Applications of He's semi-inverse variational method and ITEM to the nonlinear long-short wave interaction system

This work deals with exact soliton solutions of the nonlinear long-short wave interaction system, utilizing two analytical methods. The system of coupled long-short wave interaction equations is studied by two analytical methods, namely, the generalized tan ( ϕ/ 2)-expansion method and He’s semi-inverse variational method, based upon the integration tools. Moreover, in this paper, we generalize two aforementioned methods which give new soliton wave solutions. Abundant exact traveling wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play an important role in engineering and physics fields. By using these methods, exact solutions including the hyperbolic function solution, traveling wave solution, soliton solution, rational function solution, and periodic wave solution of this equation have been obtained. In addition, by using Matlab, some graphical simulations were done to see the behavior of these solutions.

The nonlinear long-short wave interaction systems with considering a general theory for interactions between short and long waves first introduced by Benney (1977). Describes of the nonlinear resonance interaction of multiple short waves with a long wave in two spatial dimension by considering a general multi-component (2 + 1)dimensional long-wave-shortwave resonance interaction system with arbitrary nonlinearity coefficients have been investigated by Sakkaravarthi et al. (2014) by applying the Hirota (1985) bilinearization method. The entangled mapping approach based on the general reduction theory was investigated by Dai and Liu (2012), in which they have derived new type of variable separation solution for the (2 + 1)-dimensional long wave short wave interaction model. By utilizing the first integral method obtained one-soliton solutions and also by help aforesaid method is used to construct exact solutions of the nonlinear long-short wave resonance equations (Jafari et al., 2015). Apart from this, study on the long-short-wave interaction system by utilizing (G′/G)-expansion method was also carried out in Bekir et al. (2013). Triki et al. (2015) studied the long-wave short-wave interaction equation by help the simplest equation approach also obtained soliton solutions as well as other solutions such as singular periodic solutions and plane waves. Later on, the nonlinear long-short wave interaction system was studied by investigating the transverse linear instability of one-dimensional solitary wave solution (Erbay and Erbay, 2012). Dias et al. (2010), proof of the global existence and uniqueness of the solution of the Cauchy problem and also proof of the convergence of the whole sequence of solutions have been studied. Finally, by applying the new modified exp (−Ω (ξ))-expansion method sets of solutions including, hyperbolic, complex, and dark soliton solutions have obtained in Baskonus et al. (2017).
It has been discovered that many models in mathematics and physics are described by nonlinear Partial differential equations. Indeed modeling physical problems using partial differential equations with the exact parameters is not always easy but also impossible in the real problems. For this purpose, one way is using integration methods for finding the exact solutions. One of the most recent approaches is using numerical methods including the multiresolution analysis (Seyedi et al., 2015), the multi-scale analysis (Seyedi et al., 2018), semi-analytical methods (Dehghan and Manafian, 2009;Dehghan et al., 2010;Rashidi et al., 2013) or analytical methods (Manafian, 2015;2018;Foroutan et al., 2018;Sendi et al., 2019;Dehghan et al., 2011a;2011b;Manafian and Lakestani, 2015a;2015b;2015c;Biswas, 2009;Bekir and Aksoy, 2012;Manafian and Lakestani, 2016a;2016c;Manafian et al., 2016a;Aghdaei and Manafian, 2016). Also, of applied methods for solving nonlinear partial differential equation is He's semiinverse variational principle, introduced by He (2006). For further information see references Kohl et al. (2009), Zhang (2007, Biswas et al. (2012aBiswas et al. ( , 2012b, Sassaman et al. (2010). So instead of using current models of partial differential equations, we can transfer PDEs to ordinary differential equations. Hence there occurs a need to use solitary wave variable that would appropriately transforms PDEs to ODEs and solve them. In recent decade, exact solutions of nonlinear differential equations have been attracted attention from all over the world. Therefore, some newly published papers can be pointed to new exact solutions in new works in which given in Refs. (Cattani et al., 2018a;Sulaiman et al., 2018;Baskonus et al., 2018b;Ciancio et al., 2018;Baskonus, 2016;2017;Baskonus and Cattani, 2018).
In this paper, a novel and high accuracy method based on the classical Galerkin method proposed by Seyedi et al. (2018). They used Alpert Wavelet basis in the spectral methods and could solve the nanofluid problems with high accuracy. Using the integration methods, we construct two analytical methods for Eq. 1.1, give corresponding algebraic equations, and show the efficiency of these schemes by the applied equation. Compared with some existed results, these methods are especially well designed for the solution of PDEs as particular the nonlinear long-short wave interaction system. The aim of this paper is to obtain analytical solutions of the aforementioned equation, and to determine the accuracy of these methods in solving this equation. The rest of the Paper is organized as follows: In Section 2, we present the He's semi-inverse variational principle method and the improved tan (ϕ/2)-expansion method. In Section 3, we use transformations for converting the nonlinear longshort wave interaction system to an ODE form. In Section 4, by help of methods applied in section 2 we drive new soliton wave solutions for the nonlinear long-short wave interaction system. Moreover, in Section 5, we give the simulation and discussion of the solutions with depicting figures. Also conclusion is given in Section 6.

