Frobenius integrable decompositions for PDEs

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1 Frobenius integrable decompositions for PDEs GAO Liang Department of Applied Mathematics, School of Science, Northwestern Polytechnical University Xi an, Shaanxi 71019, P.R. China Joint work with Wen-Xiu Ma and Wei Xu

2 Outline 1. Introduction. Specific PDEs possessing Frobenius integrable decompositions (FIDs) 3. Conclusions

3 1. Introduction Many PDEs exhibiting soliton phenomena appeared in many science subjects fluid physics, solid physics, elementary particle physics, biological physics, superconductor physics, It is a quite fascinating research topic that how to solve such PDEs to obtain interesting solutions including solitons, which attracts much attention of mathematicians, physicists and dynamicists.

4 In the past several decades, there have been many efficient methods for constructing exact solutions to nonlinear PDEs. Though the solving methods are diverse, appropriate reductions similarity reductions, symmetry constraints, travelling wave reductions To reduce given PDEs to simpler PDEs and/or integrable ODEs.

5 W.X. Ma, H.Y. Wu, J.S. He, Partial differential equations possessing Frobenius integrable decompositions, Physics Letters A, 364 (007) 9-3. They presented FIDs for two classes of nonlinear evolution equations (NEEs) with logarithmic derivative Bäcklund transformations in soliton theory. u ψ = (ln φ) x =, φ u λφ ψ = (ln φ) xx =. φ The discussed NEEs are transformed into systems of Frobenius integrable ODEs with cubic nonlinearity.

6 F.C. You, T.C. Xia, J. Zhang, Frobenius integrable decompositions for two classes of nonlinear evolution equations with variable coefficients, Modern Physics Letters B, 3 (1) (009) They obtained two classes of PDEs with variable coefficients possessing FIDs, including the KdV equation the potential KdV equation the Boussinesq equation the generalized BBM equation,

7 Puu (,, u, u, ) = 0 t x xt What is FIDs? u = v( Φ ) = v( Φ( x, t)), Φ =Α( Φ), Φ =Β( Φ), x t Then we say that the equation possesses a FID.

8 FIDs generalize compatible time-space decompositions requiring Hamiltonian structures, which aim to guarantee the Liouville integrability ([1, ]). [1] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, [] W.X. Ma, in: A. Scott (Ed.), Encyclopedia of Nonlinear Science, Taylor & Francis, New York, 005, pp Through FIDs, a PDE problem can be transformed into two associated ODE problems. Thus, the existence of solutions can be guaranteed easily by the theory of ODEs.

9 Our work Present two classes of PDEs of specific type possessing FIDs by introducing some general ansatzes on FIDs, motivated by the works about FIDs by Ma and You et al. Two kinds of functions for Bäcklund transformations are taken in our constructive computation algorithm, and the associated Frobenius integrable ODEs possess higher-order nonlinearity.

10 . Specific PDEs possessing FIDs Ruu (,, u, u, u, u, u, u, u, u, u, ) = 0. x xx xxx xxxx 5x 7x 9x t tt xxt Many interesting wave equations belong to this class of PDEs, the KdV, the potential KdV, the generalized seventh-order KdV, the Burgers, the spatially periodic third-order dispersive PDE, the b-equation,...

11 Based on the symmetry constrains theory [1-4], [1] W.X. Ma, W. Strampp, Physics Letters A, 185 (1994) 77. [] W.X. Ma, Journal of the Physical Society of Japan, 64 (1995) [3] W.X. Ma, Z.X. Zhou, Journal of Mathematical Physics, 4 (001) [4] Y.B. Zeng, W.X. Ma, Journal of Mathematical Physics, 40 (1999) 656. we consider the following case

12 T T Φ= ( φψ, ) = ( φ( xt, ), ψ( xt, )), which satisfies two Frobenius integrable systems: a real parameter φ = ψ, ψ = λφ, x x undetermined polynomials φ = θ ( φ, ψ), ψ = θ ( φ, ψ). t 1 t It is easy to see that the equation can be generated from the Schrödinger spectral problem with zero potential.

