On the exact solutions of nonlinear evolution equations by the improved $$\tan (\varphi /2)$$
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© Indian Academy of Sciences
On the exact solutions of nonlinear evolution equations by the improved tan(ϕ/2)-expansion method† ˘ YESIM ¸ SAGLAM ÖZKAN
∗
and EMRULLAH YASAR ¸
Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, 16059 Bursa, Turkey ∗ Corresponding author. E-mail: [email protected] MS received 29 May 2019; revised 23 September 2019; accepted 1 October 2019 Abstract. In this paper, the improved tan(ϕ/2)-expansion method (ITEM) is proposed to obtain more general exact solutions of the nonlinear evolution equations (NLEEs). This method is applied to the generalised Hirota– Satsuma coupled KdV (HScKdV) equation and (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) system. We have obtained four types of solutions of these equations such as hyperbolic, trigonometric, exponential and rational functions as an advantage of this method. These solutions include solitons, rational, periodic and kink solutions. Moreover, modulation instability is used to establish stability of the obtained solutions. Keywords. Improved tan(ϕ/2)-expansion method; generalised Hirota–Satsuma coupled KdV equation; (2 + 1)dimensional Nizhnik–Novikov–Veselov system. PACS Nos 04.20.Jb; 02.30.Jr
1. Introduction The nonlinear evolution equations (NLEE) are useful because many ideas in NLEEs are more easily understood in terms of simpler equations which model more complicated systems in various aspects. NLEEs appear not only in applied mathematics but also in theoretical physics. The general solutions of these types of equations in areas such as engineering, biology, chemistry, finance and mechanics, help scientists to obtain quite favourable information about the character of equations. Thus, these equations are critical in solving real-world problems and it is important to reach general solutions to make sense of this physical phenomenon. Following the progress on computeraided calculations and the development of non-linear sciences based on algebraic systems, the application fields of NLEEs have also expanded. Examples of these fields include: optical fibres [1], fluid dynamics and condensed matter physics [2], plasma physics [3] and so on. Due to the efficiency, reliability, and ease of use of symbolic software packages such as
† Dedicated
to Prof. Mehmet Cagliyan on the occasion of his 70th birthday.
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Mathematica or Maple, many powerful techniques such as the (G /G)-expansion method [4,5], the simplest equation method [6], the Jacobi elliptic function method [7,8], the homotopy perturbation method [9,10], the variational iteration method [11], the sine–cosine method [12], the tanh–coth method [13,14], the expfunction method [15], the homogeneous balance method [16], first integral methods [17,18], the Lie symmetry method [19], exp(−(ξ )) expansion method [20], the Hirota bilinear method [21,22] and so on have been constructed and developed. All these methods are effective methods to obtain travelling wave solutions of NLEEs. Several important results have emerged for the twod
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