# Exact Solutions of Non-Linear Lattice Equations by an Improved Exp-Function Method

^{*}

## Abstract

**:**

## 1. Introduction

_{1}, x

_{2}, ⋯, x

_{s}and dependent variable u:

_{n}, b

_{m}, k

_{i}and w are undetermined constants, f, p, g and q can be determined by using Equation (2) to balance the highest order non-linear term with the highest order derivative of u in Equation (1). It is He and Wu [20] who first concluded that the final solution does not strongly depend on the choices of values of f, p, g and q. Usually, f = p = g = q = 1 is the simplest choice. More recently, Ebaid [41] proved that f = p and g = q are the only relations for four types of nonlinear ordinary differential equations (ODEs) and hence concluded that the additional calculations of balancing the highest order derivative with the highest order non-linear term are not longer required. Ebaid’s work is significant, which makes the exp-function method more straightforward. The present paper is motivated by the desire to prove that f = p and g = q are also the only relations when we generalize the exp-function method [20] to solve non-linear lattice equations. Thus, the exp-function method can be further improved because it is not necessary to balance the highest order derivative with the highest order non-linear term in the process of solving non-linear lattice equations.

## 2. Generalized Exp-Function Method for Non-Linear Lattice Equations

_{n}, u

_{n}

_{−}

_{θ}(θ = ±1, ±2, ⋯) and the various derivatives of u

_{n}. Otherwise, a suitable transformation can transform Equation (3) into such an equation.

_{n}and U

_{n}

_{−}

_{θ}(θ = ±1, ±2, ⋯) determined by Equation (6) into Equation (5) and then balance the highest order derivative with the highest order nonlinear term in Equation (5) to obtain the integers f, p, g and q. Finally, we determine the coefficients a

_{−}

_{f}(x, t), ⋯, a

_{g}(x, t), b

_{−}

_{p}, ⋯, b

_{q}, d and c(x, t) by solving the resulting equations from the substitution of U

_{n}and U

_{n}

_{−}

_{θ}(θ = ±1, ±2, ⋯) along with the obtained values of f, p, g, q into Equation (5).

_{−}

_{f}(x, t) and a

_{g}(x, t), and the constants b

_{−}

_{p}and b

_{q}are all nonzero coefficients. Therefore, we can easily obtain N(U

_{n}

_{−}

_{θ}) = p−f and P (U

_{n}

_{−}

_{θ}) = g −q. For the derivatives of U

_{n}, we have a general formula:

_{r}(x, t) and σ

_{r}(x, t) are functions of x and t, δ

_{r}and ς

_{r}are constants, and r ≥ 1 is an integer. If τ

_{r}(x, t), σ

_{r}(x, t), δ

_{r}and ς

_{r}are nonzero coefficients, then $N({U}_{n}^{(r)})=p-f$ and $P({U}_{n}^{(r)})=g-q$.

_{1}, j

_{1}, i

_{2}, j

_{2}, ⋯, i

_{z}, j

_{z}, l

_{1}, l

_{2}, ⋯, l

_{s}are nonnegative integers which satisfy

**Remark 1.**If we let a

_{−}

_{f}(x, t), ⋯, a

_{g}(x, t) be nonzero constants and take c(x, t) as a linear function kx + lt, k and l are undetermined constants, then the generalized exp-function method described in this section is also effective for non-linear lattice equations with constant coefficients. So the starting point of this paper is to generalize the exp-function method [20] to solve Equation (3) with variable coefficients. In the next section, we shall further improve this generalized exp-function method.

## 3. Theorem and Improvement

**Theorem 1.**Suppose that Equations (8) and (10) are respectively the highest order derivative and the highest order nonlinear term of Equation (5), then the balancing procedure using the exponential function ansätz (6) leads to f = p and g = q.

**Proof.**By contradiction, we suppose that f ≠ p and g ≠ q. Then a computation shows that τ

_{r}(x, t), σ

_{r}(x, t), δ

_{r}, and ς

_{r}in Equation (8) are all nonzero coefficients. Using Equations (6) and (8), we have

## 4. Application

_{n}= u(n, t), α(t) is an arbitrary differentiable function of t. When α(t) = 0, 1, α(const.), Equation (28) can give three known constant-coefficient versions of the mKdV lattice equation.

