The Lie symmetries, conservation laws, and exact solutions of a generalized nonlinear system and a (2+1)-dimensional generalized Nizhink-Novikov-Veselov (NNV) equation, arising in the study of hydrodynamics, are investigated. The multiplier approach is employed to compute the conservation laws for systems under consideration. The Lie point symmetries are derived and the association between symmetries and conserved vectors are established using symmetries conservation laws relationship. The double reduction theory is utilized which results in the reduction and exact solutions of models under investigation. All cases are discussed in detail and new solutions are determined.

1. Introduction

Noether [1] formulated a general approach for constructing conservation laws for Euler-Lagrange equations when Lie-Bäcklund symmetries came to the fore. Noether’s approach not only established a connection between conservation laws and symmetry properties of a physical system, but also provided a basis for the application of double reduction method. However, its applicability remained limited to those physical systems which have a Lagrangian formulation. Kara and Mahomed [2] relaxed this condition by introducing the concept of partial Lagrangian and associated Noether type symmetries with conservation laws. However, a connection between other types of symmetries and conservation laws remained elusive. Sjöberg [3, 4] developed the theory of double reduction for two independent variables. Bokhari et al. [5] then generalized this idea of reductions using conservation laws for independent variables.

Symmetries play an important role in many areas of physical sciences. In particular, hydrodynamics has remained an area of great interest owing to its applications in physics. Most of the studies have focused on providing an underlying mathematical theory by establishing empirical and semiempirical laws derived from the flow measurement of fluids and using them to solve real world problems. Mathematical models thus obtained provide vital information about various hydrological processes including the designing of dams, harbours, and bridges. These models usually entail systems of nonlinear partial differential equations (PDEs). Consequently, developing techniques for exact solutions is crucial to the analysis of such systems.

A system involving two nonlinear PDEs,where , , and are constants, arises in various physical phenomena in physics and engineering. As noted by Congy et al. [6], for , , and , the corresponding Kaup-Boussinesq (KB) system models the physics of shallow water waves. In this context, denotes the local height of the water layer and denotes the local mean flow velocity. The corresponding KB system is in fact a “positive dispersion” system, which also models the propagation of capillary waves on top of a thin fluid layer [7]. System (1) also serves as an approximation to various equations arising in physics and engineering. For instance, (1) provides an approximation to the Landau-Lifshitz equation, which models the propagation of magnetization waves in easy plane magnets. Similarly, the choice of coefficients , , and models a (1+1)-dimensional dispersive long wave equation [8], and the values , , yield the Kaup-Boussinesq system [9]. System (1) also models the long wave phenomena in shallow water, where the depth of water is considered to be uniform and and denote the elevation of water waves and the surface velocity of water along -direction, respectively. Further applications of (1) can be found in [1013] and some more recent works are given in [14] where soliton solutions are determined using Hirota’s direct method and in [15] where symmetric analysis is used.

Another PDE that arises in the study of fluid dynamics iswhere denotes the integral with respect to the subscripts, represents a physical field, and , and are specified constants. The integro-partial differential equation (2) is the well-known generalized Nizhnik-Novikov-Veselov (GNNV) equation [16, 17] and is essentially a symmetric generalization of the famous KdV equation to the dimensions [18], which has numerous applications in engineering and physics including the electrical transmission lines, surface gravity waves, internal solitons in the ocean, nonlinear acoustics of bubbly liquids, voidage slugs in fluidized beds, and the “Great Red spot” in Jupiter. For a detailed account of these applications, we refer the reader to [19]. Eq. (2) can be simplified using the transformations and , where and are functions of , and . This yields the systemSystem (3) also arises in applications in plasma physics, optics, and condense matter physics and has been studied extensively. For instance, Boiti et al. [20] used inverse scattering method to procure solutions to (2), and Radha et al. [21] employed Painleve analysis and bilinear method to find multi-dromion solutions to (2). Some recent works on the GNNV equation can be found in [22, 23].

In this paper, we provide a double reduction analysis of the system of PDEs (1) and (3). The double reduction theory not only reduces the order and the number of independent variables involved but also makes use of symmetry, if it exists, to reduce a PDE further. Some of the solutions to (1) and (3) are of a “travelling wave” type and are based on a well-known substitution, which is possible only if translational symmetries exist. In double reduction, not only are travelling wave type solutions obtained by making use of translational symmetries but also solutions are obtained by forming an association of the symmetry present with conserved vectors of the equation. This association is not only limited to the translational symmetries and it may exist for a wide range of symmetries like rotations, dilatations, inversions. Here, we exploit the dilation symmetry of the systems to extract new exact solutions. The interested reader is referred to some recent works in the field of symmetries, conservation laws, and the double reduction theory (see, e.g., [24, 25]).

