Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions

Accepted Manuscript Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions Xianguo Geng, Ruomeng Li PII: DOI: Reference:

S0003-4916(15)00253-5 http://dx.doi.org/10.1016/j.aop.2015.06.017 YAPHY 66881

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Annals of Physics

Received date: 15 December 2014 Accepted date: 28 June 2015 Please cite this article as: X. Geng, R. Li, Darboux transformation of the Drinfeld–Sokolov–Satsuma–Hirota system and exact solutions, Annals of Physics (2015), http://dx.doi.org/10.1016/j.aop.2015.06.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Darboux transformation of the Drinfeld-Sokolov-Satsuma-Hirota system and exact solutions Xianguo Geng, Ruomeng Li School of Mathematics and Statistics, Zhengzhou University, 100 Kexue Road, Zhengzhou, Henan 450001, People’s Republic of China

Abstract A Darboux transformation for the Drinfeld-Sokolov-Satsuma-Hirota system of coupled equations is constructed with the aid of gauge transformations between the Lax pairs. As an application, several types of solutions of the Drinfeld-Sokolov-Satsuma-Hirota system are obtained, including soliton solutions, periodic solutions, rational solutions and others. Keywords: Drinfeld-Sokolov-Satsuma-Hirota system, Darboux transformation, exact solutions 1. Introduction The study of soliton equation is one of the most prominent subjects in the field of nonlinear science. A fairly satisfactory understanding has been obtained for the integrable models over recent decades. There are many ways to obtain explicit solutions of soliton equations such as the inverse scattering transformation [1, 2], the bilinear transformation [3], the B¨acklund and Darboux transformations [4] and the algebrogeometric method [5] and so on. A variety of types of solutions have been found, the most important among which are pure-soliton solutions, quasi-periodic solutions and rational solutions It is well known that Darboux transformation is a powerful tool for solving soliton equations [4, 6–9]. In the present paper, we are going to construct a Darboux transformation for the Drinfeld-Sokolov-Satsuma-Hirota (DSSH) system of coupled equations ut = 21 uxxx − 3uux + 3vx , vt = −vxxx + 3uvx

(1)

with the aid of a gauge transformation between the corresponding Lax pairs, by which some exact solutions of the DSSH system are obtained including one-soliton, twosoliton, periodic solution, rational solution and others. Email address: [email protected] (Ruomeng Li)

Preprint submitted to Elsevier

June 23, 2015

The DSSH system (1) was proposed, independently, by Drinfeld and Sololov [10], and by Satsuma and Hirota [11]. In [10], the DSSH system (1) was introduced as one of numerous examples of nonlinear equations possessing Lax pairs of a particular form. In [11], the system (1) was found as a special case of the four-reduction of the KP hierarchy, and its one-soliton soluton was given. The Ref. [12] found a recursion operator and a bi-Hamiltonian structure for system (1), which provided the system with an infinite algebra of generalized symmetries and an infinite set of conservation laws. In [13], the truncated singular expansion method was used to construct an explicit B¨acklund transformation, from which special solutions of this system are derived. This B¨acklund transformation, as was predicted in [13], is in fact a superposition of two simpler B¨acklund transformations, as was shown in [14]. Based on the Painlev´e analysis for integrability of partial differential equations, a class of sixth-order nonlinear wave equations was discussed, which contains the DSSH system (1) as a special case [15]. In [16], using three distinct methods (the Cole-Hoph transformation, the tanh-coth method, and the exp-function method), multiple soliton solutions and plane periodic solutions for this system were obtained, including multiple singular soliton solutions and singular periodic solutions. In [17], the DSSH hierarchy of nonlinear evolution equations was derived from a 4 × 4 matrix spectral problem. It was shown that the DSSH hierarchy of nonlinear evolution equations possesses the bi-Hamiltonian structures and was completely integrable in the Liouville sense The present paper is organized as follows. In section 2, we first introduce a gauge transformation between the Lax pairs of the DSSH system (1) and discuss the condition for the existence of the gauge transformation. Then we construct successfully a Darboux transformation of the DSSH system (1). In section 3, by the application of the Darboux transformation, some explicit exact solutions of the DSSH system (1) are derived, including one-soliton, two-soliton, periodic solution, rational solution and others. All the solutions given here are analytic for all x and t ∈ R. Moveover, the determinants of Darboux matrices were always assumed to be polynomials in the spectral parameter with constant distinct roots, which is not always true and this paper provides a counterexample. 2. Darboux Transformation In this section, we shall construct a Darboux transformation for the DSSH system (1), which comes from the condition of the compatibility for the matrix spectral problem ϕxx = (U + λ σ )ϕ , (2) and an auxiliary problem

