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Related content Construction of type-II Bäcklund transformation for the mKdV hierarchy J F Gomes, A L Retore and A H Zimerman - Geometric aspects of Painlevé equations Kenji Kajiwara, Masatoshi Noumi and Yasuhiko Yamada - Soliton splitting in quenched classical integrable systems O Gamayun and M Semenyakin - Recent citations A new spectral problem and the related integrable nonlinear evolution equations Xiao Yang and Dianlou Du - Quantum integrability and quantum groups: a special issue in memory of Petr P Kulish Nikolai Kitanine et al - This content was downloaded from IP address 186.217.236.55 on 26/07/2019 at 16:36 https://doi.org/10.1088/1751-8113/49/50/504003 http://iopscience.iop.org/article/10.1088/1751-8113/48/40/405203 http://iopscience.iop.org/article/10.1088/1751-8113/48/40/405203 http://iopscience.iop.org/article/10.1088/1751-8121/50/7/073001 http://iopscience.iop.org/article/10.1088/1751-8113/49/33/335201 http://iopscience.iop.org/article/10.1088/1751-8113/49/33/335201 http://dx.doi.org/10.1016/j.aml.2017.08.010 http://dx.doi.org/10.1016/j.aml.2017.08.010 http://iopscience.iop.org/1751-8121/51/11/110201 http://iopscience.iop.org/1751-8121/51/11/110201 http://iopscience.iop.org/1751-8121/51/11/110201 https://oasc-eu1.247realmedia.com/5c/iopscience.iop.org/523341230/Middle/IOPP/IOPs-Mid-JPA-pdf/IOPs-Mid-JPA-pdf.jpg/1? Miura and generalized Bäcklund transformation for KdV hierarchy J F Gomes1, A L Retore and A H Zimerman Instituto de Física Teórica-UNESP Rua Dr Bento Teobaldo Ferraz 271, Bloco II 01140-070, São Paulo, Brazil E-mail: jfg@ift.unesp.br Received 6 June 2016, revised 7 October 2016 Accepted for publication 13 October 2016 Published 21 November 2016 Abstract Using the fact that Miura transformation can be expressed in the form of gauge transformation connecting the KdV and mKdV equations, we discuss the derivation of the Bäcklund transformation and its Miura-gauge transformation connecting both hierarchies. Keywords: integrable models, solitons, Bäcklund transformation 1. Introduction Integrable models are characterized by an infinite number of conservation laws which are responsible for the stability of soliton solutions. In fact, these conservation laws may be regarded as hamiltonians generating time evolutions within a multi-time space. Each of these time evolutions, in turn, are associated to a nonlinear equation of motion and henceforth constitute an integrable hierarchy of equations with a common set of conservation laws. It has become clear in the past few years that integrable hierarchies may be systematically con- structed from a graded algebraic structure involving affine Kac–Moody algebras. The equations of motion are constructed systematically from a graded algebra embedded within a zero curvature representation. The most well known example are the sine (sinh)-Gordon and mKdV equations, both underlined by the same sl 2ˆ ( ) affine algebra in the principal gradation (see [1] for a review). Another peculiar feature of integrable models is the existence of Bäcklund transforma- tions which relate two different field configurations of certain nonlinear differential equation. These Bäcklund transformations, among other applications, generate an infinite sequence of soliton solutions from a nonlinear superposition principle (see [2] for review). Bäcklund transformations have also been employed to describe integrable defects [3] in the sense that two solutions of an integrable model may be interpolated by a defect at certain Journal of Physics A: Mathematical and Theoretical J. Phys. A: Math. Theor. 49 (2016) 504003 (19pp) doi:10.1088/1751-8113/49/50/504003 1 Author to whom any correspondence should be addressed. 1751-8113/16/504003+19$33.00 © 2016 IOP Publishing Ltd Printed in the UK 1 mailto: jfg@ift.unesp.br http://dx.doi.org/10.1088/1751-8113/49/50/504003 http://crossmark.crossref.org/dialog/?doi=10.1088/1751-8113/49/50/504003&domain=pdf&date_stamp=2016-11-21 http://crossmark.crossref.org/dialog/?doi=10.1088/1751-8113/49/50/504003&domain=pdf&date_stamp=2016-11-21 spatial position. The Bäcklund transformation connecting the two field configurations is the key ingredient to preserve the integrability of the system. Under such formulation the canonical energy and momentum are no longer conserved and modifications to ensure its conservation are required in order to take into account the contribution of the defect [4]. Well known (relativistic) integrable models as the sine (sinh)-Gordon, Tzitzeica [3], Lund–Regge [5] and other (non-relativistic) models as nonlinear Schroedinger (NLS), mKdV, etc have been studied within such context [6]. In [7] we have constructed Bäcklund transformation to the mKdV hierarchy by assuming that two field configurations of the same equation of motion were related by a gauge trans- formation and dubbed from now on Bäcklund-gauge transformation. As a result it was sub- sequently shown that such Bäcklund-gauge transformation acted in a universal manner in all members of the hierarchy, i.e., the same for all models within the mKdV hierarchy. In this paper we extend the results of [7] to the KdV hierarchy. We first consider the transformation from the principal to the homogeneous gradation by a global gauge trans- formation g1. Next, by a second, local gauge transformation