Hamiltonian structures for the Ostrovsky–Vakhnenko equation

Commun Nonlinear Sci Numer Simulat 18 (2013) 56–62

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Hamiltonian structures for the Ostrovsky–Vakhnenko equation J.C. Brunelli a,⇑, S. Sakovich b a b

Departamento de Física, CFM, Universidade Federal de Santa Catarina, Campus Universitário, Trindade, C.P. 476, CEP 88040-900, Florianópolis, SC, Brazil Institute of Physics, National Academy of Sciences, 220072 Minsk, Belarus

a r t i c l e

i n f o

Article history: Received 16 April 2012 Accepted 27 June 2012 Available online 14 July 2012 Keywords: Integrable models Nonlinear evolution equations Miura-type transformations Bi-Hamiltonian systems

a b s t r a c t We obtain a bi-Hamiltonian formulation for the Ostrovsky–Vakhnenko (OV) equation using its higher order symmetry and a new transformation to the Caudrey–Dodd– Gibbon–Sawada–Kotera equation. Central to this derivation is the relation between Hamiltonian structures when dependent and independent variables are transformed. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction The Whitham equation

ut þ uux þ

Z

þ1

Kðx  yÞuy dy ¼ 0

ð1Þ

1

was introduced in [1] to model a wave equation containing both breaking and peaking. A good discussion of Eq. (1) can be found in [2]. Here we are interested in Eq. (1) with the kernel

Kðx  yÞ ¼

1 jx  yj; 2

which yields

ut þ uux þ @ 1 u ¼ 0;

ð2Þ

or in a local form

ðut þ uux Þx þ u ¼ 0:

ð3Þ

Eq. (3) also follows as a particular limit of the following generalized Korteweg-de Vries (KdV) equation

ðut þ uux  buxxx Þx ¼ cu;

ð4Þ

derived by Ostrovsky [3] to model small-amplitude long waves in a rotating fluid (cu is induced by the Coriolis force) of finite depth. For b ¼ 0 (no high-frequency dispersion) the Eq. (4) is known under different names in the literature, such as the reduced Ostrovsky equation, the Ostrovsky–Hunter equation, the short-wave equation and the Vakhnenko equation. From ⇑ Corresponding author. E-mail addresses: [email protected] (J.C. Brunelli), [email protected] (S. Sakovich). 1007-5704/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cnsns.2012.06.018

J.C. Brunelli, S. Sakovich / Commun Nonlinear Sci Numer Simulat 18 (2013) 56–62

57

now on we will call (3) the Ostrovsky–Vakhnenko (OV) equation. This OV equation describes the short-wave perturbation in a relaxing medium [4]. This equation has a purely dispersive term and although it has the same nonlinearity of the KdV equation the dispersive terms are different. In [4] and in a series of papers [5–7] it was established its integrability by deriving explicit solutions. Also, in [8] the integrability via inverse scattering method was derived via a third-order eigenvalue problem obtained after a Bäcklund transformation. However in [9] this Lax pair was written in its original variables as a zero curvature condition

A1;t  A0;x  ½A0 ; A1  ¼ 0; with

0 0 1B A1 ¼ @ 1 3 ux

ux =k 1=k 0

0

1

0

1

0

1B C A A0 ¼ @ 3

0

3  uux =k u=k

u

0

3k þ uux

2u

1

C 3 A: 0

Also, they obtained the following third order Lax pair for the OV equation

wxxx þ

  1 1 W ¼ 0; uxx þ 9k 3

wt  9kwxx þ uwx  ux w ¼ 0;

ð5Þ

whose compatibility condition yields the x-derivative of the OV Eq. (3). It turns out that the OV equation can be obtained as a short-wave limit of another integrable equation, the DegasperisProcesi (DP) equation [10–12]

ut  uxxt þ 4uux ¼ 3ux uxx þ uuxxx :

ð6Þ

The authors of [11,12] found a Lax pair, derived two infinite sequences of conserved charges for (6) and proposed a bi-Hamiltonian formulation. Hone and Wang [9] have shown and explored the fact that the OV equation can be obtained as a limit of the DP equation through the transformation

