A Method for Obtaining Darboux Transformations

Transcription

1 Journal of Nonlinear Mathematical Physics 998, V.5, N, Letter A Method for Obtaining Darbou Transformations Baoqun LU,YongHE and Guangjiong NI Department of Physics, Fudan University, 00433, Shanghai, P.R. China Institute of Mathematics, Fudan University, 00433, Shanghai, P.R. China mailing address Department of Applied Mathematics, the Central South University of Technology of China, 40083, Changsha, P.R. China Received December, 997; Accepted February 7, 998 Abstract In this paper we give a method to obtain Darbou transformations DTs of integrable equations. As an eample we give a DT of the dispersive water wave equation. Using the Miura map, we also obtain the DT of the Jaulent-Miodek equation. Introduction For integrable equations which can be solved by the Inverse Scattering Transform, there eist Bäcklund transformations BTs []. These transformations were first discovered for the Sine-Gordon equation at the end of the 9th century. Usually they are treated as nonlinear superpositions, which allow one to create new solutions of a nonlinear evolution equation from a finite number of known solutions. In practice, BTs are not very straightforward to apply in the construction of multisolutions. On the other hand, the Darbou transformation DT is a very convenient way of constructing new solutions of nonlinear integrable equations []; the algorithm is purely algebraic and can be continued successively. Therefore, it is interesting to transform BTs into DTs. Many integrable equations of the form u t = Ku. possess the recursion operator Φ with the property called hereditary symmetry [3, 4, 5, 6], and they possess a La pair Φσ = λσ, σ t = K u σ.. Copyright c 998 by B. Lu, Y. He and G. Ni

2 A Method for Obtaining Darbou Transformations 4 Here K u is the Fréchet derivative of K with respect to u. Two interesting questions are raised: How is the DT related to the La pair.? and What happens to the symmetry σ under a BT? In this paper, we will study the above problems. Section gives the general method to obtain DTs of integrable equations by using symmetry. As an eample, Section 3 gives the DT of the dispersive water equation. In Section 4, we obtain the DT of the Jaulent- Miodek equation by using the Miura map. These DTs are not easily obtained by other well-known methods. The Method Suppose that the equation. has a BT of the form Bu, u[] = 0.. Now, we suppose that u u [][σ ]=0,. which means that the symmetry σ is transformed into 0 under the BT, where σ is the eigenfunction of. with λ = λ. Then taking Fréchet derivative of B =0,wehave B u [σ ]=B u [σ ]+B u[] [u u []σ ]=0..3 Fuchssteiner and Aiyer have showed that the KdV equation, the Burgers equation, and the CDGSK equation admit this relation [7, 8, 9]. This formula gives the transformation relation between u, σ and u[]: u[] = F u, σ..4 At this point we can directly check whether.4 is a BT. If so, we can conclude that.3 is true, and we also have the transformation for eigenfunctions σ[] = u u [][σ] =F u [σ]+f σ [σ u σ]..5 Relations.4 and.5 is called the DT of.. Remark.. Here we give a method to calculate σ u [ ]. Because Φσ = λσ,wehave σ u [ ] = Φ λ Φ u [ ]σ. We can apply the factorization method to calculate Φ λ. Remark.. When.4 and.5 is the DT Φu[]σ[] = λσ[], and.5 is the symmetry of u t [] = Ku[], the result [0] shows that.4 and.5 is a DT for the hierarchy u t =Φ n Ku. Remark.3. Relation.4 reveals the connection among the BT, symmetry, and strong symmetry operator. We conjecture that.4 may be right for all equations which possess

3 4 B. Lu, Y. He and G. Ni strong symmetry operators. usual La pair For some + -dimension equations u t = Ku, their Lφ = λφ, φ t = Aφ can be transformated into Ψσ = µσ, σ t = K u σ by a transformation σ = fφ. We can then obtain the DT with respect to the usual La pair. 3 The DT of the Dispersive Water Wave Equation In this section we study the dispersive water wave equation DWW [, ] v t = Kv, 3. where v =q, r T, Kv = qr q, r + r +q and T denotes the transpose of vectors. The DWW equation was studied systematically by Kupershmidt []. System 3. has the following La representation []. T Lφ = λφ, 3. in which φ t = L + φ, 3.3 L = D +D r q, D =, D D = D D = and + is the projection to the purely differential part of the operator, L + = D +q. Here we denote the operator A acting on the operator B by A B, and the operator A action on a function f by Af. The system 3. possesses a strong symmetry operator D + r q + q D Φv = D + D r D. 3.4 So 3. is the following integrable condition Φσ = λσ, σ t = K v σ,

