21-dimensional new bi-hamiltonian integrable system Symmetries Noethers theorem and integrals of mot

Abstract

Keywords: Bi-Hamiltonian; Integrals of Motion; Lie Point Symmetries; Noether's Theorem; Symmetry Reduction

Introduction

Evolutionary Hirota type equations in (3 + 1)-dimensions have the form:

(1) where 𝑢 is an unknown that depends on the coordinates (𝑧1, 𝑧2, 𝑧3, 𝑡) and 𝑓, 𝑔 are smooth functions of 𝑢𝑖𝑗 (𝑖, 𝑗 = 1,2,3,𝑡). The subscripts 𝑖, 𝑗 of 𝑢 denote partial derivatives with respect to the designated variables, such as 𝑢𝑡2 = 𝜕2 𝑢 ⁄ 𝜕𝑡𝜕𝑧2, 𝑢𝑡2 = 𝜕2𝑢/𝜕𝑡𝜕𝑧2. In [1], these types of equations

equations have the Monge-Ampère form, where the only nonlinear terms consist of minors of the Hessian matrix of 𝑢. In this paper, it is sufficient for our purposes to restrict ourselves to a particular case of such an equation, namely:

*Corresponding author. *E-mail address: devrimyazici@gmail.com

This paper was recommended for publication in revised form by Editor-in-Chief Ahmet Selim Dalkilic Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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Here, 𝑎11, 𝑐4, 𝑐5, 𝑐8 and 𝑐10 are arbitrary constants. This equation, denoted as System-I , is expressed in the two component form:

(3) where 𝑎11 and the condition 𝑐10𝑐8 = 𝑐5𝑐9 is imposed. [1,2]. The operator: (4)

is introduced for brevity, where 𝐷𝑖 denotes the total derivative with respect to 𝑧𝑖. The explicit form of (3) is given as: (5) In [1, 2], the bi-Hamiltonian structure of (5) was discovered, demonstrating that this system is integrable in the sense of Magri [3,4]. In four dimensions, the evolutionary Hirota-type equations (1) exhibit the symplectic MongeAmpère property, as demonstrated in former studies [5,6]. These equations find applications in various fields, particularly in gravitational physics. For instance, they are relevant to Plebanski’s so called heavenly equations which simplify the complex Einstein field equations governing self-dual gravitational fields [7]. In this study, we find out if the two-component system (3) could be reduced into a (2 + 1)-dimensional bi-Hamiltonian system. We perform the reduction using the method previously applied in [8,9,10]. We choose a specific linear combination of symmetries that is critical to the success of the reduction. Upon obtaining the (2 + 1)-dimensional system in two-component form, we employ the method used before in [11-16] to construct the bi-Hamiltonian system. In order to obtain the first Hamiltonian structure, we use Dirac’s constraint analysis [17]. The skew-factorized form of the symmetry condition is reduced from the (3 + 1)-dimensional system [1] to obtain the recursion operator. The second Hamiltonian operator is obtained by applying the recursion operator to the first. Magri’s theorem [3,4] is then employed to determine whether the (2 + 1)-dimensional system forms a bi-Hamiltonian system, indicating its integrability. Completely integrable systems are intriguing because they present many symmetries and conserved densities in their solutions, although finding them is often challenging. We employ tools of Lie symmetry analysis to conduct symmetry reduction and discover first integrals. Recent papers such as [18-20] have used this powerful approach, where the authors have adopted power series expansion to find exact solutions of some nonlinear equations. In addition to well-known analytical methods like Darboux [21],

Bäcklund transformations [22] and the recently discovered Kudryashov method [23], as well as the generalized auxiliary equation technique [24]; numerical methods also play a crucial role in this research field. Historically, the wellknown KdV equation was initially solved through a numerical study [25]. Recently, new numerical approaches, such as the Fractional Iteration Algorithm [26] and Variational Iterational Algorithm [27] have been employed to obtain exact solutions for some nonlinear evolution equations. In this paper, we adopt Dirac constraint analysis which is very powerful in handling variational problems when the Lagrangian density is linear in velocity. However, in any other case, such as when the Lagrangian density is quadratic in velocity, this approach is not applicable. Magri made valuable contributions to the field of Hamiltonian systems by proving a theorem stating that evolutionary systems may have a multi-Hamiltonian structure. The Magri theorem, along with Dirac constraint analysis, has paved the way for discovering new integrable Hamiltonian systems, as evidenced in [28-33]. Besides the theoretical realm of science, Hamiltonian systems find utility in applied engineering problems as demonstrated in [34]. This paper is organized as follows: In section 2, we define the symmetries of the system (3) and conduct symmetry reduction to obtain the reduced system in two-component form. In section 3, we verify that the system is in Euler-Lagrange form and determine the degenerate Lagrangian density belonging to the system. Starting from the degenerate Lagrangian density, we construct the first Hamiltonian structure of the reduced system. In section 4, we obtain the recursion operator using the skew-factorized method for the symmetry condition. In section 5, we compose the recursion operator with the first Hamiltonian operator to get the second Hamiltonian operator. Then, we apply Magri’s Theorem to establish the second Hamiltonian structure of the reduced system. In section 6, we identify Lie point symmetries and obtain the Lie Algebra of the reduced system. We determine the symmetry characteristics and apply these results in Noether’s Theorem to identify new conserved densities of the system. Once we obtain the new conserved densities, we validate their legitimacy by casting them into total divergence form. SYMMETRY REDUCTION AND THE (2 + 1) -DIMENSIONAL SYSTEM In [2], the generators of point symmetries for (3) were identified as follows; (6)