ϕt = −Vx ϕ + 2(V + λ σ )ϕx , where λ is a constant spectral parameter, ϕ = (ϕ1 , ϕ2 )T , ( ) ) ( ) ( 1 u v v 1 0 −2u U= , σ= , , V= 1 u 0 −1 1 − 12 u 2

(3)

and u, v are the two potentials. It is easy to see that the Lax pair, (2) and (3), can be written as two 4 × 4 matrix spectral problems

where ψ

ψx = U ψ ,

(4)

ψt = V ψ ,

(5)

= (ϕ T , ϕxT )T ,

I is the 2 × 2 unit matrix, ( ) 0 I U = , U +λσ 0 ( −Vx V = −Vxx + 2VU + uσ λ + 2λ 2 I

) 2(V + λ σ ) . Vx

In fact, a direct calculation shows that the zero-curvature equation, Ut − Vx + [U , V ] = 0, implies the DSSH system (1) In order to derive a Darboux transformation of the DSSH system, we introduce a gauge transformation between the corresponding Lax pair, (2) and (3), by T : ϕ 7→ ϕ¯ , that is ϕ¯ = (A + λ σ )ϕ + cϕx (6) or, in operator form, where

T = A + λ σ + ∂x I, (

) a b A= , 1 a

and a, b and c are scalar functions to be determined in (x,t). We define a Darboux matrix T by ( ) A+λσ cI T = . Ax + cU + λ cσ A + cx I + λ σ

Then it is easy to see that the Darboux matrix T maps ψ = (ϕ T , ϕxT )T to ψ¯ = (ϕ¯ T , ϕ¯xT )T . In the following, we are trying to find two new potentials u¯ and v¯ such that

where or equivalently

where

ϕ¯xx = (U¯ + λ σ )ϕ¯ ,

(7)

ϕ¯t = −V¯x ϕ + 2(V¯ + λ σ )ϕ¯x ,

(8)

U¯ = U|u=u,v= ¯ v¯ ,

V¯ = V |u=u,v= ¯ v¯ ,

U¯ T = Tx + T U ,

(10)

V¯ T = Tt + T V ,

(11)

U¯ = U |u=u,v= ¯ v¯ ,

V¯ = V |u=u,v= ¯ v¯ .

3

Theorem 1. Equation (10) holds if and only if the following systems hold: u¯ = u + 2cx , and

v¯ = −v + 2b

2a − c2 + cx = 0, bx − (b − v)c = 0, (c2 u + cax − a2 − acx − b)x = 0.

(12)

(13)

P ROOF. Comparing the coefficients of λ j in (10) we obtain (12) for λ 1 and ua ¯ + v¯ = axx + 2cx u + cux + au + b, ub ¯ + va ¯ = bxx + 2cx v + cvx + av + bu, b + ua ¯ = axx + 2cx u + cux + au + v, a + u¯ = 2cx + a + u, cu¯ = cu + 2ax + cxx , cv¯ = cv + 2bx

(14)

for λ 0 . A little more straightforward calculation shows that constraints (14) can be reduced to equation system (13) by means of using transformation (12). □ Lemma 1. Let

γ 2 = c2 u + cax − a2 − acx − b.

(15)

det T = (λ 2 − γ 2 )2

(16)

If system (13) is true, then γ is a constant,

and rank T |λ =γ = 2. P ROOF. With the aid of the well-known identity P Q −1 R S = det(QR − QSQ P)

for square matrices P, R, S and non-degenerate square matrix Q, it can be verified without effort. □ Theorem 2. Let γ be a nonzero constant, and let Φ± , respectively, be a non-degenerate 2 × 2 matrix, whose column vectors are solutions of the spectral problem (2) and (3) with λ = ±γ . Let (a, b, c) be a solution (if there is any) of the following overdetermined system, (A + γσ )Φ+ + cΦ+,x = 0, (17) (A − γσ )Φ− + cΦ−,x = 0. Then (u, ¯ v) ¯ determined by transformation (12) is a solution of the DSSH system (1). The transformation (12) is a Darboux transformation of the DSSH system (1). P ROOF. The proof is divided into two parts: Lemma 2 for condition (10) and Lemma 3 for condition (11). □ 4

