Conservation Laws and Soliton Solutions of the (1+1)-Dimensional Modified Improved Boussinesq Equation

1 Introduction

The notion of conservation laws plays an important role for the integrability, devoloping numerical methods, and behaviour of solutions for differential equations arising in nonlinear phonomena. A systematic way to determine the conservation laws of partial differential equations (PDEs) is the famous Noether’s theorem. This theorem is linked to the knowledge of a classical Lagrangian [1]. There are differential equations that do not admit of a Lagrangian, e.g. scalar evolution equations. Several methos have been devoloped in the literature, which do not admit of a Lagrangian such as multiplier method, direct method, partial Lagrangian method, and non-local conservation method [2–5].

On the other hand, there is plainly a tendency in the modern nonlinear science community to obtain exact solutions for nonlinear equations. In recent years, many methods were developed for finding exact solutions of nonlinear evolution equations (NEEs) as the tanh−sech method [6, 7], extended tanh method [8, 9], sine−cosine method [10], homogeneous balance method [11, 12], first integral method [13], Jacobi elliptic function method [14], (G′/G)-expansion method [15], exp-function method [16], and F-expansion method [17]. It is well known that wave phenomena of optical fibres and nonlinear dispersive media are modelled by dark-shaped tanh^p solutions or by bright-shaped sech^p solutions.

Soliton was first discovered in 1834 by John Scott Russell [18], who observed that a canal boat stopping suddenly gave rise to a solitary wave that travelled down the canal for several miles, without breaking up or losing strength. Russell named this phenomenon the “soliton”. A soliton is a special travelling wave that after a collision with another soliton eventually emerges unscathed. Solitons are solutions of PDEs that model phenomena like water waves or waves along a weakly anharmonic mass-spring chain. The soliton has exponential tails, which are the basic character of solitary waves. This property allows the exponential function to describe its solution. The soliton obeys a superposition-like principle: solitons passing through one another emerge unmodified. In addition, theories of dispersion-managed solitons, quasi-linear pulses have also been developed. It is known that dark solitons are more stable in the presence of noise and spreads more slowly in the presence of loss, in the optical communication systems, as compared to bright solitons [19–21].

We consider the following one-dimensional generalised IMBq equation (modified improved Boussinesq equation) [22]:

(1)utt − uxxtt − uxx − (f(u))xx = 0. (1)

The generalised IMBq equation governs various physical models like nonlinear wave in weakly dispersive medium (in this case, f(u)=u² [23–25]) and longitudinal variation wave in elastic rod (f(u)=u³ or f(u)=u⁵ [26]), etc. For the local or global existence of solution of IMBq equation, see Refs. [27–29], and for the small amplitude solution and scattering, see Refs. [28, 30]. Yang and Wang [31], example of (f(u)=u²) the blowup of solutions is obtained numerically. If we take f(u)=u³, (1) becomes

(2)utt − uxxtt − uxx − (u3)xx = 0. (2)

The article is organised as follows: in Section 2, we present the fundemental operators for the determination of conservation laws. In Section 3, we introduce the partial Lagrangian method and multiplier approach and apply both of them to IMBq equation. In Section 4, we investigate the bright soliton solutions. In last section, we briefly make a summary to the results that we have obtained.

2 Methods to Derive Conservation Laws of (2)

In this section, we implement two distinct methods for obtaining the conserved vectors of (2). The first one is the partial Lagrangian approach, which is developed by Kara and Mahomed [3]. If the standard Lagrangian does not exist or is difficult to find of the considered PDEs, then we write its partial Lagrangian and derive the conservation laws by the partial Noether approach. Omitting the straightforward calculations, we present only the conserved vectors such as

(3)T1x = − F(t)ux − 3F(t)u2ux − 2F(t)uttx − utxFt,T1t = −uFt + F(t)ut + F(t)utxx,T2x = u3Gx + uGx−G(x)ux−G(x)3u2ux + G(x)uttx,T2t = G(x)ut − 2G(x)utxx − utxGx, (3)

where F(t) and G(x) are arbitrary constants of t and x, respectively.

Secondly, we applied the multiplier approach to (2). For the theoretical aspects and applications of the multiplier approach, one can resort to Refs. [32–35]. Using this method, we find out the four multipliers and four conserved vectors as follows (with the help of [36]):

(4)Q1 = x,Q2 = 1,Q3 = xt,Q4 = t. (4)

(5)T1x = − 3xu2ux + u3 + u − xux,T1t = xut − xutxx,T2x = − 3uxu2 − ux,T2t = ut − utxx,T3x = − 3xtuxu2 + tu3 + tu − xtux,T3t = − ux + xtut + xuxx − xtutxx,T4x = − 3tuxu2 − tux,T4t = − u + tut + uxx − tutxx. (5)

Thus, using the multiplier method, we obtained one trivial and three nontrivial conservation laws of IMBq equation.

It is well known that in order to obtain the physical meanings of the considered underlying equation, conservation laws are the key instruments. Some well known conservation laws in physics are the conservation of energy, mass (or matter), momentum (linear or angular) and Hamiltonian. For instance, the conservation of energy follows from the time invariance of physical systems. Therefore, in (5), T2t and T2x represent the conservation of energy of (2). The other conserved vectors might correspond to the momentum and Hamiltonian.

3 Exact Solutions

3.1 The Bright (Non-topological) Soliton Solution

Using the ansatz method for the bright (non-topological) 1-soliton, solution of (2) is taken to be given by the form [37, 38]

(6)u(x, t) = λsechpτ, (6)

and

(7)τ = B(x − vt). (7)

Here, λ is the soliton amplitude, v is the soliton velocity, and B is the inverse width of the soliton. The unknown p will be determined during the course of derivation of the solutions of (2). From the ansatz (6), it is possible to obtain necessary derivatives. Then, the obtained derivatives are substituted in (2), and we collect all terms with the same order of sechξ. Then, by equating each coefficient of the resulting polynomial to zero, we obtain a set of algebraic equations:

(8)−p(p + 1)(p + 2)(p + 3)λB4v2 + 3p(3p + 1)λ3B2 = 0, (8)

(9)p2λv2B2 − p4λB4v2 − p2λB2 = 0. (9)

Finally, solving the system of equations, we can get

(10)v = ± λ22B, (10)

(11)B = ± λλ2 + 2. (11)

Equating the exponents of sechξ leads to

(12)p = 1. (12)

Thus, finally, the 1-soliton solution of (2) is given by

(13)u(x, t) = λsech{± λλ2 + 2(x ∓ λ22Bt)}, (13)

which exists provided that B≠ 0.

Figures 1 and 2 are the profile of a 1-soliton solution of IMBq equation for λ= 1/2, p= 1 and λ= 1, p= 1, respectively.

Figure 1:

λ=1 and p=1.

Figure 2:

λ=−1 and p=1.

4 Conclusions

In this study, we studied the conservation laws and soliton-type exact solutions of (1+1)-dimensional IMBq equation. The infinitely many nonlocal conservation laws for the considered equation were computed via the partial Noether approach. In addition, the local conservation laws were constructed with the help of the multiplier method and maple GEM package [36]. The multiplier method, when applied to the IMBq equation, gave rise to four multipliers of the form Q(x, t, u), and thus, four local conserved vectors were obtained in each case. The conserved vectors obtained here can be used in reductions and solutions of the underlying equation.

Furthermore, we have derived the exact bright soliton solutions of the IMBq equation. This has been realised by using the solitary wave ansatz method. In view of the analysis, we see that the used method is an efficient method of integrability for constructing exact soliton solutions.