Reciprocal transformations and reductions for a 2+1 integrable equation

Transcription

1 Reciprocal transformations and reductions for a 2+1 integrable equation P. G. Estévez. Departamento de Fisica Fundamental. Universidad de Salamanca SPAIN Cyprus. October 2008

2 Summary An spectral problem in 2+1 that includes equation previously studied Integrable cases Hodograph transformation I Reduction to 1 + 1: The Degasperi-Procesi and Vakhnenko-Parkes equations Hodograph transformation II Reduction to 1 + 1: The Vakhnenko-Parkes equation

3 Bibliography. Reciprocal Transformations Rogers C and Nucci M C. On reciprocal Bäcklund transformations.... Physica Scripta 33 (1986) Hone A. Reciprocal transformations, Painlevé property and solutions of.... Physics Letters A 249. (1998) Euler N et al. Auto-hodograph transformations for.... Jour. of Math.Anal. Aplic. 257 (2001) Ferapontov E V and Pavlov M V. Reciprocal transformations of Hamiltonian operators.... Jour. Math Phys. 44 (2003) Rogers C and Schief W K. Hidden symmetry in subsonic gasdynamics. Reciprocal.... Int.Jour. Eng. Sci. 42 (2004) Abenda S and Grava T. Reciprocal transformations and flat metrics on Hurwitz spaces. J. Phys.A: Math. Theor. 40 (2007)

4 Bibliography. Degasperis Procesi and Vakhnenko-Parkes equations Vakhnenko V and Parkes J The calculation of multisoliton.... Chaos, solitons and fractals 13 (2002) Vakhnenko V and Parkes J A Bäcklund transformation.... Chaos, solitons and fractals 17 (2003) Degasperis A, Holm D D and Hone A. A new integrable.... Theor. Math. Phys. 133 (2002) Hone A and Wang J P Prolongation algebras and... Inv. Problems 19 (2003) Vakhnenko V and Parkes J Periodic and solitary waves.... Chaos, solitons and fractals 20 (2004) Vakhnenko V and Parkes J The connection of the Degasperis.... Proc. of Ins. of Math. Of NAS of Ukraine 50 (2004) Ragnisco O. and Popowicz Z A two component.... J. Phys. A: Math.Gen.39 (2006)

5 CASE I: Estévez Prada. Journal of nonlinear Math. Phys. (2005) ( H x2 x 1 x 1 + 3H x2 H x1 3 4 (H x2 x 1 ) 2 H x2 ) x 1 = H x2 x 3 Φ x3 Φ x1 x 1 x 1 3H x1 Φ x1 3 2 H x 1 x 1 Φ = 0 Φ x1 x 2 + H x2 Φ 1 2 H x1 x 2 H x2 Φ x2 = 0 CASE II: Estévez Leble. Inverse Problems 11 (1995). (H x2 x 1 x 1 + 3H x2 H x1 ) x1 = H x2 x 3 Φ x3 Φ x1 x 1 x 1 3H x1 Φ x1 3H x1 x 1 Φ = 0 Φ x1 x 2 + H x2 Φ H x 1 x 2 H x2 Φ x2 = 0

6 There are particular cases of the equation (k 5)(k + 1) Ω = H x2 x 1 x 1 + 3H x2 H x1 12 ( ) 2 Hx2 x 1 H x2 Ω x1 = H x2 x 3 It has the Painlevé property for k = 2 (case I) or k = 1 (case II) The spectral problem can be summarized for both cases as: ( ) k 5 Hx1 x Φ x1 x 2 + H x2 Φ + 2 Φ x2 = 0 6 H x2 Φ x3 Φ x1 x 1 x 1 3H x1 Φ x1 + ( k 5 2 ) H x1 x 1 Φ = 0 Many different solutions as dromions, line solitons...etc. have been obtained for the two integrable cases in the above references

7 RECIPROCAL TRANSFORMATION I dx 1 = α(x, t, T ) (dx β(x, t, T )dt ɛ(x, t, T )dt ) x 2 = t, x 3 = T x 1 = 1 α x, = x 2 t + β x, x 3 = T + ɛ x α t + (αβ) x = 0 α T + (αɛ) x = 0 β T ɛ t + ɛβ x βɛ x = 0

8 We select the transformation in the following form: H x2 (x 1, x 2, x 3 ) = α(x, t, T ) k (k + 1)(k 2) = 0 H x1 (x 1, x 2, x 3 ) = 1 3 ( Ω(x, t, T ) α k k α ) xx α 3 + (2k 1)α2 x α 4 yielding the set of equations ( ) Ω t = βω x kωβ x + α k 2 α x kβ xxx + (k 2)β xx α + 3kαk α x Ω x = kα k+1 ɛ x α t + (αβ) x = 0, α T + (αɛ) x = 0 β T ɛ t + ɛβ x βɛ x = 0

9 LAX PAIR Φ(x 1, x 2, x 3 ) = ψ(x, t, T )α 2k 1 3 ψ xt = βψ xx (k 2)β x ψ x + ( ) 2k 1 β xx α k+1 ψ 3 ψ T = ψ xxx α 3 ( ( )) +2(k 2)α x Ω αxx α 4 ψ xx+ ɛ + (k 2) αk+1 α 4 4α2 x α 5 ψ x (k + 1)(k 2) = 0