In Section 2, we provide a double reduction analysis of (1). We procure exact solutions to (1) by finding Lie symmetries and conservation laws. In Section 3, we extend this work to system (3). The results are discussed in Section 4. The conclusions are presented in Section 5.

2. Generalized Nonlinear System

We use the theory of double reduction to compute the exact solutions to (1) by first finding the conservation laws using the variational derivative approach. Many software packages are available to compute Lie symmetries of differential equations. For instance, the mathematical software Maple can be employed to obtain the Lie symmetries to (1). This yieldsWe investigate the conservation laws of (1) using the multiplier approach. The multiplier satisfies (see [26, 27])where is the conserved vector andis the total derivative operator. The repeated indices throughout indicate the summation convention. The total derivative operator for two independent variables and two dependent variables takes the form

The variational derivative of (5) yieldswhere is the Euler operator and is given by

The Euler operator for two independent variables and two dependent variables leads towhere total derivative operators and are defined above.

Relation (8) giveswhere and . Equation (11) results inThe separation of (12) with respect to the derivatives of and yields the systemso thatwhere , , , and are arbitrary constants. Four sets of multipliers can be obtained by setting one of the constants equal to 1 and the rest to zero. This yieldsThe conserved vectors corresponding to these multipliers areThe above analysis shows that system (1) admits four symmetries and four conserved vectors.

We now compute the exact solutions of (1) by utilizing the double reduction theory. A Lie-Bäcklund symmetry generator is associated with a conserved vector of system (1) if it satisfiesThe association between symmetry and conservation laws of system (1) is presented in Table 1.

Case 1 (solution of (1) using ). Let be any symmetry generator and be a conserved vector associated with . Under a similarity transformation of , there exists such that , where can be calculated (see [5]) from the systemwhereRelation (17) shows that the Lie symmetry is associated with and . The change of variablesreduces the generator to the canonical form . For two independent variables, (18) reduces to the conserved formwhereThe -components of and are obtained by exploiting relations (21) and (22). This givesso thatwhere and are constants. Equations (24) and (25) yieldConsequently, the solution to (1) is

Case 2 (solution of (1) using and ). The symmetries and are associated with , , and (see Table 1). We derive the reduced conserved form by considering a combination of and , namely, . The generator has a canonical form whenThe characteristic equation (28) yieldsThe conserved vectors , and , using (29) and (21)-(22), reduce toThe conserved vectors , , and satisfy the reduced conserved formConsequently (30), (31), and (32) givewhere , , and are arbitrary constants. Any pair of equations in the reduced conserved form (34)-(36) can be used to construct a solution to (1) with the help of Maple. For instance, (34) and (35) giveandAnother exact solution to (1) can be gleaned using (34) and (36). This givesandIt can be verified directly that solutions (39) and (40) satisfy (1). The same solution is obtained from (35) and (36) by setting and solving the equations simultaneously.

The transformation introduced in (29) describes one-dimensional travelling wave, where is the wave number and is the wave speed. If we choose , and , then system (1) is identified as dispersive long wave system [8]. Here is a nonlinearity parameter and is dispersive parameter. If we set , then the exact solution (39) and (40) corresponds for internal waves. If the wave speed is chosen to be and , then the elevation in (39) of waves is shown in Figure 1.

If , then the solution (39) represents the rogue waves. This situation is depicted in Figure 2, where we assume and .

The elevation function of (39) is the same as (40).

Case 3 (solution of (1) using ). Relation (17) shows that the symmetry is associated with and . Canonical coordinates of are Equations (21) and (22) along with (41) can be used to procure the reduced conserved form of vectors and . This yieldsandSince (42) and (43) do not admit any symmetry, the double reduction theory can not be applied to find the exact solution of (1). However, numerical methods can be employed to obtain approximate solutions (Figure 3).

3. (2+1)-Dimensional GNNV Equation

In this section, we compute exact solutions to the (2+1)-dimensional system (3) by following a pattern similar to that used in Section 2. This gives the Lie point symmetries admitted by (3) and the following set of determining equations for multipliers , where The last set of equations giveso that the conserved vectors corresponding to these multipliers areThe association between symmetries and conservation laws is shown in Table 2.