_{0}is the phase, we transform Equation (28) into

_{n})(j = 0, ±1, ±2, ±3) to zero yields a set of equations for a

_{1}(t), a

_{0}(t), a

_{−}

_{1}(t), b

_{1}, b

_{0}, b

_{−}

_{1}and c(t). Solving the system of equations by the use of Mathematica, we have:

_{1}and b

_{−1}are arbitrary constants.

_{n}= dn + 2 tanh(d) ∫ α(t)dt + η

_{0}. If set b

_{1}= 1, then solutions (38) become the known solutions [42].

_{1}= −15, b

_{−1}= −2, d = 1, η

_{0}= 0. Figs. 1(a)–(d) show that the amplitude of wave changes periodically in the process of propagation. It is shown in Figure 1c that the “breather”-like phenomena has occurred at the location n = 0. In Figure 2, we show the structures of solutions (39) with (+,+) branch, where α(t) = 1 + secht, b

_{1}= 15 and the other parameters are same as those in Figure 1. From Figure 2c, we can see that u

_{0}has a singularity in the interval t ∈ (0, 1). It is easy to see that when b

_{1}= 15 and b

_{−1}= −2, solutions (39) are unbounded. Such unbounded solutions develop singularity at a finite time, i.e. for any fixed n = n

_{0}, there always exists t = t

_{0}at which these solutions “blow-up”. In view of the physical significance, they do not exist after “blow-up”. In the actual experimental physical system, there is no “blow-up”, but a sharp spike [43]. Thus, the finite time “blow-up” can provide an approximation to the corresponding physical phenomenon.