g v,2 ( ), we realize the Miura transformation relating the fields v of the mKdV to those   = ¶ - = J v v , 1x 2 of the KdV hierarchies. In fact, the Miura transformation displays a sign ambiguity represented by the  -factor, i.e., solutions of the mKdV hierarchy generate two towers of solutions of the KdV hierarchy labeled by the  -sign. In section 2 we review the algebraic construction of mKdV hierarchy and construct explicitly the Miura-gauge transformation that maps the equations of the mKdV into equations of the KdV hierarchy. We display the first few positive grade time evolution equations for both hierarchies. In section 3, from the Bäcklund-gauge transformation con- structed in [7] we discuss its Miura extension and propose Bäcklund-gauge transformation for the KdV hierarchy. This, however displays a sign ambiguity inherited from the Miura transformation. Each solution, v x t, N( ) of the mKdV hierarchy is mapped into two solutions,   = ¶ - = J x t v v, , 1N x 2( ) of the KdV hierarchy. One of the main results of our construction is to show that a pair of Bäcklund solutions of the mKdV hierarchy, v v,1 2 have to be mapped into a pair of KdV Bäcklund solutions of opposite  -signs, = ¶ - = -¶ -+ -J v v J v v,x x1 1 1 2 2 2 2 2 in order to satisfy the KdV Bäcklund equations. In section 4 we discuss the composition of gauge-Bäcklund transformations for both hierarchies. In fact we start by considering two consecutive gauge-Bäcklund transformations for the mKdV hierarchy, K1 and K2 and we show that the product, =K K KII 2 1 generate the so called Type-II Bäcklund transformation proposed in [3]. We follow the same philosophy to construct the Type-II Bäcklund transformation for the KdV hierarchy. First by direct product composition of two Type-I, =K K KII 2 1˜ ˜ ˜ and the second by Miura-gauge transforming the Type-II Bäcklund transformation of the mKdV hierarchy. Consistency is shown that the two alternative constructions indeed agree. 2. The algebraic formalism for KdV and mKdV hierarchies Following the algebraic formalism described in [7] we recall that the nonlinear equations of the mKdV hierarchy can be derived from the zero curvature representation, ¶ + ¶ + =A A, 0. 2.1x x t tN N [ ] ( ) Underlined by an affine sl 2ˆ ( ) centerless Kac–Moody algebra generated by l l l= = Î Îa a h h E E C m Z, , ,m m m m( ) ( ) satisfying J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 2 =  =a a a a  + - +h E E E E h, 2 , , . 2.2m n m n m n m n[ ] [ ] ( )( ) ( ) ( ) ( ) ( ) ( ) Here = +A E Ax 1 0 ( ) , = + + +- A D D Dt N N 1 0 N ( ) ( ) ( ), l= +a a-E E E1( ) and2 u=A x t h, N0 ( ) are constructed according to the principal gradation, l= + l Q h2 1 2p d d which decomposes the affine = åsl 2 i i ˆ ( ) algebra into graded subspaces according to powers of the spectral parameter λ,     = Î Î+Q a a b Z, , , , , , 2.3p a a a b a b[ ] [ ] ( ) where the subspaces  m2 and  +m2 1 contains the following generators   l l l l l = = = + -a a a a+ - - h h E E E E , , . 2.4 m m m m m m 2 2 1 { } { ( ) ( )} ( ) ( ) Here ÎD i i ( ) . The zero curvature (2.1) decomposes according to the graded structure as = + + ¶ = = + ¶ - ¶ = -   E D E D A D D A D D A , 0 , , 0 , 0, 2.5 N N N x N x t 1 1 1 0 0 0 0 0N [ ] [ ] [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) which allows solving for = D i N, 0,i( ) and the last equation in (2.5) yields the time evolution for fields A0. We should point out that D i( ) are constructed systematically for each value of N and so is AtN . The first few explicit equations are (see for instance [7, 9]), ¶ - ¶ ¶ - =v v v m4 2 0 KdV, 2.6t x x 2 3 3 ( ) ( ) ¶ - ¶ ¶ - ¶ - ¶ + =v v v v v v v16 10 10 6 0, 2.7t x x x x 4 2 2 2 5 5 ( ( ) ( ) ) ( ) ¶ - ¶ ¶ - ¶ ¶ - ¶ - ¶ ¶ + ¶ ¶ - ¶ - ¶ + =  v v v v v v v v v v v v v v v v 64 70 42 56 14 140 70 20 0 etc. 2.8 t x x x x x x x x x x x 6 2 2 2 2 3 2 4 3 2 4 2 7 7 ( ( ) ( ) ( ) ( )( )) ( ( ) ( ) ) ( ) Consider now the global gauge transformation generated by ⎛ ⎝⎜ ⎞ ⎠⎟ z z z l= - =g 1 1 , 2.91 2 ( ) which transforms ⎜ ⎟ ⎛ ⎝ ⎞ ⎠l = + = - A E v x t h v v , 1 ,x m N, KdV princ 1 ( )( ) into ⎛ ⎝⎜ ⎞ ⎠⎟ z z = = + = - - -A g A g g E v x t h g v v , , 2.10x m x m N, KdV hom 1 , KdV princ 1 1 1 1 1 1( ) ( ( ) ) ( )( ) i.e., transforms the principal into homogeneous gradation, z= z Qhom d d . A subsequent local Miura-gauge transformation [10, 11] ⎜ ⎟⎛ ⎝ ⎞ ⎠   z= - - +g v v v , 1 2 , 2.112 ( ) ( ) 2 = - h 1 0 0 1( ), =aE 0 1 0 0( ) and =a-E 0 0 1 0 .( ) J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 3 transforms A Ax m x, KdV ,KdV. i.e., ⎛ ⎝⎜ ⎞ ⎠⎟    z z = - ¶ = - - - -A g v A g v g v g v J , , , , 1 . 2.12x x m x,KdV 2 , KdV hom 2 1 2 2 1( ) ( ) ( ) ( ) ( ) And realizes the Miura transformation,  = ¶ - =J v v , 1. 2.13x 2 2 ( ) We should emphasize that for each solution v x t, N( ) of the evolution equations for the mKdV hierarchy, the Miura transformation (2.13) generates two towers of solutions,  = J x t, , 1N( ) , of the KdV hierarchy [11]. The zero curvature under the homogeneous gradation3 ¶ + ¶ + =A A, 0, 2.14x x t t,KdV ,KdVN N [ ] ( ) with     = + + + Î- A ,t N N j j ,KdV 1 0 N ˜ ˜ ˜ ˜ ˜( ) ( ) ( ) ( ) yields the KdV hierarchy equations of motion. For instance ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ z z z z z z z = + + ¶ - - + ¶ + ¶ + - - - ¶ A J J J J J J J J J 2.15t x x x x ,KdV 3 1 2 1 4 2 1 2 2 1 2 1 4 2 1 2 2 3 1 2 1 4 3 ( ) yields the KdV equation ¶ - ¶ - ¶ = ¶ - ¶ - ¶ ¶ - =J J J J v v v v4 6 2 4 2 0. 