Cent. Eur. J. Phys. • 12(2) • 2014 • 81-89 DOI: 10.2478/s11534-014-0430-6

VERS ITA

Central European Journal of Physics

Nonlinear self-adjointness and invariant solutions of a 2D Rossby wave equation

Review Article

Rodica Cimpoiasu, Radu Constantinescu*

University of Craiova, 13A.I.Cuza, 200585 Craiova, Romania

Received 3 September 2013; accepted 5 December 2013

Abstract: The paper investigates the nonlinear self-adjointness of the nonlinear inviscid barotropic nondivergent vor-

ticity equation in a beta-plane. It is a particular form of Rossby equation which does not possess variational structure and it is studied using a recently method developed by Ibragimov. The conservation laws associated with the infinite-dimensional symmetry Lie algebra models are constructed and analyzed. Based on this Lie algebra, some classes of similarity invariant solutions with nonconstant linear and nonlinear shears are obtained. It is also shown how one of the conservation laws generates a particular wave solution of this equation.

PACS C200SJ: 05.45.-a; 11.30.Na; 02.20.Sv; 47.10.ab

Keywords: nonlinear self-adjointness • Rossby waves • conservation laws • invariant solutions

© Versitasp. zo.o.

1. Introduction

The concepts of symmetry, Invariants and conservation laws are fundamental in the study of dynamical systems, providing a clear connection between the equations of motion and their solutions. There are many reasons for computing symmetries and conservation laws corresponding to systems described by differential equations. In recent years, a remarkable number of mathematical models occurring in various research domains have been studied from the point of view of symmetry group theory [1-3]. The Lie symmetry approach is now an established route for the reduction of differential equations. The method centers on the algebra of one parameter Lie group of transformations admitted by the PDEs. Once known, the reduction of the PDE is standard and may lead to exact (symmetry invari-

*E-mail: rconsta@yahoo.com (Corresponding author)

ant) solutions [4-8].

There are a number of reasons to find conserved densities of PDEs. Some conservation laws are physical (e.g., conservation of momentum, mass, energy, electric charge) and others facilitate analysis of the PDE and predict integra-bility. Conservation laws play an important role in the development of soliton theory, in the theory of non-classical transformations [9], [10] and in the theory of normal forms and asymptotic integrability [11]. The knowledge of conservation laws is also useful in the numerical integration of PDEs [12, 13], for example, to control numerical errors.

Although Noether's approach provides an elega nt algorithm for finding conservation laws, it possesses a strong limitation: it can only be applied to equations that have variational structure. Finding methods for constructing conservation laws for equations without variational structure has been subject of intense research. For example, in [14] a nice relationship is established between symmetries and conservation laws for self-adjoint differential

Springer

equations, an Identity which does not depend on the use of a Lagrangian. Another interesting result concerns a direct link between the components of a conserved vector for an arbitrary partial differential equation and the Lie-Backlund symmetry generator associated to the conserved vector's components [15].

Recently, [16] demonstrated a new algorithm for finding conserved vectors associated to any symmetry of nonlinear self-adjoint evolutionary equation. Extensive research has been carried out in order to find self-adjoint and quasi-self-adjoint classes of equations and their conservation laws. For example, the necessary and sufficient conditions for a general fourth-order evolution equation to be self-adjoint is determined in [17], the quasi-self-adjointness of a generalized Camassa-Holm equation was obtained in [18], a quasi self-adjointness classification of quasilinear dispersive equations was carried out in [19]. The purpose of this paper is to apply the recent Ibrag-imov's approach to the study of two-dimensional Rossby waves. The study of Rossby waves is one of the basic important problems in geophysical fluid dynamics such as atmospheric and oceanic circulation dynamics [20-22]. The Rossby waves studied in this paper are restricted on the following dimensionless inviscid barotropic nondivergent vorticity equation in a beta-plane [23]:

Aut + J(u, Au) + ßux = 0,

where u is the dimensionless stream function, A = d2Idx2 + d2ldy2 denotes the 2—dim Laplacian operator, J(u, Au) = uxAuy — UyAux is the Jacobian. We shall characterize the Earth's rotation effects through the quantity ft = W0 cos(^o)(uj), where R0 is the Earth's radius, w0 is the angular frequency of the Earth's rotation, (p0 is the latitude, L and U are the characteristic horizontal length and velocity scales.