## 5. Conclusions

_{1}and b

_{−1}, which provide enough freedom for us to describe rich spatial structures of these obtained solutions. Applying the improved exp-function method to some other non-linear lattice equations with variable coefficients are worthy of study. This is our task in the future.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Fermi, E.; Pasta, J.; Ulam, S. Collected Papers of Enrico Fermi II; University of Chicago Press: Chicago, IL, USA, 1965. [Google Scholar]
- Toda, M. Theory of Nonlinear Lattics, 2nd ed; Springer: Berlin, Germany, 1989. [Google Scholar]
- Garder, C.S.; Greene, J.M.; Kruskal, M.D.; Miura, R.M. Method for solving the Korteweg–de Vries equation. Phys. Rev. Lett.
**1965**, 19, 1095–1097. [Google Scholar] - Miurs, M.R. Bäcklund Transformation; Springer: Berlin, Germany, 1978. [Google Scholar]
- Hirota, R. Exact solution of the Korteweg–de Vries equation for multiple collisions of solitons. Phys. Rev. Lett.
**1971**, 27, 1192–1194. [Google Scholar] - Wang, M.L. Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A.
**1996**, 213, 279–287. [Google Scholar] - Malfliet, M. Solitary wave solutions ofnonlinear wave equations. Am. J. Phys.
**1992**, 60, 650–654. [Google Scholar] - Liu, S.K.; Fu, Z.T.; Liu, S.D.; Zhao, Q. Jacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations. Phys. Lett. A
**2001**, 289, 69–74. [Google Scholar] - Abdel-Salam, E.A-B.; Al-Muhiameed, Z.I.A. Exotic localized structures based on the symmetrical lucas function of the (2+1)-dimensional generalized Nizhnik–Novikov–Veselov system. Turk. J. Phys.
**2011**, 35, 241–256. [Google Scholar] - Tabatabaei, K.; Celik, E.; Tabatabaei, R. The differential transform method for solving heat-like and wave-like equations with variable coeffcients. Turk. J. Phys.
**2012**, 36, 87–98. [Google Scholar] - Fan, E.G. Travelling wave solutions in terms of special functions for nonlinear coupled evolution systems. Phys. Lett. A
**2002**, 300, 243–249. [Google Scholar] - Fan, E.G. An algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations. J. Phys. A: Math. Gen.
**2003**, 36, 7009–7026. [Google Scholar] - Dai, C.Q.; Wang, Y.Y.; Tian, Q.; Zhang, J.F. The management and containment of self-similar rogue waves in the inhomogeneous nonlinear Schrödinger equation. Ann. Phys.
**2012**, 327, 512–521. [Google Scholar] - Dolapci, I.T.; Yildirim, A. Some exact solutions to the generalized Korteweg–de Vries equation and the system of shallow water wave equations. Nonlinear Anal. Model. Control.
**2013**, 18, 27–36. [Google Scholar] - He, J.H. Asymptotic methods for solitary solutions and compactons. Abstr. Appl. Anal.
**2012**, 2012, 916793, 130pages. [Google Scholar] - Dai, C.Q.; Wang, X.G.; Zhou, G.Q. Stable light-bullet solutions in the harmonic and parity-time-symmetric potentials. Phys. Rev. A
**2014**, 89, 013834. [Google Scholar] - Dai, C.Q.; Zhu, H.P. Superposed Kuznetsov–Ma solitons in a two-dimensional graded-index grating waveguide. J. Opt. Soc. Am. B
**2013**, 30, 3291–3297. [Google Scholar] - Liu, Y.; Gao, Y.T.; Sun, Z.Y.; Yu, X. Multi-soliton solutions of the forced variable-coefficient extended Korteweg–de Vries equation arisen in fluid dynamics of internal solitary vaves. Nonlinear Dyn.
**2011**, 66, 575–587. [Google Scholar] - Zhang, S.; Cai, B. Multi-soliton solutions of a variable-coefficient KdV hierarchy. Nonlinear Dyn.
**2014**, 78, 1593–1600. [Google Scholar] - He, J.H.; Wu, X. H. Exp-function method for nonlinear wave equations. Chaos Solitons Fractals.
**2006**, 30, 700–708. [Google Scholar] - He, J.H.; Abdou, M.A. New periodic solutions for nonlinear evolution equations using Exp-function method. Chaos Solitons Fractals
**2006**, 34, 1421–1429. [Google Scholar] - He, J.H.; Zhang, L.N. Generalized solitary solution and compacton-like solution of the Jaulent–Miodek equations using the Exp-function method. Phys. Lett. A
**2008**, 372, 1044–1047. [Google Scholar] - Ebaid, A. Exact solitary wave solutions for some nonlinear evolution equations via Exp-function method. Phys. Lett. A
**2007**, 365, 213–219. [Google Scholar] - Zhang, S. Exp-function method for solving Maccari’s system. Phys. Lett. A
**2007**, 371, 65–71. [Google Scholar] - Zhang, S. Application of Exp-function method to a KdV equation with variable coefficients. Phys. Lett. A
**2007**, 365, 448–453. [Google Scholar] - Zhang, S. Exp-function method for constructing explicit and exact solutions of a lattice equation. Appl. Math. Comput.
**2008**, 199, 242–249. [Google Scholar] - Dai, C.Q.; Zhang, J.F. Application of He’s exp-function method to the stochastic mKdV equation. Int. J. Nonlinear Sci. Num. Simul.
**2009**, 10, 675–680. [Google Scholar] - Marinakis, V. The exp-function method and n-soliton solutions. Z. Naturforsch. A
**2008**, 63, 653–656. [Google Scholar] - Zhang, S. Application of Exp-function method to high-dimensional nonlinear evolution equation. Chaos Solitons Fractals
**2008**, 38, 270–276. [Google Scholar] - Ebaid, A. Generalization of He’s exp-function method and new exact solutions for Burgers equation. Z. Naturforsch. A
**2009**, 64, 604–608. [Google Scholar] - Ebaid, A. Exact solutions for the generalized Klein–Gordon equation via a transformation and Exp-function method and comparison with Adomian’s method. J. Comput. Appl. Math.
**2009**, 223, 278–290. [Google Scholar] - Mohyud-Din, S.T.; Khan, Y.; Faraz, N.; Yildirim, A. Exp-function method for solitary and periodic solutions of Fitzhugh–Nagumo equation. Int. J. Numer. Methods H
**2012**, 22, 335–341. [Google Scholar] - Bekir, A.; Aksoy, E. Exact solutions of extended shallow water wave equations by exp-function method. Int. J. Numer. Method. H
**2013**, 23, 305–319. [Google Scholar] - Chai, Y.Z.; Jia, T.T.; Hao, H.Q.; Zhang, J.W. Exp-function method for a generalized mKdV equation. Discrete Dyn. Nat. Soc.
**2014**, 2014, 153974. [Google Scholar] - He, J.H. Exp-function method for fractional differential equations. Int. J. Nonlinear Sci. Num. Simul.
**2013**, 14, 363–366. [Google Scholar] - Yan, L.M. Generalized exp-function method for non-linear space-time fractional differential equations. Therm. Sci.
**2014**, 18, 1573–1576. [Google Scholar] - Malik, S.A.; Qureshi, I.M.; Amir, M.; Malik, AN; Haq, I. Numerical solution to generalized Burgers’–Fisher equation using exp-function method hybridized with heuristic computation. PLoS ONE
**2015**, 10, 1–15. [Google Scholar] - Kudryashov, N.A.; Loguinova, N.B. Be careful with the Exp-function method. Commun. Nonlinear Sci. Numer. Simul.
**2009**, 14, 1881–1890. [Google Scholar] - Aslan, I.; Marinakis, V. Some remarks on exp-function method and its applications. Commun. Theor. Phys.
**2011**, 56, 397–403. [Google Scholar] - Aslan, I. Some remarks on exp-function method and its applications-a supplement. Commun. Theor. Phys.
**2013**, 60, 521–525. [Google Scholar] - Edaid, A. An improvement on the Exp-function method when balancing the highest order linear and nonlinear terms. J. Math. Anal. Appl.
**2012**, 392, 1–5. [Google Scholar] - Zhang, S.; Zhou, Y.Y. Kink-type solutions of the mKdV lattice equation with an arbitrary function. Adv. Mater. Res.
**2014**, 989–994, 1716–1719. [Google Scholar] - Zhang, S.; Zhang, H.Q. Discrete Jacobi elliptic function expansion method for nonlinear differential-difference equations. Phys. Scr.
**2009**, 80, 045002. [Google Scholar]

