New exact solutions to the Fitzhugh–Nagumo equation

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Abstract

By the first integral method, a series of new exact solutions of the Fitzhugh–Nagumo equation have been obtained. It is shown that this method is one of the most effective approaches to obtain the exact solutions of the nonlinear evolution equations, especially for nonintegrable models.

Introduction

The Fitzhugh–Nagumo equationut-uxx=u(u-α)(1-u),is an important nonlinear reaction–diffusion equation and usually used to model the transmission of nerve impulses [1], [2]; also used in circuit theory, biology and the area of population genetics [3] as mathematical models. When α = −1, the Fitzhugh–Nagumo equation reduces to the real Newell–Whitehead equation. By using Hirota method, Kawahara and Tanaka [4] have found new exact solutions of Eq. (1); by applying the nonclassical symmetry reductions approach, Nucci and Clarkson [5] have obtained some new exact solutions with Jacobbi elliptic function. Some other solutions of Eq. (1) have been given by several authors [6], [7]. In this paper, using a new method that is called the first integral method, we obtain a series of new exact solutions of Eq. (1).

The first integral method is due to Feng [8], [9], [10], [11]. By applying the theory of commutative algebra, he proposed this very concise and effective approach to search the solutions of nonlinear partial differential equations. By means of this new method, Feng studied the compound Burgers–KdV equation [8], Burgers–KdV equation [9], two-dimensional Burgers–KdV equation [11] and an approximate sine-Gordon equation in (n + 1)-dimensional space [10]. Naranmandula [12] followed this line to investigate a nonlinear dispersive–dissipative equation.

Assume that Eq. (1) has travelling wave solutions in the formu(x,t)=u(ξ),ξ=x-vt(vR),where v is the velocity of the travelling wave. Substituting (2) into Eq. (1) yieldsu=-vu-u(u-α)(1-u).Let x = u, y = uξ, and then Eq. (3) is equivalent tox=y,y=-vy-αx+(1+α)x2-x3.If we can find two first integrals of (4) under the same conditions, the general solutions of (4) can be obtained directly. But, in general, it is rather difficult to realize this object, even for one first integral. The excellent idea of the first integral method is using the division theorem to seek one first integral of (4), which can reduce Eq. (3) to a first-order integrable ordinary differential equation, and then we can obtain the exact solutions of Eq. (1) by using the direct integral method.

Section snippets

Exact solutions to the Fitzhugh–Nagumo equation

According to the first integral method, we suppose that x = x(ξ) and y = y(ξ) are the nontrivial solutions of (4) and q(x,y)=i=0mai(x)yi is an irreducible polynomial in the complex domain C[x, y] such thatq[x(ξ),y(ξ)]=i=0mai(x)yi=0,where ai(x) (i = 0, 1,  , m) are polynomials of x and am(x)  0. Eq. (5) is also called the first integral to (4). We start our study by assuming m = 2 in Eq. (5). Note that dq/dξ is a polynomial in x and y, and that q[x(ξ), y(ξ)] = 0 implies dq/dξ = 0. Due to the division theorem,

Conclusion

Applying the first integral method, we have found a series of new exact solutions of the Fitzhugh–Nagumo equation including some new special solutions. It is shown that this method is one of the most effective approaches to obtain the exact solutions of the nonlinear evolution equations, especially for nonintegrable models.

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