New Solutions for ( 1 + 1 )-Dimensional and ( 2 + 1 )-Dimensional Ito Equations

Using the extended F-expansionmethod based on computerized symbolic computation technique, we find several new solutions of 1 1 -dimensional and 2 1 -dimensional Ito equations. These solutions contain hyperbolic and triangular solutions. It is shown that the power of the extended F-expansion method is its ease of use to determine shock or solitary type of solutions. In addition, as an illustrative sample, the properties for the extended F-expansion solutions of the Ito equations are shown with some figures.


Introduction
The nonlinear wave phenomena can be observed in various scientific fields, such as plasma physics, optical fibers, fluid dynamics, and chemical physics.The nonlinear wave phenomena can be obtained in solutions of nonlinear evolution equations NEEs .The study of NLEEs appear everywhere in applied mathematics and theoretical physics including engineering sciences and biological sciences.These NLEEEs play a key role in describing key scientific phenomena.For example, the nonlinear Schr ödinger's equation describes the dynamics of propagation of solitons through optical fibers.The Korteweg-de Vries equation models the shallow water wave dynamics near ocean shore and beaches.Additionally, the Schr ödinger-Hirota equation describes the dispersive soliton propagation through optical fibers.These are just a few examples in the whole wide world of NLEEs and their applications, see, for instance, 1-4 .While the above mentioned NLEEs are scalar NLEEs, there is a large number of NLEEs that are coupled.Some of them are two-coupled NLEEs such as the Gear-Grimshaw equation 2 , while there are several others that are three-coupled NLEEs.An example of a three-coupled NLEE is the Wu-Zhang equation 4 .These coupled NLEEs are also studied in various areas of theoretical physics as well.
The exact solutions of these NEEs play an important role in the understanding of nonlinear phenomena.In the past decades, many methods were developed for finding exact solutions of NEEs such as the inverse scattering method 5, 6 , improved projective Riccati equations method 7, 8 , Cole-Hopf transformation method 9 , exp-function method 10-16 , bifurcation theory method 17 , G /G -expansion method 18, 19 , homotopy perturbation method 20 , tanh function method 20-24 , and Jacobi and Weierstrass elliptic function method 25, 26 .Although  have obtained some exact periodic solutions to some nonlinear wave equations, they use the Weierstrass elliptic function and involve complicated deducing.A Jacobi elliptic function JEF expansion method, which is straightforward and effective, was proposed for constructing periodic wave solutions for some nonlinear evolution equations.The essential idea of this method is similar to the tanh method by replacing the tanh function with some JEFs such as sn, cn, and dn.For example, the Jacobi periodic solution in terms of sn may be obtained by applying the sn-function expansion.Many similarl repetitious calculations have to be done to search for the Jacobi doubly periodic wave solutions in terms of cn and dn 30 .
Recently, F-expansion method 31-34 was proposed to obtain periodic wave solutions of NLEEs, which can be thought of as a concentration of JEF expansion since F here stands for every function of JEFs.The objectives of this work are twofold.First, we seek to extend others works to establish new exact solutions of distinct physical structures for the nonlinear equations 1.1 and 1.2 .The extended F-expansion EFE method will be used to achieve the first goal.The second goal is to show that the power of the EFE method is its ease of use to determine shock or solitary type of solutions.In this paper, we study two wellknown PDEs, namely, generalized 1 1 -dimensional and generalized 2 1 -dimensional Ito equations.Many studies are concerning the 1 1 -dimensional Ito equation and the 2 1dimensional Ito equation 35-42 .
The history of the KdV equation started with experiments by John Scott Russell in 1834, followed by theoretical investigations by Lord Rayleigh and Joseph Boussinesq around 1870, and, finally, Korteweg and de Vries in 1895 39 .The KdV equation was not studied much after this until Zabusky and Kruskal 1965 40 discovered numerically that its solutions seemed to decompose at large times into a collection of "solitons": well-separated solitary waves.Ito 41, 42 obtained the well-known generalized 1 1 -dimensional and generalized 2 1 -dimensional Ito equations by generalization of the bilinear KdV equation as In this paper, we extend the EFE method with symbolic computation to 1.1 and 1.2 for constructing their interesting Jacobi doubly periodic wave solutions.It is shown that soliton solutions and triangular periodic solutions can be established as the limits of Jacobi doubly periodic wave solutions.In addition, the algorithm that we use here is also a computerized method, in which we are generating an algebraic system.