2.16t x x x t x x 3 2 3 3 3 ( )[ ( )] ( ) Similarly from At5 and for At7, given in the appendix, we find the Sawada–Kotera equation [13]  ¶ - ¶ - ¶ ¶ - ¶ - ¶ = ¶ - ¶ - ¶ ¶ - ¶ - ¶ + = J J J J J J J J v v v v v v v v 16 20 10 30 2 16 10 10 6 0, 2.17 t x x x x x x t x x x x 5 2 3 2 4 2 2 2 5 5 5 ( )[ ( ( ) )] ( ) and4  ¶ - ¶ - ¶ ¶ - ¶ ¶ - ¶ - ¶ - ¶ ¶ - ¶ - ¶ = ¶ - ¶ - ¶ ¶ - ¶ ¶ - ¶ - ¶ ¶ - ¶ + ¶ + ¶ - = J J J J J J J J J J J J J J J J v v v v v v v v v v v v v v v v v 64 70 42 70 14 280 70 140 2 64 70 42 56 14 140 70 20 0 2.18 t x x x x x x x x x x x x t x x x x x x x x x x 7 2 3 4 3 5 2 2 3 3 6 2 2 2 2 3 2 4 3 2 4 2 7 7 7 ( ) ( )( ( ( ) ( ) ( ) )) ( ) respectively. Equations (2.16)–(2.18) are displayed as explicit examples as illustration of the formalism. Higher flows (time evolutions) can be systematically constructed for generic N from the same formalism. 3. Bäcklund transformation 3.1. mKdV In this section we start by noticing that the zero curvature representation (2.1) and (2.14) are invariant under gauge transformations of the type 3 Notice that under the homogeneous gradation the decomposition of the affine Lie alge- bra   z z z= å Î = a a-sl a Z h E E2 , , , , .a a a a a aˆ ( ) ˜ ˜ { } 4 In general, we find = ¶ -J v m vKdV 2 KdVx( ) ( ) ( ). J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 4 f f¶  = + ¶m m m m - -A A K A K K K, , , 3.1x 1 1( ) ˜ ( ) where Aμ stands for either AtN or Ax. The key ingredient of this section is to consider two field configurations f1 and f2 embedded in fmA 1( ) and fmA 2( ) satisfying the zero curvature representation and assume that they are related by a Bäcklund-gauge transformation generated by f fK ,1 2( ) preserving the equations of motion (e.g, zero curvature (2.1) or (2.14)), i.e., f f f f f f f f= + ¶m m mK A A K K, , , . 3.21 2 1 2 1 2 1 2( ) ( ) ( ) ( ) ( ) ( ) If we now consider the Lax operator = ¶ +L Ax x for mKdV case within the principal gradation, ⎡ ⎣⎢ ⎤ ⎦⎥ f l f = + = ¶ - ¶ A E A x t x t , 1 , 3.3x m x N x N , KdV 1 0 ( ) ( ) ( )( ) is common to all members of the hierarchy defined by (2.1). We find that f f f f f f f f= + ¶K A A K K, , , , 3.4x m x m x1 2 , KdV 1 , KdV 2 1 2 1 2( ) ( ) ( ) ( ) ( ) ( ) where the Bäcklund-gauge generator f fK ,1 2( ) is given by [7, 9] ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥f f = - - b l f f b f f - + + K e e , 1 1 3.51 2 2 2 1 2 1 2 ( ) ( ) ( ) ( ) and b is the Bäcklund parameter. Equation (3.4) is satisfied provided f f b f f¶ - = - +sinh . 3.6x 1 2 1 2( ) ( ) ( ) For the sinh-Gordon (s-g) model, the equations of motion f f¶ ¶ = =a2 sinh 2 , 1, 2t x a a are satisfied if we further introduce the time component of the Bäcklund transformation, f f b f f¶ + = - 4 sinh . 3.7t 1 2 2 1( ) ( ) ( ) Equation (3.7) is compatible with (3.2) for =mA AtN with ⎡ ⎣⎢ ⎤ ⎦⎥ l= f f - - A e e 0 0 . 3.8t s g, 1 2 2 ( )‐ For higher graded time evolutions the time component of the Bäcklund transformation can be derived from the appropriated time component of the two-dimensional gauge potential. Several explicit examples within the positive and negative graded mKdV sub-hierarchies were discussed in [7]. We now give a general argument that the Bäcklund Transformation derived from the gauge transformation (3.4) for arbitrary N provides equations compatible with the equation of motion. Consider the zero curvature representation for certain field configuration, namely f1, i.e., f f¶ + ¶ + =A A, 0. 3.9x x t t1 1N N [ ( ) ( )] ( ) Under the gauge transformation, f f f f f f f f f f ¶ + ¶ + = ¶ + ¶ + = ¶ + ¶ + = - - - K A A K K A K K A K A A , , , , , 0, 3.10 x x t t x x t t x x t t 1 2 1 1 1 2 1 1 1 1 1 2 2 N N N N N N ( )[ ( ) ( )] ( ) [ ( ( )) ( ( )) ] [ ( ) ( )] ( ) where the last equality comes from our assumption (3.4). J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 5 The gauge transformation of the first entry in the zero curvature representation implies the x-component of the Bäcklund transformation (3.6). Since the zero curvature (3.9) and (3.10) implies that both f1 and f2 satisfy the same equation of motion, the gauge transfor- mation (3.4) for =mA AtN of the second entry in (3.10) generates the time component of BT which, by construction has to be consistent with the equations of motion with respect to time tN. 3.2. KdV In order to extend the same philosophy to the KdV hierarchy recall the fact that the two- dimensional gauge potential Ax,KdV can be obtained by Miura-gauge transformation from the homogeneous mKdV gauge potentials AmKdV hom as in (2.12), i.e.,    = - ¶- - -A J g v g A v g g v g v g v, , , , , 3.11x x m x,KdV 2 1 , KdV 1 1 2 1 2 2 1( ) ( ) ( ( )) ( ) ( ) ( ) ( ) where f= ¶v x t,x N( ). By assuming (3.2) for the KdV hierarchy, i.e., = + ¶K J J A J A J K J J K J J, , , . 3.12x x x1 2 ,KdV 1 ,KdV 2 1 2 1 2˜ ( ) ( ) ( ) ˜ ( ) ˜ ( ) ( ) the Bäcklund-gauge transformation for the KdV hierarchy K J J,1 2˜ ( ) constructed in terms of f fK ,1 2( ) can be written as  f f= - -K g v g K g g v, , , . 