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Integrability test of a higher-order (2+1)-dimensional KdV-type equation
Commun Nonlinear Sci Numer Simulat 17 (2012) 489–490
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Short communication
Integrability test of a higher-order (2+1)-dimensional KdV-type equation P.R. Gordoa, A. Pickering ⇑ Departamento de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, C/ Tulipán s/n, 28933 Móstoles, Madrid, Spain
a r t i c l e
i n f o
Article history: Received 14 March 2011 Accepted 4 April 2011 Available online 13 April 2011
a b s t r a c t The so-called KdV6 equation has recently generated much interest. Here we show that a recently considered (2+1)-dimensional extension of this equation is in fact nonintegrable. Ó 2011 Elsevier B.V. All rights reserved.
Keyword: Painlev´e test
1. The WTC Painlevé test for a (2+1)-dimensional equation The so-called KdV6 equation [1] has recently generated much interest, and many of its properties have been studied. Here we consider a (2+1)-dimensional extension of this equation that was proposed in [2],
K½u uxxxxxx þ 20ux uxxxx þ 40uxx uxxx þ 120ðux Þ2 uxx þ uxxxt þ 8ux uxt þ 4ut uxx þ uyyy ¼ 0: x
ð1:1Þ
We show that this equation does not pass the Weiss–Tabor–Carnevale (WTC) Painlevé test and so is presumably nonintegrable. The WTC test [3], simplified using Kruskal’s ‘‘reduced ansatz’’ [4], consists of seeking a solution of the form
u ¼ up
1 X
u j uj ;
where uj ¼ uj ðy; tÞ and u ¼ x þ wðy; tÞ
ð1:2Þ
j¼0
and requires a choice of expansion family or branch, that is, a choice of leading order exponent p, leading order coefficient u0, ^ and corresponding dominant terms K½u. For each family there is a set of indices, or resonances, R ¼ fr 1 ; . . . ; rN g, which give the values of j at which arbitrary data are introduced in the expansion (1.2) or in a suitable modification thereof, and at which there are compatibility conditions to be satisfied in order to avoid the introduction of logarithmic terms in (1.2). For Eq. (1.1) and resonance polynomials, in each case the dominant terms we find the following leading order behaviours b ½u ¼ uxxxxxx þ 20ux uxxxx þ 40uxx uxxx þ 120ðux Þ2 uxx : being K x
u u1 ; 1
u 3u ;
R ¼ f1; 1; 2; 5; 6; 7; 8g; R ¼ f3; 1; 1; 6; 7; 8; 10g:
ð1:3Þ ð1:4Þ
For the first family, we find that the compatibility conditions at j = 1 and j = 2 are satisfied, but that the compatibility condition at j = 5 requires
wy wyy ¼ 0
⇑ Corresponding author. Tel.: +34 91 6647447; fax: +34 91 488 7338. E-mail address: [email protected] (A. Pickering). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.006
ð1:5Þ
490
P.R. Gordoa, A. Pickering / Commun Nonlinear Sci Numer Simulat 17 (2012) 489–490
and that at j = 6 requires
wyyy ¼ 0:
ð1:6Þ
The compatibility condition at j = 7 is then satisfied, but that at j = 8 requires
3ðu2;y wy Þy þ u2 wyyy þ u1;yyy ¼ 0:
ð1:7Þ
For the second family, the compatibility condition at j = 1 is satisfied, that at j = 6 requires
wyyy ¼ 0;
ð1:8Þ
that at j = 7 is satisfied, that at j = 8 requires
20u1;yyy wt wyyy ¼ 0;
ð1:9Þ
and that at j = 10 requires
36wy ðwyt Þ2 þ 210wtt wy wyy þ 48wt wyt wyy þ 240u1;yt wyy 2295ðwy wyy Þ2 þ 48wt wy wyyt
ð1:10Þ
þ 240wy u1;yyt 6ðwt Þ2 wyyy þ 80u1;t wyyy 430ðwy Þ3 wyyy ¼ 0:
ð1:11Þ
The above failed compatibility conditions imply that (1.1) is in fact nonintegrable. 2. Discussion We have shown that a recently proposed (2+1)-dimensional extension of the KdV6 equation fails the WTC Painlevé test and so is presumably nonintegrable. It is worth noting that the failed compatibility conditions are satisfied in the y-independent case. This is related to the fact that in [2] multiple soliton solutions are not in fact found for Eq. (1.1) but for the reduction obtained under u = w(n, t),n = x + y (the coefficients of x and y in the exponentials used are assumed equal); this reduced equation is equivalent to the y-independent case of (1.1) under u(x, t) = w(n, t) + (t/4), n = x. Acknowledgements This work is supported in part by the Ministry of Science and Innovation of Spain under contract MTM2009-12670 and by the Junta de Castilla y León under contract SA034A08. References [1] Karasu-Kalkanlı A, Karasu A, Sakovich A, Sakovich S, Turhan R. A new integrable generalization of the Korteweg-de Vries equation. J Math Phys 2008;49:073516. [2] Wazwaz A-M. Multiple soliton solutions for a (2+1)-dimensional integrable KdV6 equation. Commun Nonlinear Sci Numer Simulat 2010;15:1466–72. [3] Weiss J, Tabor M, Carnevale G. The Painlevé property for partial differential equations. J Math Phys 1983;24:522–6. [4] Jimbo M, Kruskal MD, Miwa T. Painlevé test for the self-dual Yang-Mills equation. Phys Lett A 1982;92:59–60.