Lemma 2. Let all the conditions be the same as those in Theorem 2. Then function ϕ¯ defined by (6) satisfies the spectral problem (7). P ROOF. Noticing the identities (17) and Φ±,xx = (U ± γσ )Φ± , a direct calculation shows that (12) and (15) or (13) hold, which implies that function ϕ¯ defined by (6) satisfies the spectral problems (7). □ Lemma 3. Let all the conditions be the same as those in Theorem 2. Then function ϕ¯ defined by (6) satisfies the spectral problem (8). P ROOF. Firstly, letting K± = Φ±,x Φ−1 ± , one can easily deduce from Φ±,xx = (U ± γσ )Φ± that γ ̸= 0 implies tr(σ K+ ) ̸= 0 and tr(σ K− ) ̸= 0. And, solving (17), we can obtain c = − tr(σ2Kγ + ) , c = tr(σ2Kγ − ) , A=γ which implies that is non-degenerate. Let

(

2K+ tr(σ K+ )

) −σ ,

A = −γ

(

2K− tr(σ K− )

) −σ ,

(18)

K+ − K− = − 2cγ σ

(19)

D = V¯ T − (Tt + T V ).

(20)

With the aid of transformation (12) and constraints (13), and through tedious calculations, one can show that D are in the form of ( ) D1 D2 D= , (21) D3 D4 + λ D2 σ where D1 , D2 , . . . , D4 are second-order matrix-valued functions in (x,t) to be determined. For notation simplicity, let us denote ( ) Φ± Ψ± = . Φ±,x Equation (17) defines A and c, thus it defines T . From (17) we can know that the first two rows of T |λ =γ Ψ± is zero. Noticing that the first two rows of [T |λ =γ Ψ+ ]x is the same as the last two rows of T |λ =γ Ψ+ , we arrive at T |λ =γ Ψ+ = 0.

(22)

Recalling the definition of Φ+ we can see that Ψ+ satisfies that Ψ+,t = V |λ =γ Ψ+ .

(23)

D|λ =γ Ψ+ = [V¯ T − (Tt + T V )]λ =γ Ψ+ = V¯ T |λ =γ Ψ+ − (Tt |λ =γ Ψ+ + T |λ =γ Ψ+,t ) = 0,

(24)

Then it is easy to calculate that

5

that is, D1 + D2 K+ = 0, For λ = −γ , we have similarly D1 + D2 K− = 0,

D3 + (D4 + λ D2 σ )K+ = 0.

(25)

D3 + (D4 − λ D2 σ )K− = 0.

(26)

Noticing that (K+ − K− ) is non-degenerate we obtain from (25) and (26) that D1 = D2 = D3 = D4 = 0, that is, D = 0. Therefore (11) hold, which implies that function ϕ¯ defined by (6) satisfies the spectral problem (8). This completes the proof. □ Theorem 3. Let Φ+ be a non-degenerate 2 × 2 matrix, whose column vectors are solutions of the spectral problem (2) and (3) with λ = γ . Then there exists a non-degenerate 2 × 2 matrix Φ− , whose column vectors are solutions of the spectral problem (2) and (3) with λ = −γ , satisfying −1 † Φ+,x Φ−1 + = [Φ−,x Φ− ] ,

(27)

where † is an operator that swaps the (1, 1)- and (2, 2)-entries of a 2 × 2 matrix, that is, ( )† ( ) m1 m2 m4 m2 = m3 m4 m3 m1

for all scalars m1 , m2 , . . . , m4 . This means that overdetermined system (17) is compatible. P ROOF. Let K = Φ+,x Φ−1 + . Consider the spectral problem

ϕx = K ϕ , ϕt = −Vx ϕ + 2(V + γσ )ϕx .

(28)

Vx − 2VU + Kt − (Vx K + KVx )+ 2K(V + λ σ )K − uσ γ − 2γ 2 I = 0

(29)

The two column vectors of Φ+ are solutions of equation (28), so it is compatible. Thus the compatibility condition

holds. Acting operator † on equation (29), we can obtain Vx − 2VU + Kt† − (Vx K † + K †Vx )+ 2K † (V − γσ )K † + uσ γ − 2γ 2 I = 0

(30)

in view of U † = U,V † = V,UV = VU, σ † = −σ and (M1 M2 )† = M2† M1† , which is the compatibility condition for the following system

ϕx = K † ϕ , ϕt = −Vx ϕ + 2(V − γσ )ϕx .