8 t > < x ! x  3 ; T : t ! t; > : u ! u  13 2 ; in the short wave limit

 ! 0,

T

DP ! OV  2 ðut þ 4uux Þ: Using this short wave limit Hone and Wang obtained the scalar linear problem (5) from the corresponding Lax pair of the DP equation. Also, from the bi-Hamiltonian study of the DP equation performed in [11] they proposed two Hamiltonian operators for the OV equation but not in the original variable u. In this paper our main interest is to investigate the Hamiltonian integrability for the OV equation directly in its original variable and evolutionary nonlocal form (2). As far as we known this is the simplest equation involving nonlinearity an nonlocality and will provide a good ‘‘laboratory’’ for study of nonlocal equations. Results on the OV equation frequently rely on the well known reciprocal transformation to the special case of Ito’s equation which can be written in Hirota’s bilinear form [6]. We point out that a transformation between the OV equation and the Bullough–Dodd–Tzitzeica equation is also known [13] but not explored in the literature. In this paper we will introduce and explore a new third transformation. In Section 2 we find a fifth order symmetry to the OV equation and we show in Section 3 that the evolution equation associated to this symmetry can be transformed to the Caudrey–Dodd–Gibbon-Sawada–Kotera equation (CDGSK) [14–16] via a chain of Miura-type transformations. In Section 2 we also find the first Hamiltonian structure for the OV hierarchy. The transformations introduced in Section 3 transform dependent as well as independent variables and we give in Section 4 a formula relating the Hamiltonian structures in this situation. This formula is used in Section 5 to obtain the second Hamiltonian structure of the OV equation via the well known CDGSK’s Hamiltonian structure. Our results are purely algebraic and no analytical justifications are provided. Issues about wave breaking, blow-up rates, boundary conditions, well-posedness and so on can be found, for instance, in [17] and references within. 2. Symmetries and a first Hamiltonian structure First let us look for some symmetries of the OV Eq. (2). It is well known [18,19] that a symmetry r of the evolution equation ut ¼ KðuÞ should satisfy

@r þ ½r; K ¼ 0; @t where

ð7Þ

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½r; KðuÞ  r0 ðuÞ½KðuÞ  K 0 ðuÞ½rðuÞ and

F 0 ðuÞ½X ¼

d Fðu þ XÞj¼0 d

ð8Þ

is the Fréchet or directional derivative of FðuÞ in the direction of the vector field X. In the case that Fðu; v Þ depends on two variables we write F u and F v for the Fréchet derivative with respect to u and v, respectively. If H is a functional then

H0 ðuÞ½X ¼ hgrad H; Xi ¼

  dH ;X ; du

ð9Þ

where h; i denotes the usual scalar product between the dual spaces. In what follows, we indicate by ‘‘’’ the formal adjoint of an operator with respect to the pairing (9). From (7) it is easy to check that

r1 ¼ KðuÞ; r2 ¼ ux ; r3 ¼ 

i 1h ð1 þ 3uxx Þ2=3 xxx 2

ð10Þ

are symmetries for the autonomous OV Eq. (2). This means that the OV equation and the local nonlinear equation

 2=3 3 1 ut ¼  D3x uxx þ ; 2 3

ð11Þ

where the right-hand side is the generator r3 of the higher symmetries and where the coefficient 3=2 is taken for simplicity, are members of the same hierarchy of equations and share many properties such as integrability. As far as we know this Ostrovsky–Vakhnenko fifth-order equation (OVF) (11) is a new integrable equation but we will show that it can be transformed to the CDGSK equation in the next Section. By construction the OV Eq. (2) and the OVF Eq. (11) share the same conserved charges as well Hamiltonian structures. Let us find the first Hamiltonian structure for these equations [20]. Introducing the Clebsch potential u ¼ /x the Eq. (2) can be written as

/xt þ /x /xx þ / ¼ 0: R This equation can be obtained from a variational principle, d dtdx L, with the Lagrangian density

L¼

1 1 1 / / þ /3  /2 : 2 t x 6 x 2

ð12Þ

This is a first order Lagrangian density and we can use, for example, Dirac’s theory of constraints to obtain the Hamiltonian and the Hamiltonian operator associated with (12). The Lagrangian is degenerate and the primary constraint is obtained to be

1 2

U ¼ p  /x ; where

ð13Þ

p ¼ @L=@/t is the canonical momentum. The total Hamiltonian can be written as HT ¼