3.132 2 2 1 1 2 1 1 2 1 1 1˜ ( )( ( ) ) ( ) ( ) At this stage we should recall that for each solution of the mKdV hierarchy v, the Miura transformation (2.13) generates two solutions,   = ¶ - = J v v , 1i x i i 2 i satisfying the associated equation of motion of the KdV hierarchy. This is precisely why we assume 1 and 2 in equation (3.13) independent. In terms of mKdV variables f= ¶vi x i, K̃ is given for the particular case where   = - º1 2 and denote5   = = - ºK J J K, ,1 2 1 2˜ ( ) ˜ ( )           z b z b z z b z b z z z b z b z = - - - - + = - = - + + - + - - + - = - - - + - - - - - K J J v e e K J J K J J v v e v v e v v K J J v e e , 1 4 1 4 1 , 1 , 4 1 1 4 1 1 , 1 4 1 4 1 , 3.14 p p p p p p 1 2 11 1 1 2 12 1 2 21 1 2 1 2 1 2 1 2 22 2 ˜ ( ) ( ) ( ) ˜ ( ) ˜ ( ) ( ( ) ( )) ( ( ) ( )) ˜ ( ) ( ) ( ) ( ) where f f= +p 1 2. Substituting in equation (3.12) we find the following equations: • Matrix element 11: z b- ¶ - - + =- -J v v e e v v: 1 2 0. 3.15x p p1 1 1 1 1 2( ) ( ) 5 Notice that K̃ is given in terms of mKdV variables v v,1 2 and we need to rewrite it in terms of KdV variables J J,1 2. This requires solving Riccati equation =v v J( ) (2.13). J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 6 • Matrix element 12: z b - + - =- -v v e e: 2 0. 3.16p p1 1 2 ( ) ( ) • Matrix element 21:     z b b b b + + + - - - - + + + = - - J J v e v e v e v e v v : 2 1 2 1 2 1 2 1 2 0, 3.17 p p p p 0 1 2 1 1 2 2 1 2 ( ) ( ) ( ) ( ) ( )   z b - - ¶ - ¶ - - = - - J v J v v v v v v v e e : 1 2 0. 3.18 x x p p 1 1 2 2 1 1 2 2 1 2 ( ) ( ) ( ) • Matrix element 22: z b - - ¶ - - =- -J v v e e 2 0. 3.19x p p1 2 2 2 ( ) ( ) Using the mixed Miura transformation, i.e.,   = - º2 1 ,  = ¶ - = - ¶ -J v v J v v, 3.20x x1 1 1 2 2 2 2 2 ( ) together with the mKdV Bäcklund transformation (3.6) b - = - - -v v e e 2 , 3.21p p 1 2 ( ) ( ) we find that equations (3.15), (3.16), (3.18) and (3.19) are identically satisfied. Defining the new variable Q and taking into account the Bäcklund equation (3.21) we find the following equality    b b = + - + +-Q v e e 1 2 4 1 4 1 3.22p p 1 ( ) ( ) ( )    b b = + + + --v e e 4 1 4 1 , 3.23p p 2 ( ) ( ) ( ) Eliminating v1 and v2 from equations (3.22) and (3.23) we find    b b = - - - +-v Q e e 2 4 1 4 1 , 3.24p p 1 ( ) ( ) ( )    b b = - + - --v Q e e 2 4 1 4 1 3.25p p 2 ( ) ( ) ( ) and henceforth     b b b - + + - + - - + - = - -v v e v v e v v Q 4 1 1 4 1 1 4 4 . 3.26 p p 1 2 1 2 1 2 2 2 ( ( ) ( )) ( ( ) ( )) ( ) Equation (3.17) then becomes b + = -J J Q 2 2 . 3.271 2 2 2 ( ) J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 7 From (3.22) and (3.23) we find that  b = + + + -Q v v e e 2 . 3.28p p 1 2( ) ( ) ( ) Acting with ¶x in (3.28) and using (3.20) and (3.21),   b w w ¶ = ¶ + + + - = ¶ + - - + = - = ¶ - -Q v v v v e e v v v v v v J J 2 , 3.29 x x p p x x 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ( ) ( )( ) ( ) ( )( ) ( ) ( ) where we have used º ¶ =J w i, 1, 2i x i . It therefore follows that = -Q w w 3.301 2 ( ) and the Bäcklund transformation for the spatial component of the KdV equation becomes, b + = ¶ = - - = +J J P w w P w w 2 2 , . 3.31x1 2 2 1 2 2 1 2 ( ) ( ) Which is in agreement with the Bäcklund transformation proposed in [12] and with [14]. In the new variable Q defined in (3.22) and (3.23) we rewrite the gauge-Bäcklund transformation K J J,1 2˜ ( ) in (3.14) as ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟b z z z = - - + + +b- K J J Q Q Q , , 1 1 . 3.321 2 1 2 4 1 4 2 1 2 2 ˜ ( ) ( ) Other cases with  = = 11 2 lead to trivial Bäcklund transformations in the sense that (3.12) for  K 1, 1˜ ( ) is trivially satisfied for mKdV Bäcklund and Miura transformations (3.6) and (2.13). There is no new equation relating the two KdV fields J1 and J2. From now on we shall only consider + - ºK K1, 1˜ ( ) ˜ given in (3.32) and Miura transformation given by (3.20). We now discuss the extension of the Bäcklund transformation to the time component of the KdV hierarchy. Notice that in the zero curvature representation the spatial component of the two-dimensional gauge potential Ax is the same for all flows and therefore universal among the different evolution equations. They differ from the time component AtN written according to the algebraic graded structure and parametrized by the integer N.     = + + + Î- A , . 3.33t N N j j ,KdV 1 0 N ˜ ˜ ˜ ˜ ˜ ( )( ) ( ) ( ) ( ) The Bäcklund-gauge transformation (3.32) acting on the potentials At ,KdV3 , At ,KdV5 and At ,KdV7 given by equations (A.2)–(A.4) of the appendix leads to the following Bäcklund equations respectively ¶ = - ¶ + ¶ + ¶P Q Q Q P4 1 2 3 , 3.34t x x x 2 2 2 3 (( ) ( ) ) ( ) ¶ = - ¶ + ¶ ¶ + ¶ ¶ + ¶ - ¶ + ¶ ¶ + ¶ P Q Q Q Q P P P Q P P Q 16 5 1 2 5 5 2 3 , 3.35 t x x x x x x x x x x 4 3 3 2 2 2 2 2 2 5 ( ( ) ( ) ) (( ) ( ) ) ( ) J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 8 ¶ =- ¶ + ¶ ¶ + ¶ ¶ - ¶ ¶ + ¶ ¶ + ¶ ¶ ¶ + ¶ ¶ ¶ + ¶ + ¶ + ¶ ¶ + ¶ + ¶ ¶ + ¶ + ¶ ¶ + ¶ + ¶ P Q Q Q Q P P Q Q P P Q Q P P Q Q P Q P P Q P P Q P Q P Q 64 7 14 35 35 21 2 1 2 35 2 35 2 105 4 35 8 35 8 , 3.36 t x x x x x x x x x x x x x x x x x x x x x x x x x x x 6 5 5 2 4 2 4 2 2 3 3 2 3 2 2 2 2 2 3 2 2 2 2 4 2 7 ( ) ( ) (( ) ( ) ) (( ) ( ) ) ( ) ( ) ( ) ( ) ( ) where ¶ = +J JP 1 2. Equations (3.31) and (3.34) coincide with the Bäcklund transformation proposed in [12] for the KdV equation. Equations (3.31) and (3.35) correspond to those derived for the Sawada–Kotera equation in [14]6. In the appendix we have checked the consistency between the spatial, (3.31) and time components (3.34)–(3.36) of the Bäcklund transformations for N = 3, 5 and 7. By direct calculation, using software Mathematica, we indeed recover the evolution equations (2.16)–(2.18). We would like to point out that our method is systematic and provides the Bäcklund transformations for arbitrary time evolution in terms of its time component 2D gauge potential At ,KdVN in terms of graded subspaces = D i N, 0,i˜ ( ) . The examples given above for t t,3 5 and t7 just illustrate the potential of the formalism. 