(31)

Let Φ− be a fundamental solution matrix to system (31). Then −1 † Φ+,x Φ−1 + = [Φ−,x Φ− ]

holds. This completes the proof. 6

□

Theorem 4. Let Φ+ be a non-degenerate matrix, whose column vectors are solutions of spectral problem (2) and (3) with λ = γ , where γ is any nonzero constant. Let ( ) p+q r Φ+,x Φ−1 = . (32) + s p−q Then there exists a solution (a, b, c) of equation (17) if and only if q − γ s = 0. □

P ROOF. Through direct calculations. 3. Exact solutions

In this section, we shall apply the Darboux transformation to construct exact solutions of the DSSH system (1). Solution 1. Substituting the trivial solution u = 1, v = −25 of the DSSH system (1) into (2) and (3), it is easy to calculate that {ϕ1 , ϕ2 , . . . , ϕ4 } is a fundamental system of solutions of system (2) and (3) with λ = γ = 5, where ( ) 2 cosh(t − x) − 5(3t + x) sinh(t − x) ϕ1 = , −(3t + x) sinh(t − x) ( ) 3 sinh(t − x) + 5(3t + x) cosh(t − x) ϕ2 = , sinh(t − x) + (3t + x) cosh(t − x) ( ) 25(3t + x) sinh(t − x) ϕ3 = , 5(3t + x) sinh(t − x) + 2 cosh(t − x) ( ) −25 sinh(t − x) − 25(3t + x) cosh(t − x) ϕ4 = . −7 sinh(t − x) − 5(3t + x) cosh(t − x) Let Φ+ be a 2 × 2 matrix, whose two column vectors are, respectively,

ϕ1 + ϕ3

and

9ϕ2 + ϕ4 .

One can easily verify that Φ+ fits the condition in Theorems 2 and 4. Using the Darboux transformation (12), we arrive at a rational solution (see Figure 1) of the DSSH system (1) [4(3t + x)2 + 9]2 − 96 , [4(3t + x)2 + 1]2 32 . v1 = 4(3t + x)2 + 1

u1 =

7

(33a) (33b)

u1

v1

u1

v1

Figure 1: u1 and v1 , and their sections

Solution 2. Choose a trivial solution u = 1, v = −1 of the DSSH system (1), γ = 1, and ) ( 2 sinh η22 − ξ2 cosh η22 ξ2 sinh η22 − cosh η22 Φ+ = , cosh η22 + ξ2 sinh η22 −ξ2 cosh η22 where ξ2 = 3t + x, η2 = 2t − 2x. From the Darboux transformation (12), we obtain a two-soliton solution (see Figure 2) of the DSSH system (1) N2 , 2(cosh η2 + 2ξ22 + 1)2 16 − 1, v2 = cosh η2 + 2ξ22 + 1

u2 =

(34a) (34b)

where N2 = −4(6ξ22 + 7) cosh η2 − 64ξ2 sinh η2 + cosh 2η2 + 8ξ24 + 40ξ22 − 29. Solutions 3 and 4. Let u = 5, v = 12 and γ = 2. In a similar way, we arrive at a traveling wave solution (u3 , v3 ) (see Figure 3) of the DSSH system (1) √ 10 cosh 2ξ3 + 4 3 cosh ξ3 − 7 √ u3 = , (35a) (2 cosh ξ3 + 3)2 √ 64 3 √ , v3 = 12 − (35b) 2 cosh ξ3 + 3

8

u2

v2

Figure 2: u2 and v2 , a two-wave interaction

where ξ3 = 22t + 2x, and a non-traveling wave solution (u4 , v4 ) (see Figure 4) of the DSSH system (1) N4 , 2(4 cosh η4 + cosh ξ4 + 3)2 192 v4 = 12 − , 4 cosh η4 + cosh ξ4 + 3

u4 =

(36a) (36b)

where ξ4 = 4t − 4x, η4 = 22t + 2x, N4 = −256 sinh η4 sinh ξ4 − 240 cosh η4 cosh ξ4 + 48 cosh η4 + 80 cosh 2η4 − 132 cosh ξ4 + 5 cosh 2ξ4 − 145.

u3

v3

u3

v3

Figure 3: u3 and v3 , and their sections (lower)

9

u4

v4

Figure 4: u4 and v4 , a two-wave interaction

√ Solution 5 and 6. Let u = 0, v = 0 and γ = 2 −1. Then we obtain a periodic solution (u5 , v5 ) (see Figure 5) of the DSSH system (1) √ 8 − 8 2 cos(8t + 2x) , (37a) u5 = √ [ 2 − cos(8t + 2x)]2 √ 8( 2 + cos(8t + 2x)) √ v5 = . (37b) 2 − cos(8t + 2x) and another solution (u6 , v6 ) (see Figure 6) of the DSSH system (1) 16(sinh η6 sin ξ6 − cosh η6 + cos ξ6 ) , (cosh η6 + cos ξ6 + 2)2 32 , v6 = cosh η6 + cos ξ6 + 2