Neglecting the effects of the Earth's rotation (ft = 0), the general fluid dynamics can be described by the Navier-Stokes equation. In [24] some exact solutions of the Navier-Stokes equation have been found from the symmetry group analysis. Because of the nonintegrability and of high nonlinearity of Eq. (1), one usually studies the Rossby waves numerically or approximately [25]. This paper is organized as follows. The essential points of Ibragimov's method will be summarized and the nonlinear self-adjointness of evolution equation (1), the most important point of applying Ibragimov's method, will be investigated in Section 2. The conservation laws provided by infinite-dimensional symmetry Lie algebra admitted by the 2D Rossby equation will be constructed in Section 3. The next section of the paper will illustrate the algorithm for constructing invariant solutions of our model with respect to some one-dimensional subalgebras of the whole

Lie algebra. New such invariant solutions and a periodic solution provided by the non-trivial conservation law will be point out, respectively. Some concluding remarks will end the paper.

2. Nonlinear self-adjointness

There are many interesting results concerning the correspondence between symmetries and conservation laws. Because a large number of differential equations without variational structure admits conservation laws, an intense research has been devoted to find methods for constructing conservation laws for equations without variational structure. In this section we shall present Ibragimov's method [16] which provides an elegant algorithm for finding conserved vectors which can be applied for any differential equation (or systems of equations).

2.1. Ibragimov's method

Let us consider a partial differential equation

F = F(x, u, u(i), ...u{n)) = 0, (2)

where F is a differential function, x = (x\...,x") are the independent variables, the dependent variable is u = u(x) and u(n) is the set of all partial derivatives of u, up to n-th order.

The formal Lagrangian is introduced by the relation:

L =vF.

It involves a new dependent variable v, the so-called nonlocal variable. It is a similar approach as the use of ghost type variables [27]. Then, the adjoint equation of (2) is defined by

F*(x, u, v,.., u(„), v(„)) = ^ =0,

5 _ d 5u du

is the Euler-Lagrange operator,

_ d d d d d

' 3x' 1 du ' dv 11 duj l> dvj

■ vijk

is the total derivative operator with respect to x', i, j, k = 1,..., n, and summation over repeated indices is assumed.

Definition 1.

The equation F = 0 is said to be nonlinearly self-adjoint if there exists a function

where W = n — Z'ui is the Lie characteristic. Since £ vanishes on the solutions of Eq. (2), the term ^¡£ may be omitted in the conserved vector.

such that

V = p(x, u)

F * \v=p(xru) = XF

for some undetermined coefficient X.

If v = p(u) in (7) and (8), Eq. (2) is called quasi-self-

adjoint. If v = u, the Eq. (2) is called strictly self-adjoint.

Supposing that Eq. (2) is nonlinearly self-adjoint, then applying Ibragimov's theorem to system (2), (4) with the formal Lagrangian (3), one obtains that any Lie point, contact, generalized or nonlocal symmetry

admitted by (2) determines a conservation law DC = 0 for (2) with the components of the conserved vector given by

2.2. Strictly-self-adjointness of 2D Rossby wave equation

We will show that the nonlinear equation (1) has the remarkable property of being strictly self-adjoint. We use this feature for constructing conservation laws associated with its symmetries.

Let us apply the previous method to Eq. (2). In this case, the formal Lagrangian is given by the expression:

£ = v [Aut + J(u, Au) + ßux],

where v is the new dependent variable.

The adjoint equation to Eq. (2) is obtained by taking the variational derivative of £, namely:

C =Z£+W + Dj (W )

d£-DJd£

dui ' \duj

d£ dun

DjDk (W )

5£ 5u

Taking into account the special form (11) of £, we have:

5£ Ju

= - Dx

d£ dux

d£ du,.