**Figure 1.**Spatial structures of solution (38) with (+) branch: (a) n ∈ [−10, 10], t ∈ [−10, 10]; (b) n = −10, t ∈ [−10, 10]; (c) n = 0, t ∈ [−10, 10]; (d) n = 10, t ∈ [−10, 10]; (e) n ∈ [−10, 10], t = 0; (f) n ∈ [−10, 10], t = 2.

**Figure 2.**Spatial structures of solutions (39) with (+,+) branch: (a) n ∈ [−10, 10], t ∈ [−10, 10]; (b) n = −10, t ∈ [−10, 10]; (c) n = 0, t ∈ [−10, 10]; (d) n = 10, t = [−10, 10]; (e) n ∈ [−10, 10], t = 0; (f) n ∈ [−10, 10], t = 2.

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**MDPI and ACS Style**

Zhang, S.; Li, J.; Zhou, Y.
Exact Solutions of Non-Linear Lattice Equations by an Improved Exp-Function Method. *Entropy* **2015**, *17*, 3182-3193.
https://doi.org/10.3390/e17053182

**AMA Style**

Zhang S, Li J, Zhou Y.
Exact Solutions of Non-Linear Lattice Equations by an Improved Exp-Function Method. *Entropy*. 2015; 17(5):3182-3193.
https://doi.org/10.3390/e17053182

**Chicago/Turabian Style**

Zhang, Sheng, Jiahong Li, and Yingying Zhou.
2015. "Exact Solutions of Non-Linear Lattice Equations by an Improved Exp-Function Method" *Entropy* 17, no. 5: 3182-3193.
https://doi.org/10.3390/e17053182