where 𝑎, 𝑏, 𝑐 and 𝑒 are arbitrary smooth functions, and 𝜁 is defined as 𝜁 = 𝑐5𝑧1 − 𝑐8𝑧2. Given these symmetries, we choose the particular combination:

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(7) and get the symmetry: (8) Equation (8) leads to the characteristic equation:

Euler-Lagrange equation. Euler-Lagrange equations must satisfy the Helmholtz condition [35]. We verify that (15) possesses a Lagrangian density by checking the Helmholtz condition. Homotopy formula enables us to obtain the Lagrangian density. We present the result of our calculation after skipping the total derivative terms as follows:

(9) [35]. Integrating both sides of the first two equations given in (9) results in an invariant 𝑍1 as follows: (10)

Likewise, integrating both sides of the last two equations given in (9) leads to an invariant 𝑍2 as follows: (11)

Therefore, the invariants of 𝑋 determined by its characteristic equation (9) are: (12)

Consequently, the total derivatives undergo a transformation expressed as: (13) In equation (2), by replacing the derivatives with expressions from (13) and renaming variables:

(17) Euler-Lagrange equation using this result yields the reduced equation (15) which is in one component form. However, we want to obtain 𝐿𝑟 in two component form so that we can proceed with Dirac constraint analysis. The transformation 𝑢𝑡 = 𝑣 is applied to appropriate terms of (17) so that Euler-Lagrange equation with the new Lagrangian density results in the reduced system (16). Skipping total derivative terms, we present the new Lagrangian density as: (18) Subsequently, we obtain canonical momenta associated with the coordinates 𝑢 and 𝑣 as follows: (19)

(14) we obtain the new (2 + 1)-dimensional evolutionary equation: (15) where 𝑎 = 𝑐5 − 𝑐8, 𝑏 = 𝑐5 − 𝑐4, 𝑐 = 𝑐10 − 𝑐9 are arbitrary constants and ∆= 𝑢22 − 𝑢12. Equation (15) is represented in the two component form: (16) The superscript 𝑟 indicates that the relevant parameter is for the reduced (2 + 1)-dimensional system. Two equations presented in (16) compose the new (2 + 1)-dimensional system.

First Hamiltonian Structure Of The

(2 + 1)-DIMENSIONAL REDUCED SYSTEM Lagrangian density is the starting point for constructing the Hamiltonian structure of the new system. Thus, it is essential to verify that the reduced equation (15) is an

With the results obtained so far, the first Hamiltonian follows directly using the Legendre transformadensity tion, which in our case is expressed in the following way: (20) Substituting, (16),(18) and (19) into (20), we obtain: (21) Next, we aim to find the symplectic operator 𝐾𝑟. Lagrangian density (18) is degenerate because it is linear in velocity. Consequently, it is not possible to express velocities as a function of momenta and vice versa, as evident in (19). Dirac successfully developed a theory to analyze such cases [17]. Guided by his work, we define the second-class constraints in terms of canonical momenta (19) as: (22) and are set. The symplectic opeso that rator is defined in terms of these constraints as:

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(23) similarly as in [1, 11, 33, 36]. The Poisson Bracket of two constraints is denoted as , where the following relations hold:

is satisfied. Therefore, the 𝐾𝑟 matrix is a symplectic operator and its inverse which is given as: (33) is a Hamiltonian operator [37]. With the use of (30) and (33), we get:

(24) Here, is the discrete Dirac Delta function and 𝛿(𝑧 − 𝑧′) is the continuous Dirac Delta function. Moreover, we set 𝛱1 = 𝛱𝑢, 𝛱2 = 𝛱𝑣, 𝑢2 = 𝑣 and 𝑧 = (𝑧1, 𝑧2). Using (22) element of and (24), we can express, for instance, the the symplectic matrix as: (25) Making use of the Dirac Delta function properties, (25) results in:

Here, denotes the first Hamiltonin operator of the reduced system. The first Hamiltonian structure of the system is identified by the matrix equation: (36)

(26) Through similar but lengthy calculations, we obtain in the skew-symmetric form: (27) Since the Poisson Bracket operation is anti-symmetric, is easily found as: (28) and using the property given in (24), follows:

With these results, we obtain the symplectic matrix: (30) where is given in (27). The differential 2-form associated with 𝐾𝑟 is given in the form: (31)

Here, the summation is taken over the repeated subscripts, while Ʌ denotes the wedge product. Checking the closeness condition: (32)

of the differential 2-form (31) in a similar manner as done before in [1], reveals that the closeness condition (32)

where 𝛿𝑢 and 𝛿𝑣 are variational derivatives with respect to 𝑢 and 𝑣, respectively. By substituting equations (16), (21) and (34) into (36) and performing the calculations, we find out that equation (36) holds for the (2 + 1)−dimensional system. Therefore, the reduced system (16) exhibits a Hamiltonian structure just like the original system (3). 𝐿𝑟, 𝐻𝑟, 𝐾𝑟 and 𝐽𝑟 are obtained with identical results through direct reduction from the corresponding parameters 𝐿, 𝐻1, 𝐾, 𝐽0 given in [1] using the transformations (13).

Symmetry Condition In A Skew-Factorized FORM

We define two Lie equations: (37) where 𝜏 is the group parameter; 𝜑 and 𝜓 are symmetry characteristics. The symmetry condition of an equation is its differential compatibility with the Lie equations, and it is given as: (38) The symmetry condition of the reduced equation (15) is expressed in the following form: (39) where the operator defined in (4) is used for brevity. If the symmetry condition can be converted to the skew-factorized form: (40)

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while the commutator relations: (41) are satisfied, Lax pairs and the recursion operator can be obtained. The operators are obtained by reduction from equation (6.6) given in [1] as follows: (42) The commutator relations (41) are satisfied with these results. Lax pair is defined by:

The matrix element of the operator is obtained through the matrix multiplication (51), utilizing the properties: (52) of the operator 𝐿𝑖𝑗(𝑘) given in (4). This leads to the expression: (53)

(43) where λ is the spectral parameter. This pair yields the following results in our case with the use of (42):

We have checked that the commutator condition: (45) holds. Bringing the symmetry condition into the skew-factorized form (40) also enables us to write the recursion relations for symmetries as: (46) Using (42) in (46) and noting the relation: ,

resulting in: (56) utilizing the properties (52). For given by:

we transform the two equations in (46) into the matrix form: (48)

Direct reduction from R given in [1] results in the same 𝑅𝑟 (49).

Second Hamiltonian Structure Of The

is obtained by The second Hamiltonian operator applying the recursion operator to the first Hamilton operator as expressed by the equation: (51)

resulting in: (60) which is in skew-symmetric form. Equations (54), (56), (58) and (60) constitute the matrix representation of the second Hamiltonian operator obtained as: (61)

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where is given in (60). Similar to previous parameters, direct reduction from 𝐽1 given in [1] results in the same (61). The Hamiltonian operators and form a Hamiltonian pair if their linear combination is also a Hamiltonian operator. In this case, the linear combination is obliged to satisfy skew symmetry and the Jacobi Identity properties as it is stated by Definition 7.1 in Olver’s book [35]. It is easy to see that skew symmetry is satisfied since both and are obviously skew symmetric, i.e, 𝐽† = −𝐽 holds for both, where † denotes the adjoint operator. On the other hand, checking the Jacobi Identity condition is a complicated task. However, Theorem 7.8 suggested by Olver in his book simplifies this task. Therefore, we use Olver’s method in a similar fashion that is demonstrated in [14] and conclude that Jacobi Identity is satisfied. According to Magri’s theorem [3, 4], an evolutionary system is integrable if it satisfies the following equation:

Motion

Using the software package REDUCE 1, point symmetries of the new (2 + 1)-dimensional system (16) are identified as follows:

In the framework of Lie theory, point symmetries act as symmetry generators if they form a Lie algebra. We construct a table illustrating the Lie algebra structure of the point symmetries (64). The intersection of the 𝑖𝑡ℎ row and the 𝑗𝑡ℎ column in this table shows the result of the commutator operation [𝑋𝑖, 𝑋𝑗]. For convenience, the following notation is used in the table:

That is, the (2 + 1)-dimensional system forms a bi-Hamiltonian structure if a second Hamiltonian density satisfies (62). The second Hamiltonian density 𝐻0 of the (3 + 1)-dimensional system is given by (4.1.6) in [2]. Applying the transformations (13), we derive for the (2 + 1)-dimensional system as:

For each symmetry generator X, corresponding symmetry characteristics provide the independent variables that remain untransformed under the symmetry transformation. In [35], symmetry generators are defined in the following general form:

and the corresponding characteristics are defined in the form:

(61) and (63) into the matrix equation (62), we confirm that the equation holds. Hence, we have shown that the (2 + 1)-dimensional system admits a bi-Hamiltonian structure, analogous to the (3 + 1)-dimensional case.

(67) These equations are expressed using the Einstein summation convention. In the case of (2 + 1) −dimensional

Table 1. Commutators of point symmetry generators of reduced system X1

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system, indices 𝑖 take values: 𝑖 = 1,2,3. Additionally, we

namely 𝜑 and 𝜓, which are related to the transformations of 𝑢 and 𝑣 respectively. Using (68) in the equations (66), (67)

and replacing 𝑢𝑡 by 𝑣, 𝑣𝑡 by 𝑞 according to (16), we find the

characteristics pair (𝜑𝑖, 𝜓𝑖) of each generator 𝑋𝑖 (𝑖 = 1, 2,

We observe that the first integrals , , fail to exist. Therefore, the corresponding generators 𝑋2, 𝑋3, 𝑋5 do not count as variational symmetries. We check the time derivative of every density given in (73) along the flow (16) and obtain all the variational symmetries in total divergence form respectively as follows:

These symmetry characteristics provide a path to find new integrals of motion conserved by the flow of (16). By substituting the time variable “𝑡” with the group parameter “𝜏”, we can

employ the Lie equations provided in (37). Upon substituting

(70) This represents the Noether theorem in Hamiltonian form, providing the conserved density 𝐻𝑟 corresponding to

the given symmetry. Remarking that the first Hamiltonian operator (33) is the inverse of the symplectic operator, we arrange the matrix equation (70) into the inverse Noether theorem, taking the following form: (71) We write this matrix equation for each characteristics

Solving this equation, we determine the conserved densities, i.e., first integrals

symmetry generators 𝑋𝑖 with characteristics (𝜑𝑖, 𝜓𝑖) as follows:

We have successfully expressed the first integrals (73) in total divergence form (74). Thus, we can conclude that these integrals are indeed the constants of motion for the flow governed by the system (16). In essence, total divergences provide an independent check that the corresponding functionals 𝐻𝑟 are indeed integrals of motion subject to suitable boundary conditions.

Conclusion

We studied a symmetry reduction of the recently discovered (3 + 1)-dimensional equation of the MongeAmpere type. Our goal was to explore if it is possible to obtain a new (2 + 1)-dimensional bi-Hamiltonian system by applying symmetry reduction to a particular case of

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the (3 + 1)-dimensional equation. We used point symmetry generators of the system and proceeded by choosing a special combination of the symmetries. We determined the transformation of total derivatives under this particular symmetry, then performed the reduction accordingly. We of obtained all the parameters 𝐿𝑟, , 𝐾𝑟, , Rr, and the reduced (2 + 1)-dimensional system. Two component representation made it possible to obtain the Hamiltonian operator through Dirac constraint analysis. Being able to find the second Hamiltonian function , we state that the reduced system maintains the bi-Hamiltonian structure of the original system. We confirmed that all parameters and operators could also be obtained by direct reduction from the original system, e.g., 𝐿, 𝐻1, 𝐾, 𝐽0, 𝑅, 𝐽1, 𝐻0 with the same symmetry choice. We identified the symmetry generators of the reduced (2 + 1)-dimensional system, along with their corresponding characteristic pairs (𝜑, 𝜓). By the Noether theorem, we revealed seven new integrals of motion that define the conserved densities of the system. We also proved that the time derivatives of all variational symmetries are total divergences. Thus, we presented a new method for obtaining (2 + 1)-dimensional bi-Hamiltonian systems starting from (3 + 1)-dimensional bi-Hamiltonian systems. We have illustrated the involved procedure by an explicit example, producing a new bi-Hamiltonian system. We expect the suggested procedure to be a useful supplement to other techniques for generating (2 + 1)-dimensional bi-Hamiltonian systems.

Data Availability Statement

The authors confirm that the data that supports the findings of this study are available within the article. Raw data that support the finding of this study are available from the corresponding author, upon reasonable request.

Conflict Of Interest

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Ethics

There are no ethical issues with the publication of this manuscript.

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