Conservation laws and Exact Solutions of Phi-Four (Phi-4) Equation via the (G′/G, 1/G)-Expansion Method

1 Introduction

The concept of conservation laws has a long and profound history in physics. Whatever the physical laws considered: classical mechanics; fluid mechanics; solid-state physics; and quantum mechanics, quantum field theory, or general relativity and whatever the constituents of the theory and the intricate dynamic processes involved, quantities left dynamically invariant have always been essential ingredients to describe nature. At the mathematical level, conservation laws are deeply connected with the existence of a variational principle that admits symmetry transformations. This crucial fact was fully acknowledged by Emmy Noether in 1918 [1, 2].

Construction of conservation laws for a given system is generally a nontrivial task. However, for a system arising from a Lagrangian formulation, there exists a fundamental theorem by Emmy Noether. He proved that for every infinitesimal transformation that is admitted by the action integral of a Lagrangian system, one can constructively find a conservation law [3, 4].

There are many different approaches to the construction of conservation laws such as characteristic method, variational approach [5], symmetry and conservation law relation [6, 7], direct construction method for conservation laws [8], partial Noether’s approach [9], Noether’s approach [10], and conservation theorem [11–15]. In this work, we will use conservation theorem approach for finding Phi-4 equation’s conservation laws.

Searching exact solutions of nonlinear partial differential equations (PDEs) plays an important and significant role in the study on the dynamics of those phenomena. Many mathematical techniques have been employed to find exact solutions to these equations. These methods include the inverse scattering method [16], Hirota’s bilinear method [17], the tanh method [18], transformed rational function method [19], the sine–cosine method [20, 21], the homogeneous balance method [22], the exp-function method [23], the extended trial equation method [24], the modified simple equation (MSE) method [25, 26], the first integral method [27], the auxiliary equation method [28], the (G′/G)-expansion method [29], the (1/G′)-expansion method [30, 31], Kudryashov’s method [32], and so on.

The fundamental principle of the (G′/G)-expansion method is that the exact solutions of nonlinear PDEs can be expressed by a polynomial in (G′/G) in which G=G(ξ) satisfies the differential equation G″(ξ)+λG′ (ξ)+μG(ξ)=0. Here, λ and μ are constants [29].

The (G′/G, 1/G)-expansion method can be considered as a generalisation of the (G′/G)-expansion method. The fundamental principle of the (G′/G, 1/G)-expansion method is that the exact travelling wave solutions of nonlinear PDEs can be expressed by a polynomial in the two variables (G′/G) and (1/G), in which G=G(ξ) satisfies a second-order linear ordinary differential equation (ODE), namely G″(ξ)+λG(ξ)=μ, where λ and μ are constants. The degree of this polynomial can be determined via the homogeneous balance method. Also, the coefficients of this polynomial can be determined by solving a set of algebraic equations resulted from the process of using the method [33–36].

The rest of this article has been arranged as follows. In Section 2, basic definitions about the conservation laws and the basic ideas of the (G′/G, 1/G)-expansion method are given. In Section 3, the conservation laws of the Phi-4 equation are found. In Section 4, the method is employed for obtaining the exact solutions of Phi-4 equation. Finally in Section 5, some conclusions are presented.

2 Definitions

Firstly, it would be helpful to give some definitions and formulations of the conservation laws theory in this part of the article.

3 Conservation Laws of the Phi-4 Equation

The Phi-4 equation plays an important role in nuclear and particle physics over the decades:

where m and c are real constants. In this section, we will find conservation laws for the Phi-4 equation.

Let us construct formal Lagrangian for (19) in the following form:

where w=w(x, t) is the adjoint variable. If we substitute (20) into (7), then the adjoint equation can be verified as follows:

Now, we can see that (19) is not obtained by substituting w=u in (21). Consequently, we can say that (19) is not self-adjoint.

The Phi-4 equation admits following three Lie point symmetry generators:

In addition, above Lie point symmetry generators’ commutator table is as follows:

Now, we will find conservation laws of (19).