13 Then, we have the following mixed derivatives for the considered FIDs: and φxt = ψt = θ( φ, ψ), φtx = θ1, φψ + λθ1, ψφ, ψ xt = λφt = λθ1( φ, ψ ), ψ tx = θ, φψ + λθ, ψφ.

14 Thus, we get and accordingly, θ ( φψ, ) = θ ψ+ λθ φ, 1, φ 1, ψ λθ ( φ, ψ) = θ ψ + λθ φψ + λ θ φ + λ( θ φ+ θ ψ). 1 1, φφ 1, φψ 1, ψψ 1, φ 1, ψ the only condition on θ 1

15 To search for θ1 which satisfies λθ ( φ, ψ) = θ ψ + λθ φψ + λ θ φ + λ( θ φ + θ ψ). 1 1, φφ 1, φψ 1, ψψ 1, φ 1, ψ Take the following ansatze natural numbers θ m1 m i j 1 = bi, jφψ i= 0 j= 0, arbitrary constants

16 when m1 = m = 7 θ = λ b φ λ b φ ψ + 3λ b φ ψ + 3λ b φ ψ 3λb φ ψ 3λb φ ψ ,6 0,7 1,6 0,7 1,6 0,7 + b φψ + b ψ + λ b φ + λ b φ ψ λb φ ψ λb φ ψ + b φψ ,6 0,7 1,4 0,5 1,4 0,5 1,4 + b ψ λb φ λb φ ψ + b φψ + b ψ + b φ+ b ψ ,5 1, 0,3 1, 0,3 1,0 0,1 θ = λ( λ b φ + 6λ b φ ψ + 9λ b φ ψ 1λb φ ψ 15λb φ ψ + 6b φψ ,7 1,6 0,7 1,6 0,7 1,6 + 7b ψ + λ b φ 4λb φ ψ 6λb φ ψ + 4b φψ + 5b ψ λb φ ,7 0,5 1,4 0,5 1,4 0,5 0,3 + b φψ + 3 b ψ + b ) φ + ( 7λ b φ 6λ b φ ψ + 15λ b φ ψ , 0,3 0,1 1,6 0,7 1,6 + 1λ b φψ 9λb φψ 6λb φψ + b ψ + 5λ b φ , 7 1,6 0,7 1,6 1,4 + 4λ b φ ψ 6λb φ ψ 4λb φψ + b ψ ,5 1,4 0,5 1,4 3λb φ λb φψ + b ψ + b ) ψ. 1, 0,3 1, 1,0

17 On the other hand, we consider the following specific type of the polynomial R Ruu (,, u, u, u, u, u, u, u, u, u, ) x xx xxx xxxx 5x 7x 9x t tt xxt = d u+ d u + d u + d u + d u + d u + d uu + d uu + d uu 0,0 0,1 x 0, xx 0,3 xxx 0,4 xxxx 1,0 1,1 x 1, xx 1,3 xxx + d uu + d u + d uu + d uu + d uu + d u + d u u + d 1,4 xxxx,0 x,1 x xx, x xxx,3 x xxxx 3,0 xx 3,1 xx xxx 3, u u + d u + d u u + d u + e u + eu + e u xx xxxx 4,0 xxx 4,1 xxx xxxx 5,0 xxxx 0 5x 1 7x 9x + euu + euu + euu + fuu + fuu + fuu + fu 3 3 5x 4 7x 5 9x 0 x 1 xxx 5x 3 x + fuu + fuuu + gu+ gu + gu + gu 3 4 x 5 x xx 0 t 1 tt xxt 3 xxt.

