A new solitary wave solution of the perturbed nonlinear Schrodinger equation using a Riccati-Bernoulli Sub-ODE method

The Riccati-Bernoulli Sub-ODE method is used for the first time to investigate exact wave solution of the perturbed nonlinear Schrodinger equation with Kerr law non linearity which describes the propagation of optical solitons in nonlinear optical fibers that exhibits a Kerr law non linearity. An infinite sequence of exact solutions can be obtained according to Backlund transformation. The proposed method also can be used for many other nonlinear evolution equations.

This study shall propose a new method which does not depend on the balancing rule, namely the Riccati-Bernoulli Sub-ODE method (Emad and Mostafa, 2015) to seek traveling wave solutions of nonlinear evolution equations and according to a Backlund transformation we can generate infinite sequence of solutions of NLPDEs.

DESCRIPTION OF THE RICCATI-BERNOULLI SUB-ODE METHOD
Consider the following nonlinear evolution equation: (1) Where P is in general a polynomial function of its arguments, the subscripts denote the partial derivatives.The Riccati-Bernoulli Sub-ODE method consists of three steps.
Step 1. Combining the independent variables x and t into one variable: (2)

With (3)
Where the localized wave solution ( ) travels with speed V, by using Equations ( 2) and (3), one can transform Equation (1) to an ODE.
(4) where denotes Step 2. Suppose that the solution of Equation 4 is the solution of the Riccati-Bernoulli equation ( 5) Where a, b, c, and m are constants to be determined later.
From Equation 5and by directly calculating, we get: Remark: When ac ≠ 0 and m = 0, Equation 5 is a Riccati equation.When a ≠ 0, c = 0, and m ≠ 1, Equation 5 is a Bernoulli equation.Obviously, the Riccati equation and Shehata 81 Bernoulli equation are special cases of Equations 5.Because Equation 5 is firstly proposed, we call Equation 5the Riccati-Bernoulli equation in order to avoid introducing new terminology.Equation 5 has solutions as follows: Case 1.When m = 1, the solution of Equation 5 is Case 2. When m ≠ 1, b = 0 and c = 0, the solution of Equation 5 is: Case 3. When m ≠ 1, b ≠ 0 and c = 0, the solution of Equation 5 is Case 4. When m ≠ 1, a ≠ 0 and < 0, the solution of Equation 5 is and Case 5. When m ≠ 1, a ≠ 0 and b2-4ac > 0, the solution of Equation 5 is: and Case 6.When m ≠ 1, a ≠ 0 and b 2 -4ac = 0 the solution of Equation 5 is: Where C is an arbitrary constant.
The Riccati-Bernoulli Sub-ODE method consists of three steps.
Step 1. Combining the independent variables x and t into one variable Step 2. Suppose that the solution of Eq.(2.4) is the solution of the Riccati-Bernoulli equation where a, b, c, and m are constants to be determined later.From Eq.(2.5) and by directly calculating, we get (2.7) Remark: When ac  0and m = 0, Eq.(2.5) is a Riccati equation.When a  0, c = 0, and m  1, ( ,  ,  ,  ,  ,…..) = 0, (2.1) where P is in general a polynomial function of its arguments, the subscripts denote the partial derivatives.
The Riccati-Bernoulli Sub-ODE method consists of three steps.
Step 1. Combining the independent variables x and t into one variable Step 2. Suppose that the solution of Eq.(2.4) is the solution of the Riccati-Bernoulli equation where a, b, c, and m are constants to be determined later.From Eq.(2.5) and by directly calculating, we get (2.7) Remark: When ac  0and m = 0, Eq.(2.5) is a Riccati equation.When a  0, c = 0, and m  1, of the right-hand item of Equation 5 and setting the highest power exponents of u to equivalence in Equation 4, m can be determined.Comparing the coefficients of u i yields a set of algebraic equations for a, b, c, and V. Solving the set of algebraic equations and substituting m, a, b, c, V, and into Equations 8 to 15, we can get traveling wave solutions of Equation 1.

When
and are solutions of Equations ( 5), then Where A_1 and A_2 are arbitrary constants.
According to Equations 21, we can get infinite sequence of solutions of Equation 5 and hence we can get infinite sequence of solutions of Equation 1.

APPLICATION
This equation is well-known (Zhang et al., 2010;Moosaei et al., 2011;Eslami et al., 2013Eslami et al., , 2014;;Mirzazadeh et al., 2014;Biswas and Konar, 2007) and has the form: Where ,, and  is also a version of nonlinear dispersion (Biswas   and Konar, 2007) and (Biswas, 2003).Equation 22describes the propagation of optical solitons in nonlinear optical fibers that exhibits a Kerr law nonlinearity.Equation 22 has been discussed in Moosaei et al. (2011) using the first integral method and in (Zhang et al., 2010) using the modified mapping method and its extended.Let us now solve Equation 22 using Riccati-Bernoulli Sub-ODE method.To this end we seek its traveling wave solution of the form (Zhang et al., 2010;Moosaei et al., 2011): Where , k and   are constants, while Substituting Equation 23into Equation 22and equating the real and imaginary parts to zero, we have: With reference to Zhang et al. (2010), this equation can be rewritten as follows: Where Substituting u'' into Equation 25 we get: Setting m=0 Equation 26 becomes: Setting the coefficient of , 3, 2,1,0 i ui  to zero, we get: Solving Equations ( 28) to (31), we get: Solarity wave solution of Equations ( 34) to (37).According to Equations ( 34) and ( 35) when the parameters take the values (x=-5:5, t=-5:5), the solution represent periodic singular dark soliton; Equation (36) when the parameters take the values (x=-2:2, t=-2:2) and (u=-2:2), the solution represent kink shaped solution; and Equation (37) when the parameters take the values (x=-5:5, t=-5:5), the solution represent multiple soliton solution.
Case A: When 0, a  substituting Equations 32 and 33 into Equations 11 and 12, we obtain the solitary wave solutions: where C, A and B are arbitrary real constants.
Case B. When 0, a  substituting Equations 34 and 35 into Equations 13 and 14, we obtain the solitary wave solutions: If these solutions are substituted into Equation 21, infinite sequence of solutions can be obtained.According to the above results, its physical meaning is compatible with the corresponding physics described by the perturbed nonlinear Schrodinger equation with Kerr law nonlinearity of optical fiber, which in fact, describes it as a soliton wave with one peak such as observed in sound wave (Figure 1).

CONCLUSIONS
In this article, a new technique was introduced to obtain the exact and solitary wave solutions of the perturbed nonlinear Schrodinger equation with Kerr law of Equation ( 34) Equation ( 35) Equation ( 36 nonlinearity which agree with all nonlinear evolutions equations used in mathematical physics and does not depend on the balancing rule which fails for some nonlinear evolution equations.In addition to an infinite sequence of exact and solitary wave solutions can be generated according to a Backlund transformation of the Riccati-Bernoulli equation.It has been shown that the Riccati-Bernoulli Sub-ODE method is a powerful tool for all nonlinear evolution equations.Otherwise, the general solutions of the ODEs have been well known for the researchers.Furthermore, the new method can be used for many other nonlinear evolution equations.