Case 1: We consider

the Lie point symmetry generator. Presently, we will show that X₁ satisfies by invariance test:

Second prolonged vector of X₁ is

Here,

and

Here, i=1, 2, …. We get from the above formulations:

By substituting (28) into (25), we get X1(2) = ∂/∂x. If we substitute these values into (24), invariance condition is clearly satisfied:

X(2)(L) + L(Dx(ξx) + Dt(ξt)) = 0.

For X₁,

If we use (8), conserved vectors’ formulations are

We can find W with (29) in the following form:

As a result, conserved vectors are

for (19) with X₁=∂/∂x Lie point symmetry generator.

If we substitute (32) into (9),

is satisfied.

Case 2: Secondly, we consider

Equation (19) satisfies invariance test:

X(2)(L) + L(Dx(ξx) + Dt(ξt)) = 0,

for X₂=∂/∂t Lie-point symmetry.

Let us construct conservation laws corresponding to symmetry X₂. From X₂,

are obtained. If we use W=η−ξ^xu_x−ξ^tu_t and (35), we obtain

By substituting (36) into (30), we obtain the following conserved vectors:

For (37), (9)

is satisfied. Therefore, T2t and T2x are conservation laws for (19).

Case 3: Thirdly, we will use the following Lie point symmetry:

Equation (19) satisfies invariance test for X₃. For the operator X₃,

We have

If we substitute (41) into (30), conserved vectors are

Using (9), for (42) it is shown that

Therefore, invariance condition is satisfied.

4 Exact Solutions of Phi-4 Equation

In this section, the presented method is applied to the Phi-4 equation (7). For our purpose, we introduce the following transformation as u(x, t)=u(ξ), ξ=x−vt, where v is a constant. With this transformation, (7) reduced to following ODE:

where prime denotes the derivative with respect to ξ.

According to the homogeneous balance method, we get the balancing number as N=1. Therefore, the solution (18) takes the following form:

where a₀, a₁, and b₁ are constants to be determined. Let us examine the three cases mentioned above.

Case 1: (λ>0 Trigonometric function solutions)

Substituting (45) into (44), using (12) and (13), the left-hand side of (44) becomes a polynomial in ϕ and ψ. Equating each coefficient of this polynomial to zero yields a set of the following algebraic equations in a₀, a₁, b₁, m, v, σ, μ, and λ:

ϕ3:−2a1λ4σ2 + ca13μ4 − 2ca13λ2σμ2 + 2v2a1λ4σ2 + ca13λ4σ2 + 2v2a1μ4 +4a1λ2σμ2 + 3ca1b12λ3σ − 3ca1b12λμ2 − 4v2a1λ2σμ2 − 2a1μ4 = 0,ϕ2ψ1:−4v2b1λ2μ2 + 2v2b1μ4 − cb13λμ2 + 2v2b1λ4σ2 + 3ca12b1λ4σ2 + 3ca12b1μ4 −6ca12b1λ2σμ2 + 4b1λ2σμ2 − 2b1μ4 + cb13λ3σ − 2b1λ4σ2 = 0,ϕ2ψ0:3ca0b12λ3σ − v2b1μλ3σ + 3ca0a12λ4σ2 − 2cb13λ2μ − 3ca0b12λμ2 −b1μ3λ + b1μλ3σ + v2b1μ3λ + 3ca0a12μ4 − 6ca0a12λ2σμ2 = 0,ϕ1ψ1:−4v2b1λ2σμ2 + 2v2b1μ4 + 2v2b1λ4σ2 + 3ca12b1λ4σ2 − 6ca12b1λ2σμ2 −cb13λμ2 + 4b1λ2σμ2 − 2b1μ4 + cb13λ3σ + 3ca12b1μ4 − 2b1λ4σ2 = 0,ϕ1ψ0:3ca0b12λ3σ − v2b1μλ3σ + 3ca0a12λ4σ2 − 2cb13λ2μ − 3ca0b12λμ2 −b1μ3λ + b1μλ3σ + v2b1μ3λ + 3ca0a12μ4 − 6ca0a12λ2σμ2 = 0,ϕ0ψ1:3a1μ5 − 3v2a1μλ4σ2 + 6ca1b12λμ3 − 12ca0a1b1λ2τμ2 − 3v2a1μ5 + 6ca0a1b1λ4τ2 −6ca1b12λ3μτ + 6ca0a1b1μ4 + 3a1μλ4τ2 − 6a1μ3λ2τ + 6v2a1μ3λ2τ = 0,ϕ0ψ0:3ca02a1μ4 + 2v2a1λ5σ2 + 2v2a1λμ4 + 4a1λ3μ2 + 3ca1b12λ4σ − a03λ4σ2 −3ca1b12λ2μ2 + 3ca02a1λ4σ2 − 4v2a1λ3σμ2 − 6ca02a1λ2σμ2 − 2m2a1λ2σμ2+ m2a1λ4σ2 + m2a1μ4 − 2a1λ5σ2 − 2a1λμ4 = 0.