u6 =

(38a) (38b)

where ξ6 = 8t + 2x, η6 = 8t − 2x. 4. Conclusion In the present paper, we have constructed a Darboux transformation for the DSSH system based on a gauge transformation between the corresponding 4 × 4 matrix spectral problems with two potentials. Here a difficult problem is the compatibility for the overdetermined system (17) and the existence of its solutions. The gauge transformation T or the Darboux matrix T to be suitably chosen is the key to construct the Darboux transformation, by which the Lax pair of the DSSH system is changed into another Lax pair of the same type. As applications of Darboux transformations, a few types of new solutions of the DSSH system are given explicitly. Acknowledgements This work is supported by the National Natural Science Foundation of China (Grant Nos. 11331008 and 11171312). 10

u5

v5

u5

v5

Figure 5: u5 (panned upside-down) and v5 , and their sections

u6

v6

Figure 6: u6 (panned upside-down) and v6

References [1] M. J. Ablowitz, H. Segur, Solitons and the inverse scattering transform, vol. 4 of SIAM Studies in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981. [2] S. Novikov, S. V. Manakov, L. P. Pitaevskii, Theory of Solitons: The Inverse Scattering Method, Monographs in Contemporary Mathematics, Springer, 1984. [3] R. Hirota, The direct method in soliton theory, vol. 155 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, translated from the 1992 Japanese original and edited by Atsushi Nagai, Jon Nimmo and Claire Gilson, With a foreword by Jarmo Hietarinta and Jon Nimmo, 2004. [4] V. B. Matveev, M. A. Salle, Darboux transformations and solitons, Springer Series in Nonlinear Dynamics, Springer-Verlag, Berlin, 1991. 11

[5] E. Belokolos, A. Bobenko, V. Enol’skii, A. Its, V. Matveev, Algebro-geometric approach to nonlinear integrable equations, Springer series in nonlinear dynamics, Springer-Verlag, 1994. [6] D. Levi, On a new Darboux transformation for the construction of exact solutions of the Schr¨odinger equation, Inverse Problems 4 (1) (1988) 165–172. [7] X. Geng, Darboux transformation of the discrete Ablowitz-Ladik eigenvalue problem, Acta Math. Sci. (English Ed.) 9 (1) (1989) 21–26. [8] X. Geng, H.-W. Tam, Darboux transformation and soliton solutions for generalized nonlinear Schr¨odinger equations, J. Phys. Soc. Japan 68 (5) (1999) 1508– 1512. [9] Y. Li, W.-X. Ma, J. E. Zhang, Darboux transformations of classical Boussinesq system and its new solutions, Phys. Lett. A 275 (1-2) (2000) 60–66. [10] V. G. Drinfel’d, V. V. Sokolov, New evolution equations having an (L, A) pair, in: Partial differential equations, vol. 81 of Trudy Sem. S. L. Soboleva, No. 2, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 5–9, 1981. [11] J. Satsuma, R. Hirota, A coupled KdV equation is one case of the four-reduction of the KP hierarchy, J. Phys. Soc. Japan 51 (10) (1982) 3390–3397. [12] M. G¨urses, A. Karasu, Integrable KdV systems: recursion operators of degree four, Phys. Lett. A 251 (4) (1999) 247–249. [13] A. Karasu, S. Y. Sakovich, B¨acklund transformation and special solutions for the Drinfeld-Sokolov-Satsuma-Hirota system of coupled equations, J. Phys. A 34 (36) (2001) 7355–7358. [14] M. Yu. Balakhnev, D. K. Demskoi, Auto-B¨acklund transformations and superposition formulas for solutions of Drinfeld-Sokolov systems, Appl. Math. Comput. 219 (8) (2012) 3625–3637. [15] A. Karasu-Kalkanlı, A. Karasu, A. Sakovich, S. Sakovich, R. Turhan, A new integrable generalization of the Korteweg-de Vries equation, J. Math. Phys. 49 (7) (2008) 073516. [16] A.-M. Wazwaz, The Cole-Hopf transformation and multiple soliton solutions for the integrable sixth-order Drinfeld-Sokolov-Satsuma-Hirota equation, Appl. Math. Comput. 207 (1) (2009) 248–255. [17] B. C. Yang, X. Geng, Drinfeld-Sokolov-Satsuma-Hirota equations and generalized bi-Hamiltonian structures, J. Zhengzhou Univ. Nat. Sci. Ed. 43 (3) (2011) 31–33, 37.

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