The He's semi-inverse variational principle method
We describe the He's semi-inverse variational principle method for the given partial differential equation. First we give a description of this method, by noting the following steps: Step 1: We suppose that given nonlinear partial differential equation for u(x, t) to be in the form by the transformation ξ = kx + wt, as wave variable. Also, μ is constant to be determined later.
Step 2: According to He's semi-inverse method, we construct the following trial-functional where L is an unknown function of U and its derivatives.
Step 3: By the Ritz method, we can obtain different forms of solitary wave solutions, such as and so on. For example in this paper, we search a soliton solution in the form

Description of the ITEM
The ITEM is well-known analytical method which was improved and developed by Sendi et al. (2019).
Step 3: Determine m. This, usually, can be accomplished by balancing the linear term(s) of highest order with the highest-order nonlinear term(s) in Eq. 2.10. But, the positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms appearing in Eq. 2.10. If m = q/p (where m = q/p be a fraction in the lowest terms), we let Step 4: Substituting (2.11) into Eq. 2.10 with the value of m obtained in Step 2. Collecting the coefficients of tan (ϕ/2)k, cot (ϕ/2)k(k = 0, 1, 2, ...), then setting each coefficient to zero, we can get a set of over-determined equations for A0,Ak,Bk(k = 1, 2, ...,m) a, b, c and p with the aid of symbolic computation Maple.

The LSWI systems
In this paper, we consider the nonlinear longshort wave interaction systems  in the form Combine the real variables x and t by a compound variable ξ If we take the necessary derivations of Eq. 3.2 for Eq. 3.1, then we get the following nonlinear ODEs, Consider the complex part of Eq. 3.3 to zero, will obtain = −2 .
(3.5) By integrating Eq. 3.4 and considering Eq. 3.5, we get to (3.6) Now, when we substitute Eqs. 3.5 and 3.6 into Eq. 3.3, we obtain the NODE as 4. Test problems
By using of transformations of (3.1) and (4.38), we can obtain the following complex dark solutions for Eq. 3.1 as By using of transformations of (3.1) and (4.41), we can obtain the following complex dark solutions for Eq. 3.1 as By using of transformations of (3.1) and (4.50), we can obtain the following complex dark solutions for Eq. 3.1 as By using of transformations of (3.1) and (4.53), we can obtain the following complex dark solutions for Eq. 3.1 as

Simulation and discussion of the solutions
In this section, the numerical simulations of the the nonlinear long-short wave interaction system will be given. Now, we will discuss all possible physical significance for each parameter. By utilizing the balance principle, one can found m = 1, therefore we can write other following equations: ( ) = 0 + 1 [ + tan( /2)] + 1 [ + tan( /2)] −1 , (5.1) ′ ( ) = 1 2 ( /2) − 1 2 ( /2)[ + tan( /2)] −2 , (5.2) ′′ ( ) = 2 1 tan( /2) 2 ( /2) − 2 1 tan( /2) 2 ( / 2) [ + tan( /2)] −2 + (5.3) 2 1 4 ( /2)[ + tan( /2)] −2 where A1 ̸ = 0 and B1 ̸ = 0. When we use Eqs. 5.1 to 5.3 in Eq. 3.7, we get a system of algebraic equations from the coefficients of polynomial of tan (ϕ/2). By solving this system of algebraic equations via Maple 13 software, we can find other different style analytical solutions which can be obtained by using ITEM. We have also obtained the dark, bright and singular soliton solutions of the nonlinear long-short wave interaction system (3.1) by using He's semiinverse variational method and briefly studied their behavior dynamics. Moreover, by utilizing the ITEM, can found the exact particular solutions containing four types hyperbolic function solution (exact soliton wave solution), trigonometric function solution (exact periodic wave solution), rational exponential solution (exact singular kink-type wave solution) and rational solution (exact singular cupson wave solution). It can be said the ITEM has further merit comparing with other methods. This study will find analytical applications in nonlinear sciences, particularly in the literature we refer to the circular functions, the gravitational potential of a cylinder (Weisstein, 2002), the profile of a laminar jet (Weisstein, 2002), the Langevin function for magnetic polarization (Weisstein, 2002), the longitudinal waves such as in sound, pressure waves and musical instruments waves. In Figs. 1-12, we plot two and three dimensional graphics of absolute values of (4.33), (4.34), (4.36), (4.37), (4.54) and (4.55) by means of Section 4.2, which denote the dynamics of solutions with appropriate parametric selections. Likewise, after comparing these analytical solutions obtained via He's semi-inverse variational method and ITEM with solutions obtained by authors of (Bekir et al., 2013;Baskonus et al., 2017;Khater et al., 2010), and to the best of our current state of knowledge, we think that complex hyperbolic function, trigonometric function and rational function solutions may have been obtained here for the first time, in the literature.