Z

dxðp/t  L þ kUÞ ¼

Z

   1 1 1 dx  /3x þ /2 þ k p  /x ; 6 2 2

ð14Þ

where k is a Lagrange multiplier field. Using the canonical Poisson bracket relation

f/ðxÞ; pðyÞg ¼ dðx  yÞ;

ð15Þ

with all others vanishing, it follows that the requirement of the primary constraint to be stationary under time evolution, fUðxÞ; HT g ¼ 0, determines the Lagrange multiplier field k in (14) and the system has no further constraints. Using the canonical Poisson bracket relations (15), we can now calculate

Kðx; yÞ  fUðxÞ; UðyÞg ¼

1 1 Dy dðy  xÞ  Dx dðx  yÞ: 2 2

ð16Þ

This shows that the constraint (13) is second class and that the Dirac bracket between the basic variables has the form

f/ðxÞ; /ðyÞgD ¼ f/ðxÞ; /ðyÞg 

Z

0

dz dz f/ðxÞ; UðzÞgJðz; z0 ÞfUðz0 Þ; /ðyÞg ¼ Jðx; yÞ;

where J is the inverse of the Poisson bracket of the constraint (16),

Z

dz Kðx; zÞJðz; yÞ ¼ dðx  yÞ:

This last relation determines Dx Jðx; yÞ ¼ dðx  yÞ or Jðx; yÞ ¼ Ddðx  yÞ where

J.C. Brunelli, S. Sakovich / Commun Nonlinear Sci Numer Simulat 18 (2013) 56–62

D ¼ D1 x :

59

ð17Þ

We can now set the constraint (13) strongly to zero in (14) to obtain

HT ¼

  1 1 dx  /3x þ /2 : 6 2

Z

Using (17) and the transformation properties of Hamiltonian operators (see (37)), we get

D ¼ Dx ðDÞðDx Þ ¼ Dx and the OV Eq. (2) can be written in the Hamiltonian form as

dH1 ; du

ut ¼ D1

D1 ¼ Dx ;

H1 ¼

Z

  1 1 dx  u3 þ ð@ 1 uÞ2 : 6 2

ð18Þ

It can be easily checked that H1 is a conserved quantity for both the OV and OVF equations for rapidly decreasing or periodic boundary conditions. Taking the second x derivative of the OV equation we obtain the trivial conserved charge

Z

dx uxx

ð19Þ

and from this equation times ðuxx þ 1=3Þ2=3 we also get

H2 ¼ 

9 2

Z

 1=3 1 dx uxx þ ; 3

as conserved charge for both the OV and OVF equations. In this way the OVF equation has the following Hamiltonian representation

ut ¼ D1

dH2 ; du

D1 ¼ Dx ;

H2 ¼ 

9 2

Z

 1=3 1 dx uxx þ : 3

Multiplying the OV Eq. (2) by u and integrating we also get the following conserved charge

Z

dx u2 :

ð20Þ

3. Transformation to the CDGSK equation The link between the KdV and modified KdV equations through a Miura transformation is not an isolated result in the theory of integrable models. Very often a nonlinear equation is equivalent to a known and well studied equation via a chain of Miura-type transformations and now we will show that we can relate the OVF equation with the CDGSK equation using this procedure. We follow the methods used in [21–24]. The transformation

ðx; t; uðx; tÞÞ # ðx; t; v ðx; tÞÞ : v ¼ ðuxx þ 1=3Þ1=3

ð21Þ

relates the OVF equation with

1 2

v t ¼ v 4 D5x v 2 :

ð22Þ

The right-hand side of the transformation (21) is obtained from the separant of the OV equation. (The separant of an evolution equation ut ¼ f ðu; ux ; . . . ; uðnÞ Þ is @f =@uðnÞ , where uðnÞ is the highest derivative of u with respect to x.) Let us note that (22) is one of the non-constant separant evolution Fujimito-Watanabe equations (see [21] and references therein). The Eq. (22) has a separant v 5 and by the Ibragimov substitution

ðy; t; wðy; tÞÞ # ðx; t; v ðx; tÞÞ :



v ðx; tÞ ¼ wy ðy; tÞ; x ¼ wðy; tÞ

ð23Þ

is related with the constant separant equation 2 2 wt ¼ w5y  5w1 y w2y w4y þ 5wy w2y w3y ;

where wky ¼

@ ky w;

ð24Þ

k ¼ 2; 3; 4; 5 (and similar notations for derivatives are used in what follows). Finally, the transformation

ðy; t; wðy; tÞÞ # ðy; t; zðx; tÞÞ : z ¼ w1 y w3y ; relates (24) with the CDGSK equation

ð25Þ

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J.C. Brunelli, S. Sakovich / Commun Nonlinear Sci Numer Simulat 18 (2013) 56–62

zt ¼ z5y þ 5zz3y þ 5zy z2y þ 5z2 zy :

ð26Þ

In summary, we have schematically

OVF

Bðu;v Þ ¼ 0

!