- (D2 + D2

d£ dAut

d£ dAu,,

d£ dAu

- Dx[v(Au, + ß)] + Dy[vAux] - (D2 + D,)[Dt(v) + Dy(vux) - Dx(vu,)].

After appropriate calculations, the adjoint equation is written as:

— vt(2x) — vt{2y) — ux [v(2x)y + v(3y)] + uy{v(2y)x + v(3x)]

- fivx +2Uxyv2x — 2vxyU2x = 0. (12)

It is easy to verify that this equation becomes the 2D Rossby wave equation (1) multiplied with constant coefficient X = —1, upon the substitution v = u.It means that the equation (1) is nonlinearly self-adjoint, specifically it is strictly self-adjoint.

3. Conservation laws provided by Lie point symmetries

Noether's theorem cannot be directly applied to obtain conservation laws on the basis of the equation's symmetries. This can be overcome by applying the general concept of nonlinear self-adjointness developed by Ibragimov which enables to establish the conservation laws for any differential equation.

3.1. Lie point symmetries of the 2D Rossby wave equation

The Lie algebra of the infinitesimal symmetries of the two-dimensional Rossby wave equation (1) has been obtained in [26]. It involves two arbitrary functions of t and contains the following basis of symmetry operators:

д д д д „ Л д

t-г--х---y-г--3u — , X'i = —, X3 = —

dt дх 3 ду du 2 dt 3 ду

„J I df (t) \ d v f(t)^7. - I у) X = 9t

дt' д

When the Lie algebra is computed, the following non-vanishing relations are obtained:

[X! , X2] = -X2, [X! , X3] = X3, [X! , Xf] = Xf + X

[X1 , Xg\ = X(tg+3g), [X2, Xf] = Xr [X2, Xg\ = Xg ,

[X3,Xf ] = X-). (14)

C1 = W

-Dx(W)Dx ( д^] - Dy(W)Dy! дС

t(2y) ,

+ D2x(W) I д^) + D2y(W)' дС

C1 =3 {W(U2x + и2у) - UxDx (W) - UuDu(W) + и [D2x(W) + D2y(W)]}.

3.2.1. Dilation group

Consider the generator of the dilation group from the basis of operators (13), namely:

Xi = 4 - хд - ид - 3ид. (18)

1 дt дх ду ди v '

3.2. Conservation laws associated with symmetries

We will apply formula (10) for constructing the conserved vector associated with the symmetries (13) admitted by the 2D Rossby wave equation. Since the maximum order of derivatives involved in formal Lagrangian (11) is equal to three, this formula becomes:

Ci = W

+Dj (W )

I - D,

дС ди и

дС ди, дС

■DjDk

Dj (W )Dk (W )

¡jk (15)

where the Lagrangian containing mixed derivatives should be written in the symmetric form

It is interesting to note that the symmetry operator leaves invariant the action attached to the formal La-grangian (3). This assertion can be easily checked using the Lie equations associated to X1. For this operator, the Lie characteristic has the form:

W = -3u - tut + хих + уиу. (19)

The substitution of (19) in (17) yields: 1

C1 =3 { tu[ut(2x) + Ut(2y)] + t[ихи,х + UyU,y]

+ и[хи(3х) + уи(3у) + хих(2у) + Уи(2х)у ]

- tUt[и2х + U2y] + уи2хиу + хи2уих

- (уих + хиу)иху - 4и(и2х + U2y) + 2(u2x + U2y)}.

С = - [3рих + !

3 i-h--х ■ -Jt(2х) + их(х + ихх( + ut(2y)

+ их [и(2х)у + ихух + иухх + 3и3у] -Uy [и(2у)х + иуху + ихуу + 3и3х ]}.

yty + и yyt (16)

We modify (20) by using the identities:

-tu,u(2x) = Dx [-tu,их] + tUxUtx,

-tUtU(2y) = Dy [-tUtUy] + tUyUty.

Invoking that the analyzed Eq. (2) is strictly self-adjoint with the substitution v = u, we will replace in C' the nonlocal variables v with u, thus arriving to local conserved vectors for Rossby wave equation.