18 Theorem (a) If we take a Bäcklund transformation ψ u = (ln φ) x = φ from the Frobenius integrable systems φ = ψ, ψ = λφ, x x φ = θ ( φ, ψ), ψ = θ ( φ, ψ), t 1 t to the PDE Puu (,, u, u, ) = 0, t x xt

19 θ = λ b φ + 3λ b φ ψ 3λb φ ψ + b φψ + λ b φ λb φ ψ ,6 1,6 1,6 1,6 1,4 1,4 + b φψ λb φ + b φψ + b φ + b ψ, 4 3 1,4 1, 1, 1,0 0,1 θ = λ(6λ b φ ψ 1λb φ ψ + 6b φψ 4λb φ ψ + 4b φψ + b φψ + b ) φ ,6 1,6 1,6 1,4 1,4 1, 0,1 + (15λ b φψ 7λ b φ 9λb φψ + b ψ + 5λ b φ 6λb φψ ,6 1,6 1,6 1,6 1,4 1,4 + b ψ 3 λb φ + b ψ + b ) ψ, 4 1,4 1, 1, 1,0

20 R = (4λ e + 8λd 40λe + 4d d 6 d ) u u + f u u + f u u + f u 3 3 4,3 3 0,4,1 1,3 x 1 xxx 5 x 3 x + d uu + d uu + (3d 40λ f 60e + d 3 f 1/ f + 1 d ) uu u 1,4 xxxx 1,3 xxx, 0 3, ,4 x xx + d uu + e uu + e uu + 630e u 35e u u + d u u 1, xx 3 5x 4 7x xxxx 4 xxx xxxx,3 x xxxx + d u u + d u + d u u (56λ e + 64λ e + 16λ f + 16λ e 4 3 3, x xxx 3,0 xx,1 x xx λ f 8λ d + 4λb g + 4λd λd + b g + d ) u u / λ 1 1,4 0,1 0,3 1, 0,1 0 0,1 + (4λ e 88λ e 8λ d + b g + λd + λd 3 3 4,3 0,1 1,1 16λd0,4 1,3 + d ) uu + (105e 3/4d + 5/f 3/4 b g ) u xx u xxxx 0, x 1 xxt + gu 3 xxt. 4,0 0,1 3 + d u (140λ e + d + 10 e ) u u + d u 0,4 xxxx 4,3 3 xx xxx 0,3 xxx + d u + e u + eu + e u + d u (4λ c g 3 0, xx 9x 1 7x 0 5x 0,1 x 0, λ e + 4λ d 7λ e + 16λ e ,0 1 0 λ d + λ f λb g λd, 3 0,1 0,3 + b g + d ) u / λ + g u + g u + gu 0,1 0 0,1 x 0 t 1 tt x + d 4,0 u xxx

21 (b) If we take a Bäcklund transformation u = (ln φ) xx = λφ ψ φ θ = λ b φ + 3λ b φ ψ 3λb φ ψ + b φψ + λ b φ λb φ ψ ,6 1,6 1,6 1,6 1,4 1,4 + b φψ λb φ + b φψ + b φ + b ψ, 4 3 1,4 1, 1, 1,0 0,1 θ = λb φψ+ 3λ b φψ 3λb φψ + λb φψ λb φψ ,6 1,6 1,6 1,4 1,4 λb φ ψ + λb φ+ b ψ + b ψ + b ψ + b ψ, , 0,1 1,6 1,4 1, 0,1