ð22Þ

Bðw;v Þ ¼ 0

ð24Þ

Bðw;zÞ ¼ 0

!

CDGSK;

where the implicit transformations are



Bðw; zÞðyÞ ¼ z þ w1 y w3y ðyÞ;  Bðw; v ÞðyÞ ¼ v ðwÞ  wy ðyÞ;

Bðu; v ÞðxÞ ¼ v  ðuxx þ 1=3Þ1=3 ðxÞ:

ð27Þ

4. Hamiltonian structure behavior under change of variables Transformations between two evolution equations generate transformations between the corresponding structures such as recursion operators, Hamiltonian structures, conserved charges and so on [25,26]. Let be the change of variables among dependent and independent variables

(

y ¼ Pðx; uðnÞ Þ;

ð28Þ

v ¼ Q ðx; uðnÞ Þ;

where uðnÞ represents all derivatives of u with respect to x of order at most n. We want to relate the Hamiltonian representations

dH ¼ DðuÞ Eu ðhÞ; du e ~ e ðv Þ d H ¼ D e ðv Þ Ev ðhÞ; ¼D dv

ut ¼ DðuÞ

ð29Þ

vt

ð30Þ

where

H½u ¼ e v ¼ H½

Z Z

dx hðx; uðnÞ Þ;

ð31Þ

~ v ðnÞ Þ dy hðy;

ð32Þ

and Eu ðhÞ is the Euler operator acting on the Hamiltonian density h. The transformation (28) defines an implicit function Bðu; v Þ ¼ 0, and we have

Bu ut þ Bv v t ¼ 0 or

v t ¼ Tut ;

ð33Þ

T ¼ B1 v Bu

ð34Þ

where

and Bu and Bv are the Fréchet derivatives defined in (8). Now we use the following result (see [18, Exercise 5.49, p. 386]) for the relation between the action of the Euler operator under a change of variables

~ Eu ðhÞ ¼ O Ev ðhÞ;

ð35Þ

OðRÞ ¼ Q u ðDx P  RÞ  Pu ðDx Q  RÞ:

ð36Þ

where

From (33), (29) and (35)

~ v t ¼ TDðuÞ O Ev ðhÞ and comparing with (30) we finally get

e ðv Þ ¼ TDðuÞ O: D

ð37Þ Q u .

When the independent variable is not transformed Bu ¼ Q u ; Bv ¼ 1; T ¼ Q u and O ¼ Also, when we calculate how a recursion operator R ¼ D2 D1 1 transforms under (28) the operator O drops out. These are the usual results commonly found

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in the literature [25,26]. The generalization of (37) for a number of dependent and independent variables greater than one is straightforward. 5. Second Hamiltonian structure We use the results of the last Section to obtain the second Hamiltonian structure of the OV Eq. (2) from the known Hamiltonian structure of the CDGSK Eq. (26) given by [16]

DðzÞ ¼ D3y þ 2ðzDy þ Dy zÞ;

HðzÞ ¼

Z

dy

  1 3 1 2 z  zy : 6 2

ð38Þ

From (27) and (33) we have

zt ¼ Awt ;

3 2 1 A ¼ w1 Dx v 3 D2x ; y Dy þ wy w3y Dy ¼ v

ð39Þ

v t ¼ Bwt ;

B ¼ Dy  ¼ v Dx v ;  4=3 1 1 1 uxx þ C¼ D2x ¼  v 4 D2x : 3 3 3

ð40Þ

w1 y w2y

v t ¼ Cut ;

2

1

ð41Þ

From (39)–(41) we already see that is very convenient to use the variable v ðx; tÞ because this factorizes all the operators involved, via the relations Dy ¼ v Dx ; wy ¼ v ; w2y ¼ vv x ; w3y ¼ v ðvv x Þx ; z ¼ ðvv x Þx and uxx ¼ v 3  1=3; note also that