Let us apply the procedure to C1. Consequently, the density of the conservation law is written in the following

tUxUtx = Dx [tuu,х ]-tUUt(2х), tUyUty = Dy[tUUty]-tUUt{2y).

ихи3х = Dx[ихи2х] - 2Dx(хи2х) - Dx(ихи) + 2и2х.

uyu3y = Dy[uyu2y] - 2 Dy(yul) - Dy(uyu) + 2 Uy.

uxux(2y) = Dx[uxu2y] - xuxu2y - u'2yu, uyu(2x)y = Dy[uyu2x] - yu,u2x - u2xu.

-yuxuxy = - 2 Dy(yu2) + 2 u2

-xuyuxy = - 2 Dx (xu y) + 2 uy.

UU2x = Dx(uux) - u2x, UU2y = Dy(uuy) - Uy. (21) Substituting these In (20), we arrive to the following density of the conserved law:

C1 = C + Dx (h2) + Dx (h3), (22)

where we used the notations:

C1 = -tu(Aut ) + 3(Vu)2,

2 1 2 1 1 2

h = — 3tu,ux + 3 tuutx — 2uux + 3xu(Au) — ^*(Vu) ,

1 2 1 1 h3 = — 3 tutuy + 3 tuuty — 2uuy + 3 yu(Au) — ^ y(Vu)2.

Based on the commutativity of the total differentiations, the conserved vector C = (C, C2, C3) can be reduced to the form:

C = ( C\C2,C3)

with the components

C\ C2 = C2 + Dt(h2), C:3 = C3 + Dt(h3). (24) Remark 2.

The two-dimensional vector (C2, C3) defines the flux of the conservation laws. In the following, we will ignore the tilde in the final expressions of quantities (24).

The conservation law DiC' = 0 is trivial if and only if its density C1 evaluated on the solutions of Eq. (1), i.e. the quantity Cl = C1 ^¡satisfies the variational derivative:

5CI hi

Using the previous statement, let us verify if the analyzed conservation law is a nontrivial one.

JC1 _ 5 Ju Ju

tuJ (u, Au)+ßtuux + 3(u2 + u2y)] = -6Au = 0.

Remark 3.

The flux of the conserved vector could be obtained either by means of relations (15) or by proving that the Dt(C1) evaluated on the solutions of Eq. (1) satisfied:

Dt(C1) \,i,= Dx(P2) + Dy(P3)

with certain functions P2, P3.

We choose to derive the flux (C2, C3) of conserved vector with known density (22). In fact the relation (27) could be really obtained using the master Eq. (1) and the identities:

Dt(C1) \,i)= {(-u - tut)(Aut) - tuDt(Aut)

+6[uxuxt + uyuyt]} \(1),

uuxDy(Au) = D,

-uuyDx (Au) = -D,

u2Dy(Au)

- -u2Dxy(Au),

u2Dx (Au)

+ - u2 Dxy(Au),

tutuxDy(Au) =Dx [tuutDy(Au)] - tuutxDy(Au) - tuutDxy(Au),

-tutuxDx(Au) = - Dy [tuutDx(Au)] + tuu^Dx(Au) + tuutDxy (Au),

Dx[tßuut] = tßuxut + tßuutx, ßuxu = Dx

Dt (Aut ) \(i)= - uxtDy(Au) - uxDty(Au) + utyDx (Au)

+ uyDtx (Au) - ßuxt,

tuuxDty(Au) = Dx

-tuuyDtx (Au) = Dy

2 tu2D,y(Au)

- - tu2Dxyt(Au),

- 2 tu2Dtx (Au)

+ 2 tu2DXyt (Au),

[uxuxt + uyuyt] \(1) = Dx

1 7 1 1

uuxt + 2u Dy(Au) + 2ßu2

uuyt - 2 u Dx (Au)

. (29)

Substituting the expressions (29) in the extended right-hand side of Eq. (28), we arrive at Eq. (27) with

2 и + tUUt

[Auy +p] + -tu (Auty)+6uUxt, (30)

Hence, the invariance of Eq. (1 ) under the time translation only provides a trivial conservation law. (H) Translation of y

For the operator X3 = j-, the Lie characteristic is

дy, W = -Uy.