10 The condition of integrability can be written as: A 1 A 2 = 0, A 1 + A 2 = 1 where A 1 = k + 1 3, A 2 = 2 k 3 A 1 = 1, A 2 = 0 corresponds to the generalization of the Degasperis-Procesi equation A 1 = 0, A 2 = 1 corresponds to the generalization of the Vakhnenko-Parkes equation

11 It allows us to write the Lax pair as ψ xt = A 1 ( [ βψ xx + β xx 1 ) ] ψ + M [ βψ xx 2β x ψ x (1 + β xx )ψ] A 2 ψ T = A 1 [M ψ xxx + (M Ω ɛ) ψ x ] + A 2 [M ψ xxx + 2M x ψ xx + (M xx + Ω ɛ) ψ x ] where M = 1 α 3

12 and the equations are: MΩ t + MβΩ x + 2MΩβ x + 2Mβ xxx + 2 M x + M A 2 [Ω t + βω x Ωβ x Mβ xxx M x β xx M x ] = 0 A 1 [ ] A 1 [MΩ x + ɛ x ] + A 2 [Ω x ɛ x ] = 0 M t = 3Mβ x βm x M T = 3Mɛ x ɛm x β T ɛ t + ɛβ x βɛ x = 0

13 Reduction independent on T: ɛ = 0, Ω = a o, ψ T = λψ [ 2MA 1 β xx + a 0 β 1 ] M x A 2 [Mβ xx + a o β + M] x = 0 M t = 3Mβ x βm x A 2 = 0, a o = 1: DEGASPERIS PROCESI: β xx β = 1 M + q 0 (β xx β) t + ββ xxx + 3β x β xx 4ββ x + 3q 0 β x = 0 A 1 = 0, a o = 0: VAKHNENKO-PARKES: β xx + 1 = q 0 M [(β t + ββ x ) x + 3β] x = 0

14 Spatial part of the Lax pair DEGASPERIS PROCESI for a 0 = 1 A 2 = 0. [ A 1 ψ xxx + a o ψ x λ ] M ψ + [ A 2 ψ xxx + 2 M x M ψ xx + 1 M (M xx + a o )ψ x λ ] M ψ = 0 VAKHNENKO-PARKES for A 1 = 0. a 0 = 0 Temporal part of the Lax pair 4A 1 [λψ t + ψ xx + λβψ x + (a 0 λβ x )ψ] + A 2 [λψ t + Mψ xx + (λβ + M x )ψ x + (a 0 + λβ x )ψ] = 0

15 There are two more interesting reductions that are the transformed of those studied in Estévez, Gandarias and Prada. Phys Lett A 343 (2005)

16 RECIPROCAL TRANSFORMATION II x 1 = y, x 3 = T dx 2 = η(y, z, T ) (dz u(y, z, T )dy ω(y, z, T )dt ) = x 1 y + u z, x 2 = 1 η z, = x 3 T + ω z η y + (uη) z = 0 η T + (ηω) z = 0 u T ω y uω z + ωu z = 0

17 We select the transformation in the following form: z = H(x 1, x 2, x 3 ) dz = H x1 dx 1 + H x2 dx 2 + H x3 dx 3 H x2 (x 1, x 2, x 3 ) = 1 η(y = x 1, z = H, T = x 3 ) H x1 (x 1, x 2, x 3 ) = u(y = x 1, z = H, T = x 3 ) H x3 (x 1, x 2, x 3 ) = ω(y = x 1, z = H, T = x 3 )

18 The transformed equations are A 1 [u zy + uu zz + 1 ] [ 4 u2 z + 3u G + A 2 uzy + uu zz + u 2 z + 3u G ] G y = (ω ug) z, G = Ωη u T = ω y ωu z + uω z η y + (uη) z, η T + (ωη) z A 1 A 2 = 0, A 1 + A 2 = 1

19 and the Lax pair is: A 1 [ Φ zy + uφ zz + Φ u zφ z ] + A 2 [Φ zy + uφ zz + Φ] = 0 A 1 [Φ T Φ yyy u 3 Φ zzz + 3u A 1 [ ω ug 2 u yy u2 u zz 2 + ( u y 1 ) 2 uu z + u yu z 2 uu2 z 8 3u2 2 Φ zz + 3 ] 2 u yφ ] Φ z + A 2 [ ΦT Φ yyy u 3 Φ zzz + 3uu y Φ zz ] + A 2 [ ω 2uG uyy + u 2 u zz u y u z + uu 2 z + 3u 2] Φ z = 0

20 Reduction independent on T: ω = 0, ψ T = λψ G y + (ug) z = 0, N = u zz + 2 A 1 [u zy + uu zz + 1 ] [ 4 u2 z + 3u G +A 2 uzy + uu zz + u 2 z + 3u G ] = 0 VAKHNENKO-PARKES A 1 = 0, G = constant ( A 1 [λφ zzz + G λ N z [( N A 1 G zz + N 2 2 N z N G z Lax pair ) Φ zz + ) ] Φ ( 3 2 G z N ) ] z N G Φ z + 2 A 2 [λφ zzz + GΦ zz + G z Φ zz + (u zz + 1) Φ] = 0 [ A 1 Φ y 2λ ( N Φ zz + u 2G ) ( uz Φ z N 2 + G z N A 2 [Φ y λφ zz + (u G)Φ z u z Φ] = 0 ) ] Φ + +

21 Conclusions Different reciprocal transformations for an spectral problem in 2+1 dimensions are investigated. Reductions of the transformed spectral problems are presented. The Vakhnenko-Parkes and Degasperis-Procesi equation appear as two of the different possible reductions.