3.3. Examples • Vacuum—one soliton solution Consider f = 01 and f = r r + - ln2 1 1( ), r = +e ,kx k t2 2 N N =N 3, 5, 7 two solutions of the mKdV hierarchy. The mixed Miura transformation yields f f f f= ¶ - ¶ = = -¶ - ¶+ -J J0, 3.37x x x x 1 2 1 1 2 2 2 2 2 2( ) ( ) ( ) Integrating to obtain = ¶J wx we find r = = - + +w w k k0, 4 1 2 3.381 2 ( ) Type-I Bäcklund transformation ¶ + = - -bw w w wx 1 2 2 1 2 1 2 2 2 ( ) ( ) is satisfied by (3.38) for b =  k2 • Scattering of two one-soliton solutions Consider the one-soliton of the mKdV hierarchy given by ⎛ ⎝⎜ ⎞ ⎠⎟f r r r= + - = = =+  R R i e Nln 1 1 , 1, 2 , 3, 5, 7 3.39i i i kx k t2 2 N N ( ) Miura transformation generates two one-soliton solutions of the KdV hierarchy, namely f f= ¶ - ¶+J ; 3.40x x 1 2 1 1 2( ) ( ) f f= -¶ - ¶-J ; 3.41x x 2 2 2 2 2( ) ( ) leading to r r = - + + = - - +w k R k w k R k 4 1 2 , 4 1 2 . 3.421 1 2 2 ( ) The Type-I Bäcklund transformation is satisfied for =R R1 2. Notice that although 6 Notice that there are typos in equation (45.11) of [14]. J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 9 =R R1 2, J1 and J2 correspond to different solutions due to opposite  -sings in the Miura transformation. • One-soliton into two-soliton solution Taking f1 given by the one-soliton solution (3.39) and f2 by ⎛ ⎝⎜ ⎞ ⎠⎟f d r r r r d r r r r d= + - - - - - = + - k k k k ln 1 1 , 3.432 1 2 1 2 1 2 1 2 1 2 1 2 ( ) ( ) ( ) leading to r r r r r r = - - + + - - + - - - w k k k k k k k k 2 1 1 , 3.442 1 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 ( )( )( ) ( )( ) ( ) ( ) where r = =+e i, 1, 2i k x k t2 2i i N N satisfy the Type-I Bäcklund transformation for b =  k2 2. All these verifications were made in the software Mathematica. 4. Fusing and Type-II Bäcklund transformation In this section we shall consider the composition of two gauge-Bäcklund transformations leading to the Type-II Bäcklund transformation. Let us consider a situation in which we start with a Bäcklund relation transforming solution v1 into another solution v0. A second sub- sequent Bäcklund relation transforms v0 into v2. Such algebraic relation for the mKdV hierarchy is described by =K v v v K v v K v v, , , , , 4.1II 1 0 2 2 0 0 1( ) ( ) ( ) ( ) where K v v,i j( ) is given in (3.5) with b b= ij. It leads to ⎡ ⎣ ⎢⎢⎢ ⎤ ⎦ ⎥⎥⎥ b b b b = + + - + + b b l l f f f f f b b l - - - f- K v v v e e e e e e e , , 1 1 , 4.2 q e q II 1 0 2 4 2 01 02 1 2 01 02 4 10 02 0 1 2 0 1 2 10 02 ( ) ( ) ( ) ( ) where f f= -q 1 2 and s = - b b 2 4 10 02 . Inserting the following identity b b b b b b h+ + = + +f f f f- - -e e e e e e , 4.3q q 01 02 01 02 01 021 2 1 2( )( ) ( ) ( ) where h = b b b b +10 2 02 2 10 02 . Defining f b bL = - - + -f f s- -e eln ln0 02 01 4 1 2( ) , equation (4.2) becomes ⎡ ⎣ ⎢⎢ ⎤ ⎦ ⎥⎥ h = - + + - - s l ls s ls - -L - L- K v v v e e e e e , , 1 1 . 4.4 q e q q p q II 1 0 2 1 2 2 1 p 2 2 ( ) ( ) ( ) Equation (3.4) with K v v v, ,II 1 0 2( ) leads to the following Bäcklund equations s h s ¶ = - + + -L- - -Lq e e e e 1 2 2 , 4.5x p q q p( ) ( ) s ¶ L = -L- -e e e 1 2 . 4.6x p q q( ) ( ) Equations (4.5) and (4.6) coincide with the x-component of the Type-II Bäcklund transformation proposed for the sine-gordon model in [3]. Considering now the time component of the 2D gauge potential for =t t3 , (i.e., for the mKdV equation), J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 10 l l l l= + + + ¶ - - ¶ + + ¶ - a a a a- -A E E v h v v E v v E v v h 1 2 1 2 1 4 2 4.7 t x x x 2 2 2 2 3 3 ( ) ( ) ( ) ( ) we find from equation (3.2), s h s s s s s h ¶ = + + ¶ + ¶ + ¶ + ¶ - + - ¶ + ¶ + ¶ + ¶ - + ¶ + + L- - -L - - q e e e p q p q e e p q p q e p e e 16 2 8 4 2 8 16 4.8 t p q q x x x x q p x x x x q x q q 3 2 2 2 2 2 2 2 2 2 2 3 ( )[ ( ) ( ) ] [ ( ) ( ) ] ( ) ( ) together with s¶ L = + ¶ - + ¶L- - L+ -v v e v v e4 , 4.9t x q p x q p 1 2 1 2 2 23 ( ) ( ) ( ) which is compatible with equations of motion for the mKdV model. These Type-II Bäcklund equations (4.5)–(4.9) coincide with those derived in detail in [7] where  + -x x t x, and was extended to all positive higher graded equation within the mKdV hierarchy7. In the case of the KdV hierarchy b b=K J J J K J J K J J, , , , , , , 4.10typeII 1 0 2 2 0 02 0 1 01˜ ( ) ˜ ( ) ˜ ( ) ( ) where ⎡ ⎣ ⎢⎢⎢ ⎤ ⎦ ⎥⎥⎥ b z z z = - - + - + + bK J J Q Q Q , , 1 1 .j i ij ij ij ij 1 2 4 1 4 2 1 2 ij 2 ˜ ( ) Such transformation can be interpreted as an extended Bäcklund transformation dubbed Type-II Bäcklund transformation (see [3]). Explicitly we find directly from (4.10) ⎛ ⎝⎜ ⎞ ⎠⎟ z b b z z z z b b z z b z b z z b z z = - - + + + - W = = + - - + - + W = - + - W + -W + W - + - + - W + - + - + - - K J J J Q Q Q P K J J J Q K J J J Q Q Q P K J J J Q P Q P P Q Q P , , 1 1 2 2 8 2 , , 1 2 , , 1 1 2 2 8 2 , , 4 4 2 8 4 4 4 2 , 4.11 II 1 0 2 11 2 2 II 1 0 2 12 2 II 1 0 2 22 2 2 II 1 0 2 21 2 2 2 2 2 2 [ ˜ ( )] ( ) ( ) [ ˜ ( )] [ ˜ ( )] ( ) ( ) [ ˜ ( )] ( ) ( ) ( ) where = + = -Q Q Q w w10 02 1 2, = - + W = +P Q Q w w2 ,10 02 1 2 bW = =w , 40 b b01 2 02 2 and = -Q w wij i j. Acting with K J J J, ,typeII 1 0 2˜ ( ) in (3.12) we find the Bäcklund transformation for the KdV equation, i.e., 7 Observe that the Type-II Bäcklund transformation via gauge transformation was constructed in [8] where a solution presented there was chosen to reproduce the Bäcklund transformation proposed in [3]. Here we choose a gauge transformation solution of [8] that reproduces [9]. J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 11 b b ¶ = - + W ¶ W + = - - - W + W - + 4.12 Q PQ Q P P Q P 2 1 2 , 2 2 1 4 1 4 . x x 2 2 2 ( )( ) Similarly for the time component gauge potential (A.2) we find ¶ = ¶ ¶ + ¶ W¶ + ¶ W + W¶ - ¶ - ¶ ¶ W + = ¶ + ¶ + ¶ ¶ W + ¶ W - ¶ - ¶ + ¶ W + W¶ - W¶ W 4.13 Q P Q Q Q Q P Q Q P P P Q P P P Q Q P P 1 2 1 2 1 4 1 4 8 8 2 1 4 1 4 1 2 8 8 1 4 1 4 1 2 . t x x x x x x x x t x x x x x x x x x x 2 2 2 2 2 2 2 2 2 2 2 2 3 3 ( ) ( ) ( ) ( ) ( ) Equations (4.12) and (4.13) are compatible and lead to the equations of motion (2.16). Alternatively in terms of the mKdV Bäcklund transformation (3.13), equation (4.10) can be obtained by gauge-Miura transformation, i.e.,  f f f f= - -K J J J g v g K K g g v, , , , , ,TypeII 1 0 2 2 2 2 1 2 0 0 1 1 1 2 1 1 1˜ ( ) ( ) ( ( ) ( )) ( ) where we may introduce the identity element,   = - -g g v g v g, ,1 1 2 0 0 1 2 0 0 1( ) ( ) depending upon an arbitrary  -sign, say, 0. As argued when establishing (3.13), we are considering transitions with opposite  -signs such that   = - =1 0 and   = - = -0 2 . It therefore follows that     f f f f f f = = - - - - K J J J g v g K K g g v g v g K g g v , , , , , , , , , . 4.14 TypeII 1 0 2 2 2 1 2 0 0 1 1 1 2 1 1 2 2 1 II 2 1 1 1 2 1 1 ˜ ( ) ( ) [ ( ) ( )] ( ) ( ) ( ) ( ) ( ) The equation (4.14) yields                     sz h sz s z s z sz sz h sz sz h s z sz sz h s z s z sz h sz sz h sz s z s z sz sz h = + + + + - - - - - + - - - + + + = - - + + + = - - - - + - + + + + + + - - - + + + + + - - = - + + + + - - + - - + - + + + + L- - -L - -L L- - -L L- - - -L L- - - - L- - -L L- - -L - -L L- - 4.15 K J J J e e e e e e v e v e e e K J J J e e e e K J J J e e v v e v v e e e v v e e v v e e v v e e e v v e K J J J e e e e e e v e v e e e , , 1 1 4 1 1 2 1 2 1 1 4 , , , 1 1 4 , , , 1 1 4 2 2 1 4 1 , , , 1 1 4 1 1 2 1 2 1 1 4 . p q q p q q p p q q p p q q q q p p q q q q q q p q q p p q q p q q p p q q TypeII 1 0 2 11 2 2 2 2 2 1 2 1 TypeII 1 0 2 12 2 2 TypeII 1 0 2 21 2 1 2 1 2 1 2 2 2 1 2 2 2 2 1 2 2 1 2 TypeII 1 0 2 22 2 2 2 2 2 2 2 2 ( ) [ ˜ ( )] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ˜ ( )] ( ) ( ) ( ) [ ˜ ( )] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ˜ ( )] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 12 Comparing the matrix elements of (4.10) with (4.14) we find the following relations between the mKdV and KdV variables: • matrix element 11   z s h s - = + + + - -- L- - -LQ e e e e: 1 2 1 4 1 , 4.16p q q p1 ( ) ( ) ( ) ( )     z b b s s z sz sz h - + + + - W = - - - + - - - + + + - + - - -L L- - Q Q P e e v e v e e e : 2 8 2 1 2 1 2 1 1 4 4.17 q q p p q q 2 2 2 2 2 1 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) • matrix element 21:    z b s s sz h - + - W = - - - - + - + + + + - - - -L L- - Q P e e v v e v v e e e : 4 2 1 1 4 , 4.18 q q p p q q 1 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )    z b b s s s h s - + - W + -W + W - + = + + - - - + + + + + - - - + - - - L- - -L Q P Q P P Q v v e e v v e e v v e e e v v e : 4 4 2 8 4 4 2 2 1 4 1 4.19q q q q p q q p 2 2 2 2 1 2 2 1 2 2 1 2 1 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) • matrix element 22:     z b b s s s s h - - + - + W = - + - - + - + + + + - + - - -L L- - 4.20 Q Q P e e v e v e e e : 2 8 2 1 2 1 2 1 1 4 . q q p p q q 2 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The element 12 and the element 22 with z-1 gives us the same result of (4.16). Eliminating the mKdV variables p q, and L we recover the Type-II Bäcklund transformation for the KdV hierarchy (4.12) as shown in the appendix. 4.1. Examples and solutions • Vacuum—1-soliton—vacuum The first example is to consider vacuum to 1-soliton and back to vacuum again given by the following configuration, r = = W = - + + =w w k x t k w0, 4 1 , 2 , 0 4.21 N 1 0 2( ) ( ) with r = +x t e, N kx k t2 2 N N( ) . It is straightforward to check that equations (4.12) and (4.13) are satisfied for b =- 0 and b =+ k2 2. • 1-soliton—2 soliton—1-soliton Consider now a configuration of 1-soliton transforming into a 2-solitons solution and J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 13 back to 1-soliton. It is described by r r= - + + = = +w k x t k i e 4 1 , 2 , 1, 2 , 4.22i i i N i i k x k t2 2i i N N ( ) ( ) r r r r r r W = = - - + + - - + - - - w k k k k k k k k 2 1 1 . 4.230 1 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 ( )( )( ) ( )( ) ( ) ( ) Equations (4.12) and (4.13) are satisfied for b = -- k k2 2 1 2 and b = ++ k k1 2 2 2. • Vacuum—1-soliton—2-soliton Consider the solution of equation (4.12) and (4.13) r = = W = - + +w w k x t k0, 4 1 , 2 4.24 N 1 0 1 1 1( ) ( ) and r r r r r r = - - + + - - + - - - w k k k k k k k k 2 1 1 , 4.252 1 2 2 2 1 2 1 2 1 2 1 2 1 2 1 2 ( )( )( ) ( )( ) ( ) ( ) where r = +ei k x k t2 2i i N N equations (4.12) and (4.13) are satisfied for b = -- k k1 2 2 2 and b = ++ k k1 2 2 2. 