22 R = f uu + fuu + fuuu + (44350λ e 35λ d 3584λ e 10λe + 8λ f λ f + λ f 3 4 x 1 xxx 5 x xx 4, e + 1d + 6 d ) u u + (4d 5040e + 56 e ) u u + e uu + e uu + d uu + d uu 0 1,3,1 x 4,1 4 5x 4 7x 3 5x 1,4 xxxx 1,3 xxx + d uu + (40λd 30400λe λe + 30d,3 + 18d3,1 5040e1 + 90e3 3 f1 + 1/ 4 f4 1, xx 4,1 4 3/ f ) u + d u u + d u u + d u u + d u u + d u + d u u + (3λ d x,3 x xxxx,1 x xx 3, xx xxxx 4,1 xxx xxxx 3,0 xx 3,1 xx xxx 3, 16λ d 4λ d + 6b g + λd + 6 d ) u + (6580λ e 64λ d 64λ e 16λ d ,4 3,0 0,1 1 1, 0, 4,1 4,3 3, ,3,1 0,1 0,3 x 1,4 3, 0,4 1, x 1,4 3,0 x xxx 9x 1 7x 0 5x 0,4 xxxx 0,3 xxx 0, xx 0,1 3 3, xxx 16λ d + 403λ e 16λ e + 40λe 4λd 4λd + 1b g + 1 d ) uu + (10λd 4λ d + 30d 3 / d ) u (5 / d + 3 / 4 d ) u u + e u + eu + e u + d u + d u + d u ( b g + 5/4 d ) u 4 3 (4λb g + 16λ d + 4 λd ) u (56λ e + 64λ e + 16λ e 0,1 1 0,4 0, λb0,1g 4 λd0,3 b0,1g0) ux g0ut g1utt guxxt g3uxxt.

23 Take a special reduction as follows: Ruu (,, u, u, u, u, u, u, u, u, u, ) x xx xxx xxxx 5x 7x 9x t tt xxt = d u + d u + d u + d u + d u + d uu + d uu + d uu 0,0 0,1 x 0, xx 0,3 xxx 0,4 xxxx 1,1 x 1, xx 1,3 xxx + d u + d u u + d u u + d u u + d u u + e u + eu,0 x,1 x xx, x xxx,3 x xxxx 3,1 xx xxx 0 5x 1 7x + eu + euu + fu u + fu u + fu u + fu + fuu 3 3 9x 3 5x 0 x 1 xxx 5x 3 x 4 x 5 x xx 0 t 1 tt xxt. + fuuu + gu + gu + gu

24 For the u ψ = (ln φ) x = φ The corresponding results θ1 = λ b1,6φ + 3λ b1,6 φ ψ 3λb1,6φ ψ + b1,6 φψ + λ b1,4φ λb φ ψ + b φψ λb φ + b φψ + b φ + b ψ, ,4 1,4 1, 1, 1,0 0,1 θ = λ (6λ b φ ψ 1λb φ ψ + 6b φψ 4λb φ ψ + 4b φψ + b φψ ,6 1,6 1,6 1,4 1,4 1, + b ) φ+ ( 7λ b φ + 15λ b φ ψ 9λb φ ψ + b ψ + 5λ b φ ,1 1,6 1,6 1,6 1,6 1,4 6λb φ ψ + b ψ 3 λb φ + b ψ + b ) ψ, 4 1,4 1,4 1, 1, 1,0

25 R = (8 λ d 40 λe + 4 d 6 d d ) u u + f u u 4e u u + f u 3 3,3 3 0,4 1,3,1 x 1 xxx 1 5x 3 x + ( 336λ e λ d 4λ f + λ f 6b g 6 d + d + d ) uu 1, 1 3 0,1 0,3 1,,0 + d uu + (1680λ e + 3d 60e 3 f 1/ f ) uu u + d 1, xx 1, x xx 1,3 xxx + e uu + d u u + d u u 3 5x,1 x xx, x xxx,3 x xxxx,0 x 3,3 0,1 1 0,4 1,3,1 0,,3 3 xx xxx 0, xx 0,3 xxx 0,4 xxxx 3 0 5x 1 7 x 1, 0 λ f3 λb0,1g λ d0,3 λ d,0 0,1 0 + d uu + d u + (4λ e 8λ d + b g 16λ d + λ d + λ d + d ) uu (d + 10 e ) u u + d u + d u + d u + e u + eu + (7λ e + λ d 16λ e + + b g ) u + g u xxt. x + g u + g u 0 t 1 tt x uu x