Case 1 (solutions of (3) using and ). The Lie symmetries and are associated with the conserved vectors , , , , and if and are constants. For simplicity, we assume these constants to be equal to . Consequently, the Lie symmetries reduce toand the associated conserved vectors take the formWe note that the conserved vectors , and are the same, so we only use to construct the solution of (3). A linear combination of and , given in (48), can be expressed aswhere and are constants. The generator in the last equation reduces to with the help of similarity transformationsEquation (18) giveswhereThis yieldsso that the reduced conserved vectors areandWe note that the new conservation law satisfiesThe transformed vectors , , and in (55)-(57) inherit symmetries and which are helpful in further reduction. The combination of these symmetries yields the similarity variablesand (19) can be used to procurewhich satisfiesIntegrating the last equation yieldswhere is the constant of integration. The conserved vectors (55)-(57) using (60) and (62) can be expressed aswhere , , and are arbitrary constants. The solution of system (63) iswhereIt can be verified directly that these solutions satisfy (3). Explicit solutions to (63) can be procured by setting constants , , and equal to zero and then employing the sine-cosine method [28]. This givesIn system (3), represents physical field and and describe potentials. We recover solution in (68) as reported in [17]. This solution is identified as dromion solution. Here, we obtain three additional solutions , , and of system (3) which were not reported before and their profiles are shown below.
Figure 4 represents the breather wave.

Case 2 (solution of (3) using ). As shown in Table 2, the Lie symmetry is associated with the conserved vectors and if and . This leads to the similarity transformations of , which are given byThe conserved vectors and in new variables can be expressed asThe above vectors are invariant under Lie symmetry . Consequently, the similarity variables areso thatThe reduced conserved form thus giveswhere we have replaced by and by . We note that the above system does not have any contribution from the third equation of (3), namely, (Figures 5 and 6). This equation is invariant under , and the change of variables (70) yieldsSince (75) is also invariant under the symmetry , its further reduction leads toEquations (76) and (74) in original variables give the solution of (3) (see Figures 79)We note from Table 2 that and are also associated with if and are constants but in that case and reduce to which has already been discussed above.

Case 3 (solution of (3) using , , , and ). Consider . As shown in Table 2, the symmetries and are associated with and . The symmetries are also associated with and if and are constants, but in that case they coincide with . Consequently, considering would be sufficient. The symmetry can then be expressed asIn similarity variables, we haveso that (18) givesThe reduced conserved vectors thus obtained areThe vectors and admit symmetries and , from which further reduction is possible. Using combination of symmetries and transforming it into new similarity variables yieldAgain, from (18) it is straightforward to show thatand system (3) satisfiesEquations (84) and (85) give where , , and are constants. System (86) reduces to Case 2.

4. Results and Discussion

Whereas the symmetry method only reduces the number of independent variables in the PDE involved, the double reduction method, in addition to the reduction of independent variables, reduces the order of the PDEs. The association of symmetries and conservation laws reduces the number of independent variables and the order of PDEs simultaneously. The association of a symmetry with more than one conservation law yields the same solutions (Case 2 of Sections 2 and 3), which are either implicit or in closed form. The association of scaling symmetry leads to double reduction whereas inversion symmetry yields a distinct solution to (1). The association of Lie symmetries and conserved vectors obtained in Section 3 involves arbitrary functions. The association of symmetries and conservation laws puts restriction on the form of these functions (as listed in Table 2). Under these restrictions, group invariant solutions are constructed. One solution is obtained using the relation between scaling symmetry and conservation law, while the other solutions are of travelling wave type.

Solutions (66), (67), (68), (69), (70), (74), (75), (76), and (77) are procured in terms of trigonometric functions using the sine-cosine method. An equivalent formulation to these solutions is given in [21] where the method of ansatz is employed to obtain dromion solutions in terms of exponential functions. In addition to the existing explicit solutions to (3), some new solutions including an implicit solution (64) and an explicit solution (77) to (3) have also been obtained. Moreover, we recuperated solution as reported in [17] using the combination of symmetries and conservation laws. This solution is identified as a dromion solution. In this article, we obtained three additional solutions , , and of (3) which were not reported before.

Since the double reduction approach reduces the order of the PDE and the number of independent variables involved, it makes the numerical computations of solutions easier as compared to the application of numerical techniques to the original GNNV equation. This study provides a new way of constructing variety of exact solutions for partial differential equations by forming an association of the symmetry present with conserved vectors of the equation. This association is not only limited to the translational symmetries or transformation of the form and it may exist for a wide range of symmetries like rotations, dilatations, and inversions. In this article, we exploited the dilation symmetry of the systems to extract new exact solutions as stated above.

5. Conclusion

Double reduction of a PDE is a well-known technique which could be used when a conservation law is associated with Noether symmetry of Euler-Lagrange equation. However, in many cases, the PDE system does not admit the Lagrangian. Consequently, the existence of Noether symmetry remains out of question. In this work we considered systems of evolution equations which do not admit the Lagrangian formulation. We used the variational approach to construct conservation laws and their Lie symmetries. Double reduction was then employed by investigating the association between symmetry and conservation law. An association of scaling symmetry with a conserved vector was established. This had the consequence of integrating the differential equation further, which is significant since in many cases further integration of a differential equation is not possible if it is reduced by scaling or inversion symmetry. Several explicit solutions were also obtained which is a contribution of this paper.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.