Contents lists available at ScienceDirect
Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns
Short communication
Integrability test of a higher-order (2+1)-dimensional KdV-type equation P.R. Gordoa, A. Pickering ⇑ Departamento de Matemática Aplicada, ESCET, Universidad Rey Juan Carlos, C/ Tulipán s/n, 28933 Móstoles, Madrid, Spain
a r t i c l e
i n f o
Article history: Received 14 March 2011 Accepted 4 April 2011 Available online 13 April 2011
a b s t r a c t The so-called KdV6 equation has recently generated much interest. Here we show that a recently considered (2+1)-dimensional extension of this equation is in fact nonintegrable. Ó 2011 Elsevier B.V. All rights reserved.
Keyword: Painlev´e test
1. The WTC Painlevé test for a (2+1)-dimensional equation The so-called KdV6 equation [1] has recently generated much interest, and many of its properties have been studied. Here we consider a (2+1)-dimensional extension of this equation that was proposed in [2],
K½u uxxxxxx þ 20ux uxxxx þ 40uxx uxxx þ 120ðux Þ2 uxx þ uxxxt þ 8ux uxt þ 4ut uxx þ uyyy ¼ 0: x
ð1:1Þ
We show that this equation does not pass the Weiss–Tabor–Carnevale (WTC) Painlevé test and so is presumably nonintegrable. The WTC test [3], simplified using Kruskal’s ‘‘reduced ansatz’’ [4], consists of seeking a solution of the form
u ¼ up
1 X
u j uj ;
where uj ¼ uj ðy; tÞ and u ¼ x þ wðy; tÞ
ð1:2Þ
j¼0
and requires a choice of expansion family or branch, that is, a choice of leading order exponent p, leading order coefficient u0, ^ and corresponding dominant terms K½u. For each family there is a set of indices, or resonances, R ¼ fr 1 ; . . . ; rN g, which give the values of j at which arbitrary data are introduced in the expansion (1.2) or in a suitable modification thereof, and at which there are compatibility conditions to be satisfied in order to avoid the introduction of logarithmic terms in (1.2). For Eq. (1.1) and resonance polynomials, in each case the dominant terms we find the following leading order behaviours b ½u ¼ uxxxxxx þ 20ux uxxxx þ 40uxx uxxx þ 120ðux Þ2 uxx : being K x
u u1 ; 1
u 3u ;
R ¼ f1; 1; 2; 5; 6; 7; 8g; R ¼ f3; 1; 1; 6; 7; 8; 10g:
ð1:3Þ ð1:4Þ
For the first family, we find that the compatibility conditions at j = 1 and j = 2 are satisfied, but that the compatibility condition at j = 5 requires
wy wyy ¼ 0
⇑ Corresponding author. Tel.: +34 91 6647447; fax: +34 91 488 7338. E-mail address: [email protected] (A. Pickering). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.04.006
ð1:5Þ
490
P.R. Gordoa, A. Pickering / Commun Nonlinear Sci Numer Simulat 17 (2012) 489–490
and that at j = 6 requires
wyyy ¼ 0:
ð1:6Þ
The compatibility condition at j = 7 is then satisfied, but that at j = 8 requires
3ðu2;y wy Þy þ u2 wyyy þ u1;yyy ¼ 0:
ð1:7Þ
For the second family, the compatibility condition at j = 1 is satisfied, that at j = 6 requires
wyyy ¼ 0;
ð1:8Þ
that at j = 7 is satisfied, that at j = 8 requires
20u1;yyy wt wyyy ¼ 0;
ð1:9Þ
and that at j = 10 requires
36wy ðwyt Þ2 þ 210wtt wy wyy þ 48wt wyt wyy þ 240u1;yt wyy 2295ðwy wyy Þ2 þ 48wt wy wyyt
ð1:10Þ
þ 240wy u1;yyt 6ðwt Þ2 wyyy þ 80u1;t wyyy 430ðwy Þ3 wyyy ¼ 0:
ð1:11Þ
The above failed compatibility conditions imply that (1.1) is in fact nonintegrable. 2. Discussion We have shown that a recently proposed (2+1)-dimensional extension of the KdV6 equation fails the WTC Painlevé test and so is presumably nonintegrable. It is worth noting that the failed compatibility conditions are satisfied in the y-independent case. This is related to the fact that in [2] multiple soliton solutions are not in fact found for Eq. (1.1) but for the reduction obtained under u = w(n, t),n = x + y (the coefficients of x and y in the exponentials used are assumed equal); this reduced equation is equivalent to the y-independent case of (1.1) under u(x, t) = w(n, t) + (t/4), n = x. Acknowledgements This work is supported in part by the Ministry of Science and Innovation of Spain under contract MTM2009-12670 and by the Junta de Castilla y León under contract SA034A08. References [1] Karasu-Kalkanlı A, Karasu A, Sakovich A, Sakovich S, Turhan R. A new integrable generalization of the Korteweg-de Vries equation. J Math Phys 2008;49:073516. [2] Wazwaz A-M. Multiple soliton solutions for a (2+1)-dimensional integrable KdV6 equation. Commun Nonlinear Sci Numer Simulat 2010;15:1466–72. [3] Weiss J, Tabor M, Carnevale G. The Painlevé property for partial differential equations. J Math Phys 1983;24:522–6. [4] Jimbo M, Kruskal MD, Miwa T. Painlevé test for the self-dual Yang-Mills equation. Phys Lett A 1982;92:59–60.