5. Conclusions This paper follows the line of reasoning of [7] where we have constructed Bäcklund trans- formation for the entire mKdV hierarchy from an universal gauge transformation. Such Bäcklund-gauge transformation relates two different field configurations, preserves the zero curvature and henceforth its corresponding evolution equations. The main result of this paper is the extension of such construction to the KdV hierarchy by proposing a Miura-gauge transformation denoted by the product g g2 1 given in (2.9) and (2.11) mapping the mKdV into the KdV hierarchy (see (2.12)). A subtle point is that such Miura mapping allows a sign ambiguity such that each solution of the mKdV hierarchy defines two solutions for its KdV counterpart. The Bäcklund-gauge transformation for the KdV hierarchy is constructed by Miura-gauge transforming the Bäcklund transformation of the mKdV system as shown in (3.13). An interesting fact is that the Bäcklund transformation for the KdV hierarchy is solved by mixed Miura solutions generated by the mKdV Bäcklund solutions. A few simple explicit examples illustrate our conjecture. A more general evidence of the mixed Miura solutions is shown to agree with the Bäcklund transformation proposed in [12, 14] for the first two KdV flows. The composition law of two subsequent Bäcklund-gauge transformations leading to Type-II Bäcklund transformation (see (4.1)) introduced in [3] in the context of sine-Gordon and Tzitzeica models was extended to the KdV hierarchy. Within our construction, we have employed the direct fusion of two KdV Bäcklund-gauge transformations in (4.10) and alternatively, the Miura transformation of mKdV Type-II Bäcklund transformation as shown in (4.14). These two approaches generate relations between the mKdV and KdV variables which were shown in the appendix A.2, to be consistent. Finally we should mention that the idea of an universality of the Bäcklund-gauge trans- formation is most probably valid for other hierarchies such as the AKNS and higher rank Toda theories. It would be interesting to see how such examples can be worked out technically. J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 14 It should be interesting to develop the concept of integrable hierarchies for discrete cases and investigate whether the arguments involving Bäcklund-gauge transformation employed in this paper can be extended. The relation between the integrable discrete mKdV [15] and its Miura transformation to discrete KdV equations should be understood under the algebraic formalism. Acknowledgments AHZ and JFG were partially supported by CNPq and Fapesp. ALR was supported by Fapesp under Proc. No. 2015/00025-9. We would like to thank Prof Wen-li Yang for making a few pages of the book refered below available to us. Note added in proof. After this paper was finished we were informed about the existence of the book ‘Introduction to Soliton Theory’ by Dendyuan Chen, Science Press, Beijing, (in chinese) that may contain some overlaping results. Appendix A.1. Zero curvature for KdV hierarchy Here we write down the time component of the two-dimensional gauge potential generating the first three flows for the KdV hierarchy. Let     = ¢ + ¢ + + ¢ ¢ Î ¢- A , , A.1t K V N N j j , d 1 0 N ( )( ) ( ) ( ) ( ) where we find by solving the zero curvature representation (2.14) in the homogeneous gradation. For N = 3 we have ⎜ ⎟⎛ ⎝ ⎞ ⎠     z z z z z ¢ = ¢ =- + ¢ = ¶ + ¢ =- + ¶ + + ¶ a a a a a - - - h E J E J E J h JE J J E Jh , , 1 2 1 2 , 1 2 1 4 1 2 1 4 , A.2 x x x 3 3 2 2 2 1 0 2 2 ( ) ( ) ( ) ( ) ( ) and N = 5 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠       z z z z z z z z z z z = =- + = ¶ + =- + ¶ + + ¶ = ¶ + ¶ + ¶ + = - ¶ - + ¶ + ¶ + ¶ + ¶ + ¶ + ¢ ¢ ¢ ¢ ¢ ¢ a a a a a a a a - - - - - A.3 h E J E J E J h J E J J E J h J J J E J J h J J E J J J h J J J J J E , , 1 2 1 2 , 1 2 1 4 1 2 1 4 , 1 8 3 4 1 8 3 8 1 8 3 8 1 16 3 8 1 16 3 8 1 2 3 8 x x x x x x x x x x x x 5 5 4 4 4 3 3 3 2 2 2 2 2 2 2 1 3 2 2 0 2 2 3 4 2 2 3 ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 15 and for N = 7 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠ ⎛ ⎝ ⎞ ⎠         z z z z z z z z z z z z z z z ¢ = ¢ = - + ¢ = ¶ + ¢ = - + ¶ + + ¶ ¢ = ¶ + ¶ + ¶ + ¢ = - ¶ - + ¶ + ¶ + ¶ + ¶ + ¶ + ¢ = ¶ + ¶ ¶ + ¶ + ¶ + ¶ + ¶ + ¶ + + ¢ = - ¶ - ¶ - ¶ - + ¶ + ¶ + ¶ ¶ + ¶ + ¶ + ¶ + + ¶ + ¶ ¶ + ¶ + ¶ a a a a a a a a a a a a - - - - - - - - h E J E J E J h J E J J E J h J J J E J J h J J E J J J h J J J J J E J J J J J J J E J J J J J h J J J J J E J J J J E J J J J J J J E J J J J J J J h , , 1 2 1 2 , 1 2 1 4 1 2 1 4 , 1 8 3 4 1 8 3 8 1 8 3 8 1 16 3 8 1 16 3 8 1 2 3 8 1 32 5 8 5 16 15 16 1 32 5 32 5 16 5 16 1 32 5 32 5 16 5 16 1 64 5 16 15 32 3 16 35 32 25 32 5 16 1 64 5 16 5 32 15 32 . A.4 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 7 7 6 6 6 5 5 5 4 4 2 2 4 4 3 3 3 2 2 3 2 2 2 2 3 2 4 2 2 3 2 1 5 2 3 2 4 2 2 3 0 4 2 2 3 6 2 2 3 4 2 2 2 4 5 2 3 2 ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A.2. Equivalence between mKdV and KdV variables We now verify the equivalence between mKdV and KdV variables from (4.16) we find    s h s s ¶ =- + ¶ L - + + - + ¶ - + - ¶ - L L- - L- - -L Q p e e e qe e e p e 1 2 1 2 2 1 A.5 x x p q q x p q q x p ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) and using (4.18), (4.5) and(4.6) we obtain b¶ = - + W-Q QP Q2 2 . A.6x ( ) Consider now = + = ¶ +P J J w wx1 2 1 2( ). In