26 It includes many nonlinear equations, the potential KdV equation gu 0 t + d0,3uxxx + 6d0,3ux = 0, the Burgers equation gu 0 t + d0,uux + d0,uxx = 0, b - equation 1 1 gu 0 t uxxt uux = ( 3 uu x xx + uuxxx ). λ a general case of the famous Rod equation [1] [1] H.H. Dai, Y. Huo, Proceedings of the Royal Society of London Series A, 456 (000) 331. λ

27 For the u = (ln φ) xx = λφ ψ φ The corresponding results θ = λ b φ + 3λ b φ ψ 3λb φ ψ + b φψ + λ b φ ,6 1,6 1,6 1,6 1,4 λ b φ ψ + b φψ λb φ + b φψ + b φ + b ψ, ,4 1,4 1, 1, 1,0 0,1 θ = λ b φ ψ + 3λ b φ ψ 3λb φ ψ + λ b φ ψ λb φ ψ ,6 1,6 1,6 1,4 1,4 λb φ ψ + λb φ+ b ψ + b ψ + b ψ + ψ b , 0,1 1,6 1,4 1, 1,0,

28 R = fuu + fuu + (44350λ e 10λ e + 8λ f λ f + λ f + 1 d + 6 d 3 4 x 1 xxx ,3,1 360 e ) u u + d uu + d uu + e uu + d u u + d u u + (30 d 0 x 1, xx 1,3 xxx 3 5x 3,0 xx xxx,1 x xx,3 + 18d 30400λ e 5040 e + 90 e 3 f + 1/ 4 f 3 / f ) u + e u 3 3, x 9x + d u + f uu 0,3 xxx 5 x u + d u + e u + eu + ( 30 d 3 / d ) u xx 0,4 xxxx 0 5x 1 7x 0,4 1, x + ( 4/3λ d 16 λ d ) u ( b g + 1/3 λ d ) u ( 64λ e 3 1, 0,4 0,1 1 1, xx λ e + 16λ e + 4λ c g + 4λ d + b g ) u 4 0 0,1 0,3 0,1 0 + d u u 5040 e u u + (6580λ e 3,3 x xxxx 5x 16λ d 16λ d λ e,3 3,0 1 16λ e3 4λ d1,3 4λ d,1 + 40λ e + 1b g 0 0,1 + 1 d ) uu + g u 0,3 x 0 + gu + gu 1 tt xxt. t x

29 It includes: the KdV equation gu+ 1d uu + d u = 0, 0 t 0,3 x 0,3 xxx the family of spatially periodic third-order dispersive PDE 1 b0,1g0ut b0,1g0ux + b0,1d0,3uxxx d0,3uxxt = b0,1αλ ( u + ux uuxx ) x,

30 the generalized seventh-order KdV equation 3 g u + (λf 10λe + 8 λ f ) uu + (30d + 18d f + 90e 5040e 3 f ) u + fuuu + fuu + d u u + d uu + euu + eu = 0, 3 0 t x,3 3, x 5 x xx 1 xxx 3,0 xx xxx,3 x xxxx 3 5x 1 7x which has two well-known special cases [1]: the Lax seventh-order equation the Sawada-Kotera-Ito seventh-order equation [1] M. Ito, Journal of the Physical Society of Japan, 49 (1980) 771

31 3. Conclusions Through two Bäcklund transformations of dependent variables, ninth-order PDEs possessing the FIDs have been obtained. Two special classes of such nonlinear PDEs possessing special FIDs have been presented under a reduction. The obtained PDEs contain various significant nonlinear wave equations.

32 The approach adopted here can be easily applied to nonlinear variable coefficient PDEs, which are quite intriguing in ocean dynamics, fluid mechanics, plasma physics, etc. A question? If we take the Möbius transformation u = ( aϕ + bψ)/( cϕ + dψ), ( ad bc 0) instead of the logarithmic derivative type Bäcklund transformations, what we will get?

33 Thank Prof. Wen-Xiu Ma, Prof. Xingbiao Hu and Prof. Qingping Liu! Thank you for your attention!