Bright and singular soliton solutions of Tzitzéica type equations appear in nonlinear optics

We investigate the exact travelling wave solutions of the Tzitzéica-Dodd–Bullough–Mikhailov (TDBM) and Tzitzéica-type (TT) equations, appear in nonlinear optics, using simple ansatz approach. We use the Painlevé and traveling wave transformations to convert the mentioned nonlinear equations into nonlinear ordinary differential equations. Then, the simple ansatz approach has been employed to procure wave solutions, in terms of bright, singular soliton and trigonometric functions, for the TzitzéicaDodd–Bullough–Mikhailov (TDBM) and Tzitzéica-type (TT) equations. The study shows that the simple ansatz approach is straightforward and easy to implement for solving nonlinear problems.


Introduction
Evolution equations with nonlinear terms that appear in optics consist of the Tzitzéica-Dodd-Bullough-Mikhailov(TDBM), Tzitzéica-Dodd-Bullough(TDB), and the Liouville equations.The nonlinearities in the equations in this class include exponential functions.Consider the Tzitzéica-Dodd-Bullough-Mikhailov equation firstly defined in Tzitzéica's study [1] in 1908.In the following years a variational form of the (1) was used in various studies as in [2,3] u tt + u xx + e u + e −2u = 0 (2) named Tzitzéica-type(TT) equation.The conditions of the existence of polynomial conserved densities for the TDBM equation (1) in terms of derivatives of u were listed in [4].Bäcklund transformation for the TDBM equation from the classification of Painleve-Gambier was given in details by Conte et al [5].Zhu and Geng presented Darboux transformation for the 3 × 3-matrix spectral TDBM equation problem and derived some explicit solutions [6].The inverse problem was solved and N -soliton solutions for the TDBM were found besides infinitely many conservation laws were defined for the Toda chain system equations in two dimensions [7].One should note that Mikailov [7] claimed Conte et al. [5]'s suggestion on the polynomial conserved densities was not true.Some logarithmic and exponential function type solutions to the TDBM (1) were explicitly constructed by some Fourier series approach by El-Kalaawy [8].The modification of the simple equation approach was also used to set the traveling wave solutions in terms of logarithmic functions with hyperbolic or trigonometric parameters to the TDBM (1) by employing two variable transformations to it [9].Blow-up and broken singular traveling wave solutions covering periodic, solitary wave, kink and unbounded wave forms were successfully given with their plots for both the TDBM (1) and the TT (2) equations by integral bifurcation approach [2].Soliton and periodic solutions were derived by the hyperbolic tangent method together with the Painleve property to both the TDBM (1) and the TT (2) equations [3].Parallel developments in both computer technologies and symbolic softwares have greatly contributed to solve lots of problems defined in various fields covering applied mathematics, physics and many engineering fields.A diverse class of effective methods have successfully been introduced to study this class of equations, for example [3,[10][11][12][13][14][15].On the other hand some of the commonly used approaches, for solving nonlinear evolution equations, are: The ansatz [16][17][18], modified simple equation [19], the first integral [20,21], ( G G )-expansion [22], sine-Gordon expansion [23,24].Furthermore, some other excellent works like Kudryashov methods [25], a modified form of Kudryashov and functional variable methods [26][27][28] have been done by different researchers.In [29][30][31][32], the auxiliary equation, the improved tan( φ(η) 2 )-expansion methods and the exp function approach have been explored for discrete and fractional order PDEs as well.Ali and Hassan [33], Hosseini et al. [34] and Zayed and Al-Nowehy [35] all have utilized the exp a function method to explore the exact solutions of nonlinear partial differential equations.A useful study of hyperbolic function approach has been given in [36][37][38].
The investigation for exact solutions is as follows.The description of simple ansatz approach is provided in section 2. The method is utilized to extract new exact solutions of nonlinear Tzitzéica type equations in section 3. The conclusion is provided in the last section.

A simple ansatz approach
The present section provides a succinct description of the simple ansatz approach [39].For this purpose, suppose that a nonlinear PDE can be read as The PDE (3) can be changed into the following ODE by the classical wave transform given below in which k, l are non-zero constants.Then, we assume the solution of the form given by, where, α 0 , α 1 , α 2 , ..., α N are constant parameters and N is positive natural number that will be found by the principle of homogeneous balance.ψ(ξ) is deliberately chosen to be csch(ξ) and sech(ξ) to determine hyperbolic type solution while sec(ξ) and csc(ξ) are periodic solutions, respectively.

Solutions for the TDBM equation
The DTDBM equation is read as and using Painlevé transform v = e u or u = ln(v), the Eq. ( 7) reduces to: Eq.( 8) is reduced to the following ODE by using v(x, t) = V (ξ), ξ = kx + lt: The balance between the terms V V with V 3 implies N = 2.
By inserting Eq.( 10) with its derivatives in Eq.( 9) and setting each coefficient of powers of csch(ξ) to zero, one obtains a nonlinear system of algebraic equations.The solution of this system gives We have the following hyperbolic function solutions For more hyperbolic function solution, we take ψ(ξ) = sech(ξ) By inserting Eq.( 14) into Eq.(9), and gather the terms with the same powers of sech(ξ).Then, by putting each coefficient of sech(ξ) equal to zero, we get a set of nonlinear algebraic equations.On solving, we procure the following set of solutions Thus, substitution of the above values into the Eq. ( 7), the solutions are set as follows:

Periodic solutions for the TDBM equation
For periodic solutions, we start as follows: Setting coefficients of power of sec(ξ) to zero yields an algebraic system of equations.Thus, the solution gives the following trigonometric function solutions For more periodic solutions, we take the following non-trivial solution: Inserting Eq.( 22) and its derivatives into Eq.( 9) gives Thus the following trigonometric function solutions are obtained.
trigonometric function solutions of the TDBM and the TT equations through symbolic soft computation in Mathematica.Furthermore, these solutions might have significant importance in science and engineering fields.The study also showed that method is simple and easy to implement. u