P3 = -

(Аих) - 2tu (Аи1х) + 6uuyt. (31)

In conclusion, denoting C2 = -P2, C3 = —P3, the infinitesimal symmetry Xf = t^ — x^x — yd-y — 3udjU from the basis (13) admitted by Eq. (1), provides the conserved vector C = (C\ C2, C3) with the components:

C1 = -tu(Aut ) + 3(Vu)2, 1

C2 = -

Auy + в] - 2tu (Auty) - 6иих1, (Аих) + 2tu2(Autx) - 6uuyt. (32)

The quantities (32) do not have a direct physical significance, but they can generate interesting solitary wave solutions of (1), as it will be exemplified in subsection 4.2.

3.2.2. Translation group

The one-parameter group of translations in the variables t and x is generated by the operators X2, X3 from the basis (13). We analyzed the conservation laws generated by the invariance of Eq. (1) under this group of symmetries. (i) Time translation

For the operator X2 = d^, the Lie characteristic is:

W = -u,.

Substituting it in (17) and after some appropriate calculations, the final expression for density of the local conserved vector takes the form:

C1 = - u(Aut) + Dx ' 2

3 ии1х - 3ихи1

3uuty - 3UyUt

Replacing this expression in (17), one can rewrite the flux in the equivalent form:

C1 = Dy

1 I (Vu)2 3 2

- u(Au)

Definition 4.

The conservation law is said to be trivial if its density C1 evaluated on the solutions of Eq. (1) is the divergence:

C1 = Dx (h2) + Dy(h3).

Hence, the invariance of Eq. (1) under the translation of y only provides a trivial conservation law.

3.2.3. Infinite symmetry Lie group

Similar calculations show that the symmetry operators Xf and Xg from (13), which involves two arbitrary functions of time, also give trivial conservation laws, in according with the previous condition (39). More exactly, in these two cases, the densities of conservation laws admit, respectively the divergence expressions:

C1 =DX

1 (f (t)({^ - u(Au)] - f (t)yux

f(t)(2u - yuy)

C1 =DX

>9(1:)их

g(t)uy

4. Types of solutions

Let us present some symmetry reductions and associated invariant solutions for underlying Eq. (1).

Dropping the divergent type terms, it gives: C1 = —u(Aut ).

According to Lemma 1, we have to evaluate (35) on the solutions of Eq. (1). Then, we have

^^^ [uJ(u, Au)+ виих] = 0. (36)

4.1. Invariant solutions based on symmetry transformations

For a start, let us derive the invariant solution generated by the invariance of the analyzed equation to the dilation group. We will use the assertion that the function u = Y<1>(t,x, y) is a group invariant solution of (1) if:

X:[u - Ч>^(,,х,у)] |и=Ф(,) =0,

where the operator X1 is provided by (13)

This condition is equivalent to the partial differential

equation:

3^(1) + t Y<;> — x^X1) — yV{1) = 0, (42)

which admits the solution:

^(t,x,y) = H (Xt,yt). (43)

Introducing the invariant similarity variables w = xt and z = yt and substituting (43) in Eq. (1), one arrives at the following reduced equation for H(w,z):

wAHw + zAHz — AH + J (H, AH) + $HW = 0, (44)

with A = d2/dw2 + d2/dz2. It has the solution:

Bz3 c-3(z2 — w2) H (w,z) = + - C^z + C1 w + C2, (45)

where C', i = 1, 4 are arbitrary constants. Coming back to the original variables, the invariant solution of (1) is given, in this case, by the expression:

By3 C-i(y2 — x2) 1 , , C2

u(t,x, y) = — ^ + -1 + jA(C4y + C1x) + .

For a fixed moment of time t = 10 and for the choices of the other constants /? = 10~11, C| = 0.5, cj = 1, C3 = 2, c4 = 1, this solution has the form represented In Figure 1.