4 A Method for Obtaining Darbou Transformations 43 where σ =σ,σ T. It is difficult to get a DT of 3. with respect to 3., 3.3. In fact, we did not find any DT for 3., 3.3 until now. Let us turn to 3.5, 3.6. Theorem 3.. Let σ, =,σ, T denote the solution of 3.5, 3.6 with λ = λ.we then have the DT where D q[] = q r[] = r + ln D + σ, B σ [] = σ, 3.7 B + σ σ [] = σ D + σ. B = D σ + σ D,, 3.8, 3.9, 3.0 that is, 3.7 and 3.8 is a new solution of 3.. Moreover, 3.7, 3.8 and 3.9, 3.0 satisfy 3.5, 3.6. Furthermore, this is the DT of the hierarchy v t =Φv n Kv. Proof. i From practice, we first suppose q[] = q +lnφ isapartofthebt. Substituting φ = e D q[] q into 3. with λ = λ,wefind D q[] q + q[] + λ r r + λ D q[] q =0. 3. From.3 we find D + σ, D q[] q+r + λ D + λ σ, =0. On the other hand, 3.5 gives D + σ, =λ r. These two identities imply 3.7. Suppose q[],r[] satisfy 3., then q[] q t =q[]r[] qr q[] q, 3. D D D =q[]r[] qr + σ, t D. σ, 3.3 Using 3.6, we have D q[]r[] qr + D = σ, D r +qσ, σ, + D σ, +rσ,

5 44 B. Lu, Y. He and G. Ni We note that 3.5 yields r = λ σ, +D, 3.5 q = +D. 3.6 Substituting the above two identities and 3.7 into 3.4, we obtain, after some calculations, 3.8. We can easily prove that 3.7 and 3.8 satisfy 3.5 and 3.6, so 3.7, 3.8 is a BT of 3.. ii From 3.5 and 3.6 D + D D v σ D = σ, D σ, D D σ, D + σ, +D Now, we solve the equation a σ v σ D = that is, a σ a + D a. D D + σ, D a = σ, 3.7 σ, σ, +D a a + D a = σ. 3.8 σ, D,wefind with a a D = B B = D σ + σ D. Hence a = B D D B D + D σ, σ, a = D B D D σ. σ,

6 A Method for Obtaining Darbou Transformations 45 Now we can calculate σ[] from 3.7, 3.8: σ[] = v v []σ D = σ + D D D + σ, D + σ, D = σ + D D D + σ, D + σ, B = σ D B + σ. σ, + D D v σ σ a a This completes the proof. Remark 3.. Let w = σ, then 3.5, 3.6 can be written in a more simple form: with w + rw +qw + q w = λw, w +w + rw = λw, w t =rw +w q w, w t =w +rw +w. The DT given in Theorem 3. becomes w, q[] = q, w, r[] = r + ln w, + w,, w, B w[] = w w, B + w, w,, w, + w, B = D w, w + w w,. 4 The DT of the Jaulent-Miodek equation The Jaulent-Miodek equation takes the form [3, 4] u t = Hu =Ψuu, 4.

7 46 B. Lu, Y. He and G. Ni where u = u0 u, Ψu = 0 4 D3 + u 0 D + D u 0 u D + D u D D and Ψ is a strong symmetry operator. So 4. possesses the La pair Ψuψ = λψ, ψ t = H u ψ Usually 4. is derived from the following spectral problem [, 3, 4]: Lφ = φ +u 0 + λu φ = λ φ, 4.4 where the time evolution of the wave function φ has the form φ t = Pφ = p D 4 p φ, 4.5 with p =+λu. Then L t [P, L] =p L gives rise to 4.. The BT of 4. was given by Tu [3]. It is not easy to apply this BT to construct new solutions. An invertible Miura map [] q = u u u, 4.6 r = u 4.7 brings 4. into the DWW 3.. The Miura map 4.6, 4.7 gives the relation of the eigenfunctions between 4., 4.3 and 3.5, 3.6: ψ = u v σ = 0 = Therefore, D r σ + σ r σ σ D σ = D ψ ψ + D u ψ u [] = u + ln D + σ, σ = λ u D ψ ψ, = u + E, 4.0

8 A Method for Obtaining Darbou Transformations 47 where E = ln λ u ψ, D. ψ, u 0 [] u 0 =q[] q+ r[] r r[] rr[] + r 4 = u + ψ, D + ψ, E 4 EE +u, 4. with B = D σ + σ D =D ψ, λ u D ψ ψ D ψ, λ u D ψ ψ λ u D ψ,, ψ [] = ψ F, 4. B ψ [] = ψ D ψ, F Eψ, F u + E, 4.3 B + ψ D ψ, F = D λ u D ψ, + ψ., Theorem 4.. Suppose u, ψ, satisfies 4., 4.3 with λ = λ, then the transformation defined by 4.0, 4., 4., 4.3 is a DT of 4., Conclusion In this paper, we have presented a method to obtain DTs of integrable equations. This method can be apply to the DWW equation, the KdV equation, a shallow water equation [5] and other integrable equations. We think the relation.3 is very important because it reveals the relation between BT, symmetry, and strong symmetry of the corresponding equation. We hope that there will be further study in this direction. Acknowledgements This work was supported by Shanghai Science and Technology Morning Star planning of China. References [] Rodgers C., Shadwick W., Bäcklund Transformations and their Applications, in: Mathematics in Science and Engineering, V.6, Academic, New York, 98. [] Matveev V.B. and Salle M.A., Darbou Transformations and Solitons, Springer-Verlag, Heidelberg, 99.

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