terms of Miura transformation  ¶ = ¶ + - + = ¶ - ¶ - ¶P v v v v p p q 1 2 1 2 . A.7x x x x x1 2 1 2 2 2 2 2 2( ) ( ) ( ) ( ) ) ( ) J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 16 Acting with ¶x in (4.18) and using (A.7) and (4.16) we find   s s ¶ + ¶ W = ¶ - W + ¶ - ¶ - ¶ ¶ L + + ¶ ¶ - + ¶ +L- - - Q P Q P Q p q Q p p qe e e q e e 2 2 2 2 1 2 1 2 2 , A.8 x x x x x x x x x p q q x q q 2 2 2 ( ) ( ) [( ) ( ) ] ( ) ( ) ( ) ( ) where = + = -p v v q v v,1 2 1 2. Substituting the equation (4.16) in the equation (4.19) we find ⎛ ⎝⎜ ⎞ ⎠⎟  s s b b ¶ - ¶ = ¶ - - ¶ + - -W + W - + + - - W - - + - Q p q p e e q e e Q P P Q Q P 2 2 2 2 4 4 2 . A.9 x x x q q x q q2 2 2 2 2 2 2 [( ) ( ) ] ( ) ( ) ( ) ( ) Substituting this result in (A.8) and eliminating the mKdV variables using (4.5) and (4.6) we obtain b¶ + ¶ W = - + + W - W + +P P Q P2 1 4 2 . A.10x x 2 2 2( ) ( ) Equations (A.6) and (A.10) correspond precisely to the Type-II Bäcklund for the KdV hierarchy. A.3. Consistency with equations of motion In this appendix we verify that the compatibility of Bäcklund transformations lead us to the equation of motion. We start with the spatial part which is common to all N. For the KdV equation it is given by b ¶ = -P Q 2 1 2 . A.11x 2 2 ( ) In what follows it will be useful calculate its spatial derivatives: ¶ = - ¶P Q Q; A.12x x 2 ( ) ¶ = - ¶ - ¶P Q Q Q ; A.13x x x 3 2 2( ) ( ) ( ) ¶ = - ¶ ¶ - ¶P Q Q Q Q3 ; A.14x x x x 4 2 3( )( ) ( ) ( ) ¶ = - ¶ - ¶ ¶ - ¶P Q Q Q Q3 4 ; A.15x x x x x 5 2 2 3 4( ) ( )( ) ( ) ( ) ¶ = - ¶ ¶ - ¶ ¶ - ¶P Q Q Q Q Q Q10 5 ; A.16x x x x x x 6 2 3 4 5( )( ) ( )( ) ( ) ( ) ¶ = - ¶ - ¶ ¶ - ¶ ¶ - ¶P Q Q Q Q Q Q Q10 15 6 ; A.17x x x x x x x 7 3 2 2 4 5 6( ) ( )( ) ( )( ) ( ) ¶ = - ¶ ¶ - ¶ ¶ - ¶ ¶ - ¶P Q Q Q Q Q Q Q Q35 21 7 . A.18x x x x x x x x 8 3 4 2 5 6 7( )( ) ( )( ) ( )( ) ( ) ( ) J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 17 A.3.1. N = 3 (KdV). The temporal part of the KdV BT is given by ¶ = - ¶ + ¶ + ¶P Q Q Q P4 1 2 . A.19t x x x 2 2 2 3 ( ) [( ) ( ) ] ( ) In order to verify the consistency of this transformation we act with the spatial derivative to obtain ¶ ¶ = - ¶ + ¶ ¶P Q Q P P4 3 , A.20x t x x x 3 2 3 ( )( ) ( ) eliminating the term - ¶Q Qx 3 from equation (A.14) we find ¶ ¶ = ¶ + ¶ ¶ + ¶ ¶P P P P Q Q4 3 3 . A.21x t x x x x x 4 2 2 3 ( )( ) ( )( ) ( ) Substituting ¶ = + ¶ = -P J J Q J J, . A.22x x1 2 1 2 ( ) Equation (A.21) becomes precisely the sum of two KdV equations. A.3.2. N = 5. The temporal part of the BT for N = 5 equation is given by ¶ = - ¶ + ¶ ¶ + ¶ ¶ + ¶ - ¶ + ¶ ¶ + ¶ P Q Q Q Q P P P Q P P Q 16 5 5 2 1 2 5 2 3 . A.23 t x x x x x x x x x x 4 3 3 2 2 2 2 2 2 5 ( ) ( )( ) ( )( ) ( ) ( ) ( )[( ) ( ) ] ( ) Acting ¶x in the equation (A.23) we obtain ¶ ¶ =- ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ ¶ P Q Q P P P P P P P Q P Q Q 16 10 5 15 2 15 2 15 . A.24 x t x x x x x x x x x x x x 5 2 3 4 2 2 2 2 2 5 ( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( )( ) ( ) Then we isolate the term - ¶Q Qx 5( ) from equation (A.16) to find ¶ ¶ = ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ + ¶ ¶ ¶ A.25 P P Q Q Q Q P P P P P P P P Q Q 16 10 5 10 5 15 2 15 . x t x x x x x x x x x x x x x x x 6 2 3 4 2 3 4 2 2 2 2 5 ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )[( ) ( ) ] ( )( )( ) Substituting (A.22) we obtain the sum of two equations for N = 5, i.e., equation (2.17). A.3.3. N = 7. The temporal BT for the N = 7 equation is ¶ =- ¶ + ¶ ¶ + ¶ ¶ - ¶ ¶ + ¶ ¶ + ¶ + ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ ¶ + ¶ ¶ + ¶ + ¶ ¶ ¶ + ¶ + ¶ + ¶ ¶ P Q Q Q Q P P Q Q P P Q P P P P Q Q P Q P P Q P Q Q P Q Q P 64 7 14 1 2 21 2 35 2 35 2 35 35 2 35 35 8 105 4 . A.26 t x x x x x x x x x x x x x x x x x x x x x x x x x x x x 6 5 5 2 4 2 4 3 2 3 2 3 2 3 2 3 2 2 2 2 2 2 2 2 2 2 2 7 ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( ) ( )( ) ( )( ) ( )( )( ) ( )[( ) ( ) ] ( )( ) ( ) [( ) ( ) ] ( ) ( ) ( ) Likewise we did for other values of N, acting ¶x in the above equation. Then we isolate - ¶Q Qx 7( ) from equation (A.18) to find J. Phys. A: Math. Theor. 49 (2016) 504003 J F Gomes et al 18 ¶ ¶ = ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ + ¶ ¶ ¶ + ¶ ¶ ¶ + ¶ ¶ + ¶ ¶ ¶ + ¶ ¶ + ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ + ¶ ¶ ¶ + ¶ ¶ ¶ A.27 P P P P Q Q P P Q Q P P Q Q P P Q P Q Q P P P Q Q Q Q P P Q P P Q P P Q Q Q Q P P P Q 64 7 21 21 35 35 35 2 35 70 70 35 2 105 2 35 2 35 2 105 2 105 2 . x t x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 8 6 6 2 5 2 5 3 4 3 4 4 2 2 4 2 3 2 3 2 3 2 3 2 3 2 2 2 2 3 2 3 2 2 2 2 7 ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )[( ) ( ) ] ( )( )( ) ( )[( )( ) ( )( )] ( )[( )( ) ( )( )] ( ) ( )( ) ( )( ) ( )( ) ( )( )( ) ( )( )( ) Substituting (A.22) we obtain the equation of motion for N = 7 (2.18). References [1] Gomes J F, Starvaggi F G, de Melo G R and Zimerman A H 2009 J. Phys. A: Math. 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Introduction 2. The algebraic formalism for KdV and mKdV hierarchies 3. Bäcklund transformation 3.1. mKdV 3.2. KdV 3.3. Examples 4. Fusing and Type-II Bäcklund transformation 4.1. Examples and solutions 5. Conclusions Acknowledgments Appendix A.1. Zero curvature for KdV hierarchy A.2. Equivalence between mKdV and KdV variables A.3. Consistency with equations of motion A.3.1. N = 3 (KdV) A.3.2. N = 5 A.3.3. N = 7 References