D2y þ z ¼ Dx v 3 Dx v 1 ; D3y þ 4zDy þ 2zy ¼ v 1 Dx v 3 Dx v 3 Dx v 2 :

ð42Þ

Now using (35), (36) we have

dHðwÞ dHðzÞ ¼A ; dw dz dHðwÞ dHðv Þ ; ¼B dw dv ðuÞ ðv Þ dH dH ; ¼C du dv

2 3 2 2 A ¼ A ¼ D3y w1 ; y  Dy wy w3y ¼ v Dx v Dx v

ð43Þ

B ¼ Dy wy  w2y ¼ Dx v 2 ;  4=3 1 1 1 C ¼ C  ¼  D2x uxx þ ¼  D2x v 4 : 3 3 3

ð44Þ ð45Þ

Therefore, the Hamiltonian operators transform as

Dðv Þ ¼ BDðwÞ B;

DðwÞ ¼ A1 DðzÞ A1 ;

DðuÞ ¼ C 1 Dðv Þ C 1

ð46Þ

and as result of (42), (39)–(41) and (43)–(45) we obtain the operator of order minus five 2 1 DðuÞ ¼ 9D2 Dx v 1 D3 Dx v 2 D2 x v x v x ;;

ð47Þ

where now v is not a dependent variable but just a placeholder for the expression in (21), i.e., v  ðuxx þ 1=3Þ Now, let us find the Hamiltonian HðuÞ corresponding to HðzÞ in (38). Its variational derivative is

1=3

.

dHðzÞ 1 ¼ z2y þ z2 2 dz and from (43) and (44) we can transform it into

dHðv Þ dHðzÞ ¼ B1 A ¼ dv dz ¼ v 3 v 6x þ 9v 2 v x v 5x þ 17v 2 v 2x v 4x þ 14vv 2x v 4x þ þ 5v 1 v 4x v 2x 

19 2 2 20 v v 3x þ þ34vv x v 2x v 3x  2v 3x v 3x þ vv 32x  10v 2x v 22x 2 3

5 2 6 v vx 6

and we can reconstruct the corresponding conserved charge using the homotopy formula

H ðv Þ ¼

Z

dx

Z 0

1

dkv

dHðv Þ ðkv Þ ; dv

to finally obtain

H ðv Þ ¼

Z

  1 4 1 dx  v 3 v 23x þ v 2 v 32x  2vv 2x v 22x  v 1 v 6x ; 2 3 6

ð48Þ

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J.C. Brunelli, S. Sakovich / Commun Nonlinear Sci Numer Simulat 18 (2013) 56–62

which is also HðuÞ if we make the substitution (21) to the right hand side of (48). Computationally, this is easier than to directly perform the calculations in the u variable. From (47) and (48) we obtain the following second Hamiltonian structure for the OVF Eq. (11)

dH3 ; du 2 2 1 D2 ¼ 9Dx v Dx v 1 D3 Dx v 2 D2 x v x ;   Z 1 8 1 H3 ¼  dx v 3 v 23x  v 2 v 32x þ 4vv 2x v 22x þ v 1 v 6x ; 2 3 3

ut ¼ D2

where

v is given by (21). However, the second Hamiltonian formulation for the OV Eq. (2) can be written formally as ut ¼ D2

dH4 ; du

2 1 Dx v 1 D3 Dx v 2 D2 D2 ¼ 9D2 x v x v x ;

H4 ¼ 

1 2

Z

dx uxx ;

ð49Þ

2 where we have used the trivial conserved charge (19) with 2dH4 =du ¼ D2x  1 and D2 x Dx  1 ¼ 1. This same charge and behavior appears in the pull-back of the Harry Dym equation to the Hunter-Saxton equation [27] and is due to the nonlocality of the Hamiltonian operator D2 . R From the CDGSK conserved charge HðzÞ ¼ dy z we get

Z

dx v 1 v 2x ;

for the OV equation. In fact, from (18) and (49) we have the recursion operator 2 2 1 R ¼ D2 D1 Dx v 1 D3 Dx v 2 D3 x v x 1 ¼ 9Dx v

and this could be the starting point to generate a hierarchy of equations and charges for the OV system of equations which we will explore in a future publication. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

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