Nonlocal Symmetries, Conservation Laws and Interaction Solutions of the Generalised Dispersive Modified Benjamin–Bona–Mahony Equation

1 Introduction

The generalised Benjamin–Bona–Mahony (BBM) equation is given by

where N is an integer and α denotes an arbitrary constant. The (1) has been systematically investigated by Wazwaz via the tanh-sech and sine-cosine method [1].

For N=1, (1) can be degenerated to the regularised long-wave equation

which can display the surface water waves in certain regimes. Equation (2) can be regarded as a generalised KdV equation.

In this article, we consider the case of N=2 in (1), which can be changed into the following generalised dispersive modified Benjamin–Bona–Mahony (mBBM) equation [2]

where u denotes a differential function. The generalised dispersive mBBM equation (3) has been firstly proposed to depict an approximation status for long surface wave existed in the non-linear dispersive media. Furthermore, it can display some other dynamic behaviours, such as hydro-magnetic waves in cold plasma, acoustic waves in compressible fluids media and acoustic waves in inharmonic crystals [3], [4]. The travelling wave solutions, new analytical solutions and new generalised solitary solutions have been presented in [5], [6], [7]. To the best of our knowledge, much work for the BBM-type equations have been reported. But the Lie symmetry analysis, conservation laws, and interaction solutions of the mBBM equation (3) have not been studied before.

On the one hand, by means of a non-local symmetry method, various interaction phenomena between solitary waves and others containing cnoidal waves and Painlevé waves can be presented for non-linear evolution equations (NLEEs) [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28]. For the non-local symmetry, Lou proposed that the symmetry can be considered as a residue in regard to the singular manifold. Based on that, residual symmetry is proposed, which can yield non-local symmetry. It is interesting that a direct method can be used to obtain the interaction solutions with respect to non-local symmetry, by applying the tanh function expansion approach. Then two new methods are introduced by Lou, i.e. consistent Riccati expansion (CRE) and consistent tanh expansion (CTE) [13], [14], [15], [16], [17]. The non-local symmetry method and related methods can be used to many NLEEs [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61].

On the other hand, we know that conservation laws play a very important role in non-linear fields for integrability of NLEEs, which provide an approach to derive some interesting properties of integrable system. Particularly, one can see that many results, such as existence, uniqueness, and stability of the rational solutions are determined by the conservation laws. Moreover, for constructing conservation laws of NLEEs, some direct ways have been proposed, especially, the celebrated Noether’s theorem [62]. This is considered as the most effective way if it does exist a Lagrangian form. In fact, some equations do not possess it. Besides, Ibragimov and Gandarias proposed a concept of self-adjoint equations to derive a conservation law of NLEEs [63], [64], [65]. It can be used to all systems with the number of equations with many dependent variables.

The organization of this paper is as follows. In Section 2, based on the leading order analysis and truncated Painlevé expansion, we obtain the residual symmetry and initial value problem of (3). In Section 3, by using CRE method, we demonstrate that system (3) is a CRE solvable system. Then its interaction solution among the solitons and cnoidal periodic waves is presented. Finally, some discussions are provided in the last section.

2 Residual Symmetry and its Localisation

As we all know that the truncated Painlevé analysis is one of the effective approaches to construct non-linear waves of NLEEs. Then we can obtain the following Painlevé expansion formula of (3) via a leading order analysis

where f=f(x, t) denotes a singular manifold, u₁ and u₀ denote new real functions in regard to variables x, t. Substituting above transformation (4) into (3), collecting the coefficients of different terms fⁱ, (−4, −3, −2, −1, 0), and equating them to zero, we obtain

By computing the above equations, one has

Then we can get a solution of (3) given by

which is an auto-Bäcklund transformation between u and u₀.

In regard to the non-local symmetry σu=6αfx, we can get the following initial value problem

in which ϵ denotes a small parameter. To localise the corresponding non-local symmetry σu=6αfx, we take several dependent variable transformations given by

Furthermore, (3) admits a family of linearised equations of prolonged system

We can easily find that (14), (15) provide

Then the corresponding Lie point symmetry vector is as follows

One can obtain a class of initial value problems given by

By computing (18) and (19), we obtain the initial value solutions with respect to u̅, f̅, m̅, h̅ and ρ̅

3 Consistent Riccati Expansion Method and Interaction Solution

In this section, based on the CRE approach presented in [13], [14], [15], [16], [17], we provide a new form of solution u given by

where u₁, u₀ and w are real functions about x and t, and R(w) denotes a special solution of the following Riccati representation

with R(w) being a general solution with respect to tanh(w). Substituting (22) and (23) into (3), and collecting all coefficients of different powers of R(w), then one can obtain

with δ=a12−4a0a2. Actually, when w satisfies the (25), then system (3) has a special solution as follows

At this time, R(w)=−δtanh(δw2)2a2−a12a2 can solve (23). To seek some interesting phenomena for the generalised dispersive mBBM equation, we will construct an interaction solution of system (3) with a Jaccobi elliptic function in next calculation.

