Solitary Waves for the Modified Korteweg-De Vries Equation in Deterministic Case and Random Case

The analytical solution of the linear or nonlinear partial differential equations are so interesting. For this reason, recently, many different methods for finding exact solutions to nonlinear evolution equations have been proposed, developed and extended, such as the Jacobi elliptic function method [1-3], tanh-sech method [4-6], exp-function method [7-9], sine-cosine method [10-12], homogeneous balance method [13,14], F-expansion method [15-17], extended tanh-method [18-21], trigonometric function series method [22], ' ( ) G G − expansion method [23-26]. In this article, we introduced the Riccati-Bernoulli sub-ODE technique, to find exact solutions, solitary wave solutions, of nonlinear partial differential equations. By using our technique and a traveling wave transformation, these equations can be converted into a set of algebraic equations. Solving these algebraic equations we get the exact solutions. For illustrating our method we can discuss the modified Kortewegde Vries (mKdV) equation as test case.

The modified Kortewegâ€"de Vries (mKdV) equation is Where δ is a nonzero constant. The expression u t characterize the time evolution of propagating of the wave in one direction. This equation incorporates also two adversary effects: nonlinearity expression as u 2 u x that accounts in order to seep the wave, and linear dispersion have the form u xx that presented for the spreading of the wave. Nonlinearity leads to localize the wave while propagation prevalences it out, [27].
There are many models have been investigated in the recent decades in physics, chemistry, engineering, and others are formulated as random differential equations (RDEs). In many situations, models with random inputs are better convenient in describing the actual behaviour for the components in the model than deterministic case [32][33][34]. Additionally, this work try to study the Riccat-Bernoulli Sub-ODE technique for solving the Modified Korteweg-de Vries Equation (1.1) where δ is a bounded positive random variable. So the spreading coefficient of the wave is a random variable. Some papers have been studied this equation in random case by adding the noise term [35][36][37], but here we deal with random variable input. If we get a solution of such equations, we can get an infinite solutions sequences of these equations, using a BÃ¤cklund transformation.
The rest of the paper is given as follows: The Riccati-Bernoulli sub-ODE technique is described in Section 4.2.2. In Section 3, a BÃ¤cklund transformation is given. In Section 4 the Riccati-Bernoulli sub-ODE technique is applied to solve the mKdV equation. Finally, Section 5 is devoted to conclusions.

The Riccati-Bernoulli sub-ODE technique
Here we give the description of the Riccati-Bernoulli sub-ODE method. Consider the nonlinear evolution where P is a polynomial in u(x,t) and its partial derivatives with nonlinear terms are embroiled. Now, we give the main steps of this technique [35]: First step : We use the wave transformation where the wave solution u(ξ) propagate with velocity v and c is a positive constant. Using equation (3), one can transform equation (35) into the following ODE: Where H is a polynomial in u(ξ) and its total derivatives, while and so on.
Second step : Let equation (4) has the formal solution in the following form: a, b,c and m are constants calculated later. From equation (5), we easily get Remark 2.1 When ac≠ 0 and m=0 equation (5) is a Riccati equation. When ac≠ 0 c=0 and m≠ 0 equation (5) is a Bernoulli equation. Both equations are special cases of equation (5). Because equation (5) is firstly introduced, we call equation (5) the Riccati-Bernoulli equation.

Classification Riccati-Bernoulli equation solutions
In fact one can easily check the following cases of solutions for the Riccati-Bernoulli equation (5).
Third step : Setting derivatives of u into equation (4) gives an algebraic equation of u. Note that the symmetry of the right-hand item of equation (5) Integrating equation (16) with respect to ξ and after a straight but tedious computation, we get ( ) ( ) Where A 1 and A 2 are constants.
Equation (17) is a Bäcklund transformation of equation (5). By getting a solution of this equation, we can construct infinite sequence of solutions of equation (5) by using equation (17).

Application
Here, we apply Riccat-Bernoulli Sub-ODE technique to solve the mKdV model [38], in the deterministic case and random case as we will discuss.

The mKdV in Deterministic Case
Recall the modified Korteweg-de Vries (mKdV) equation: where δ is a nonzero constant. Using the transformation Equation (18) transforms into the following ODEs, using (19): By integrating equation (20) with respect to ξ once, we have Substituting equations (6) into equation (21), we obtain Setting m=0 equation (22) is reduced to , substituting equations (28)- (30) and (19) into equations (11) and (12), we obtain the exact solutions of equation (18) as follow and 3,4 Where w and δ are arbitrary constants. The solution u 1 is depicted in Figure 1.

The mKdV equation in random case
As we study this method in deterministic case we can development the random case. Since,If we use this method for solving any random differential equation then, the Riccati-Bernoulli sub-ODE solutions are be random variables so, In this section we discuss the stability and convergence theorems for that method.
where P is a polynomial in u(β, x, t) and its partial derivatives such that the highest order derivatives as well as nonlinear terms are also involved, is a random variable input as a coefficient. Now, we give the main stability theorem for our method as follow: where δ is a random variable. By applying our technique the Riccat-Bernoulli Sub-ODE method to the random equation (36) but when δ is a bounded positive random variable i.e., 0 < 1 δ δ ≤ where δ1>0, we can make the same steps like as in the deterministic case until we find the solutions for our problem (36) as follows, When > 0 w δ , substituting equations (28)- (30) and (19) into equations (11) and (12), we can find the random travelling wave solutions of equation (36) as follow,

Conclusion
In this work, the new method, Riccati-Bernolli Sub-ODE is introduced to find the exact traveling wave solutions of the modified Korteweg-de Vries (mKdV) problem in deterministic and random cases. Expectation value of the exact random process have been shown through some random distributions.