u2 y x3  12u1x xy x  6u1x xx y ],
we can make more truncations to the Laurent series at the constant level term u 4 and it
will produce another Bäcklund transformation for Eq. (5.1).
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