By considering (25), it is not difficult to derive a solution of (3). We consider that w can be expressed by

where

with W(X) satisfying the following two derivative formulas

Equations (28) and (29)–(30) are substituted into (25). It is easy to seek

in which C₃ is a free real constant. To construct a Jacobi elliptic function interaction solution, we now assume W₁ expressed by

Integrating above W₁ related to X, one obtains

According to the (29), we can obtain several parameters presented by

where k₁, k₂, μ₀ denote arbitrary constants. Substituting these parameters into (26), one can obtain an interaction solution between solitons and cnoidal periodic waves

where W_1X, W₁ and W are, respectively, provided by (29) and (33)–(34).

The graphics of interaction solution between solitons and cnoidal periodic solution waves (38) for (3) are plotted in Figures 1 and 2 by choosing suitable parameters.

Figure 1:

Interaction solution between solitons and cnoidal periodic solution waves (38) for (3) by choosing suitable parameters: μ₀=1.2, δ=4, k₁=0.125, k₂=0.5, n=0.3, m=1.5, α=1, X₀=0. (a) Perspective view of the real part of the wave. (b) The wave propagation pattern of the wave along the x axis. (c) The wave propagation pattern of the wave along the t axis.

Figure 2:

Interaction solution between solitons and cnoidal periodic solution waves (38) for (3) by choosing suitable parameters: μ₀=0.2, δ=4, k₁=−1.5, k₂=1, n=0.5, m=1, α=5, X₀=0. (a) Perspective view of the real part of the wave. (b) The wave propagation pattern of the wave along the x axis. (c) The wave propagation pattern of the wave along the t axis.

4 Conservation Laws

Conservation laws play an important role in studying NLEEs. The conservation laws can be used to reflect the integrability of equation, in certain condition. Recently, lots of effective approaches to derive the conservation laws of NLEEs have been proposed, mainly including the multiplier method, which is related to the new conservation theorem demonstrated by Ibragimov and Gandarias [62], [63], [64], [65]. In this section, based on a greatly straightforward method, the conservation laws of the generalised dispersive mBBM equation can be easily constructed.

To obtain the conservation laws of the equation, we briefly provide main ideas for later detail computation.

We introduce a general form of systems

where X=(x₁, x₂, …, x_p) denotes p independent variables, U=(u₁, u₂, …, u_q) represents q dependent variables, and U_(r) is a class of the partial derivatives of r-th order of U. Then an adjoint system of (39) is given by

in which

Here L=Σβ=1qVβFβ(X, U, U(), …, U(r))=0 is a Lagrangian form of (39), with V=V(v₁, v₂, …, v_q) being new dependent variables about x, and δδUα being the variational derivative, i.e.

Theorem:A general infinitesimal symmetry of (39) can be expressed by

which gives a conservation law D(C^j)=0 in the system with satisfying the adjoint (40), where C^j is a conserved vector represented by

with

According to the above Theorem, we can construct the following Lagrangian form of (3)

with ũ being a new dependent variable. By using above Theorem, we provide a general vector form of system (39)

The corresponding conservation law is limited by

where C=(C^t, C^x) denotes the conserved vector of (44). After computing it, one obtains

Based on the construction of (46), C^t, C^x can be changed into

where

By means of classic group theory, it is easy to find

In what follows we discuss several different cases.

For the generator

the corresponding Lie characteristic function is

One can find the conservation vector of (3) given by

For the generator

the corresponding Lie characteristic function is given by

One can find the conservation vector of (3) given by

For the generator

the Lie characteristic function is given by

One can find the conservation vector of (3) as follows

Conservation laws have an important application in many physic fields, such as energy conservation, mass and momentum, etc. However, it is not to say that each conservation law has self-physical significance. As an indication of integrability for NLEEs, conservation laws possess an important position. From the celebrated Noether theorem, we know that certain connection always exists among symmetry and conservation law. In fact, when the spatial transformation is unchanged, the conservation law of momentum is satisfied. If the temporal transformation is fixed, then the energy conservation, and vice versa are ensured.

5 Summary and Discussion

In this article, we study the generalised dispersive mBBM equation (3). Based on the truncated Painlevé expansion method, we accurately derived its auto-Bäcklund transformation and non-local symmetry in a very natural way. Furthermore, by prolonging the generalised dispersive mBBM equation to a closed prolonged system, its residual symmetry would be localised to the corresponding Lie point symmetry. By applying the first theorem of Lie symmetry group, the symmetry transformation related to the obtained prolonged system was constructed. By employing a CRE method to the (3), we found that the equation is a CRE solvable system. Additionally, we constructed an interaction solution among the solitary waves and cnoidal periodic waves with a Jacobi sine function. Finally, we found a special relationship between the symmetry and conservation law of the equation. It was easy to find that the integrability of NLEEs is related to the corresponding conservation law. It is hoped that the results of this article can help to enrich the dynamic behaviour of the BBM-type equations. In the near future work, we will concentrate on the further symmetry properties of non-linear integrable systems.