Solving this system with the aid of Maple, we get

a0 = 0,   a1 = −(v2 − 1)2c,   b1 = −(v2 − 1)(λ2τ − μ2)2cλ,   m = − 12λ(v2 − 1),

where λ, c≠0. By substituting these values into (45), using (11) and (13), we obtain the exact solutions of Phi-4 as follows:

where

ξ = x − vt and σ = A12 + A22.

The Figure 1 of this obtained solution can be given as follows by choosing special values for arbitrary constants:

Figure 1:

Graph of the u(x, t) corresponding to the values: v=−4, A₁=1, A₂=−1, μ=2, λ=2, c=−5.

Case 2: (λ<0 hyperbolic function solutions)

Substituting (45) into (44), using (12) and (14), the left-hand side of (44) becomes a polynomial in ϕ and ψ. By equating each coefficient of this system to zero, we get a set of algebraic equations. Solving this system with Maple, we obtain

a0 = 0, a1 = −(v2 − 1)2c, b1 = (v2 − 1)(λ2τ + μ2)2cλ, m = − 12λ(v2 − 1).

where λ, c≠0. Substituting these values into (45), using (11) and (14), we get exact solutions of Phi-4 as follows:

where

ξ = x − vt and σ = A12 − A22.

The Figure 2 of this obtained solution can be represented by choosing special values for arbitrary constants.

Figure 2:

Graph of the u(x,t) corresponding to the values: v=−4, A₁=1, A₂=−1, μ=2, λ=2, c=−5.

Case 3: (λ=0 rational function solutions)

Substituting (45) into (44), using (12) and (15), the left-hand side of (44) becomes a polynomial in ϕ and ψ. By using the same procedure above, we have a set of algebraic equations. If we solve this system, we get

a0 = 0,   a1 = − (v2 − 1)2c,   b1 = − (v2 − 1)(A12 − 2A2μ)2c,   m = 0,

where c≠0. Substituting these values into (45), using (11) and (15), we get exact solutions of Phi-4 as follows:

u(ξ) = − (v2 − 1)2c(μξ + A1)μξ22 + A1ξ + A2 + (v2 − 1)(A12 − 2A2μ)2cμξ22 + A1ξ + A2,

where ξ=x –vt. We can graph the Figure 3 of the solution by choosing special values for arbitrary constants.

Figure 3:

Graph of the u(x, t) corresponding to the values: v=−4, A₁=1, A₂=−1, μ=2, c=−5.

Note that our solutions are new and more extensive than the given ones in [42]. Comparing our solutions with [42], it can be seen that by choosing suitable values for the parameters, similar solutions can be verified.

5 Conclusion

In this study, we used conservation theorem for obtaining conservation laws of the Phi-4 equation. Also through the (G′/G, 1/G)-expansion method, we have investigated exact travelling wave solutions of the Phi-4 equation. The method proposed in this work is expected to be further employed to solve similar nonlinear problems in applied mathematics and mathematical physics. This work illustrates the validity and great potential of the (G′/G, 1/G)-expansion method for nonlinear PDEs. All the obtained solutions in this article have been checked by Maple software.