Figure 1. A plot of the Rossby wave described by (46) with ß =

10-11, c, = 0.5, c2 = 1,c3 = 2,c4 = 1 at time t = 10.

We will construct the invariant solution generated by the symmetry operator ( X3+ X, ) = dy + f (t) | - ( <f- y) £.

Following the same procedure, the invariance condition is equivalent with the following partial differential equation:

+ f(t)V(? + f (t)y = 0. (47)

The solution takes the form:

^-y^ —fx (y—W) ) +c h—m ).

Replacing the previous solution in Eq. (1), one obtains the reduced equation:

f 2f — 2ff 2 — ff (1+ f 2)zC(3z) + f 2(1+ f 2)Gv(2z)

2f f C(2z) — & 3(fz + Cz) = 0, (49)

where the invariants are denoted by w = t and z = y —

x f (t).

If we choose a concrete form for the arbitrary function, namely f(w) = —2c1 w + c2, c1,c2 = const., the Eq. (49) generates the solution:

C(w,z) = ci z2 + p(w ), (50)

with arbitrary function p(w).

Consequently, a new invariant solution of Eq. (1), written for constants c-\ = —1 /2, c2 = 0, takes the particular form:

u(t,x, y) = — x (y — x ) — 1 (y — x f+ P(t). (51)

If we take into consideration the operator X3 + Xg = dy + 9(t)du and applying a similar way, one arrives at the following solution:

u(t,y) = g(t)y + r(t), (52)

with g(t) and r(t) arbitrary functions.

As an example, for the choices g(t) = cos t, r(t) = sin t,

the solution has the form plotted in Figure 2.

4.2. Particular solution from conservation laws

We will construct particular solutions of Eq. (1) by adding to this equation the differential constraints:

C1 = C1(x, y), C2 = C2(t, y), C3 = C3(t, x), (53)

Figure 2. A plot of the periodic Rossby wave described by (52) with

g(t) = cos t and r(t) = sin t.

where the components of the conserved vector C', i = 1, 2, 3 are given by the expressions (32). Thereby, a particular solution of the analyzed model provided by the conserved vector (32) are described by the system:

were pointed out. More precisely, we considered the sub-algebras generated by X1, X3+ Xf, X3+ Xg and we found the associated solutions (46), (51), (52). These solutions were derived by solving the reduced PDEs (44), (49) written with respect to the appropriate invariant similarity variables. The solutions given show us that, in real atmospheric observations (in a background zonal basic wind), the stream function u may have not only linear shears, but also nonlinear shears.

Another result of this paper is represented by the proof of strictly self-adjointness of the analyzed model, a feature which is essential in applying Ibragimov's method. Further, the construction of conservation laws for all symmetry operators from the basis (13) were investigated. In that direction, the thorough calculations proved that only the dilation group admitted by (1) generates a non-trivial conservation law, described by the conserved vector (32). The translation group and the infinite symmetry transformations involving two arbitrary functions of time, provide trivial conservation laws. The solution (55) corresponding to concrete expressions of the non-trivial conserved vector mentioned above, was also obtained. As this solution possesses a stable localized structure, it is a Rossby solitary wave.

Aut + J(u, Au) + ßux DtC1 DXC 2 DC 3

This system is solved using the Maple program. The result is a periodic-type solution of the form:

Acknowledgements

The authors are grateful for the financial support offered by the Romanian Space Agency, in the frame of the Programme "STARS 2013", as well as by the Project FPF-2012-IRHES 316338.

u(t, x, y) =k4 cos

+ k5 sin

ßk2 (k2 + k2) ki

[kit + k2x + k3y)]

References

ßk2 (k2 + k22) ki

[k t + k2x + k3y)]

where ki, i = 1, 5 are arbitrary constants.

5. Concluding remarks

In this paper we used two important approaches for finding exact solutions of the 2D inviscid barotropic nondivergent vorticity equation (1), namely the Lie symmetry and Ibrag-imov's approaches.

Using the Lie symmetry algebra (13), three types of invariant similarity solutions generated by 1D subalgebras

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