Extended F-Expansion Method
In this section, we introduce a simple description of the EFE method, for a given partial differential equation as G u, u x , u y , u z , u xy , . . .0.

2.1
We like to know whether travelling waves or stationary waves are solutions of 2.
where  O U q d p U dζ p q 1 N p, q 0, 1, 2, . . ., p 1, 2, 3, . . ., 2.7 where A, B, and C are constants, and N in 2.3 is a positive integer that can be determined by balancing the nonlinear term s and the highest order derivatives.Normally N is a positive integer, so that an analytic solution in closed form may be obtained.Substituting 2.1 -2.5 into 2.3 and comparing the coefficients of each power of F ζ in both sides, we will get an overdetermined system of nonlinear algebraic equations with respect to ν, a 0 , a 1 , . ... We will solve the over-determined system of nonlinear algebraic equations by use of Mathematica.
When m → 1. the Jacobi functions degenerate to the hyperbolic functions, that is, when m → 0, the Jacobi functions degenerate to the triangular functions, that is,

Generalized (1+1)-Dimensional Ito Equation
We first consider the generalized 1 1 -dimensional Ito equation 1.1 as follows: where by integrating once we obtain, upon setting the constant of integration to zero, if we use the transformation W V , then 3.4 can be written as follows: Balancing the term W with the term W 2 we obtain N 2 then Substituting 3.6 into 3.5 and comparing the coefficients of each power of ψ in both sides, we will get an over-determined system of nonlinear algebraic equations with respect to ν, a i , i 0, 1, −1, −2, 2. Solving the over-determined system of nonlinear algebraic equations by use of Mathematica, we obtain three groups of constants 1 3.9 The solutions of 3.1 are 3.17 0.5m 2 0.5 t

3.39
Mathematical Problems in Engineering 11

Soliton Solutions
Some solitary wave solutions can be obtained, if the modulus m approaches to 1 in 3.10 -3.39 as follows: 3.40

Triangular Periodic Solutions
Some trigonometric function solutions can be obtained, if the modulus m approaches to zero in 3.10 -3.39 as follwos:

3.41
The modulus of solitary wave solutions u 1 , u 2 , u 21 , and u 23 is displayed in

Generalized (2+1)-Dimensional Ito Equation
In this section we consider the generalized 2 1 -dimensional Ito equation 1.2 as follows: where by integrating twice we obtain, upon setting the constant of integration to zero, Balancing the term W with the term W 2 , we obtain N 2, then Proceeding as in the previous case, we obtain 1 4.9 The solutions of 4.1 are 0.5m 2 0.5 4.10

4.12
If we take m → 1, in the two Sections 3 and 4, we obtain the solutions degenerated by the hyperbolic extended hyperbolic functions methods tanh, coth, sinh, sech,. .., etc. see, for example 42 .Moreover, when m → 0, the solutions obtained by triangular and extended triangular functions methods tan, sine, cosine, sec,. .., etc. are found as disused in Sections 3.1, 3.2, 4.1, and 4.2.

Conclusion
By introducing appropriate transformations and using extended F-expansion method, we have been able to obtain, in a unified way with the aid of symbolic computation systemmathematica, a series of solutions including single and the combined Jacobi elliptic function.Also, extended F-expansion method showed that soliton solutions and triangular periodic solutions can be established as the limits of Jacobi doubly periodic wave solutions.When m → 1, the Jacobi functions degenerate to the hyperbolic functions and give the solutions by the extended hyperbolic functions methods.When m → 0, the Jacobi functions degenerate to the triangular functions and give the solutions by extended triangular functions methods.In fact, the disadvantage of extended F-expansion method is the existence of complex solutions which are listed here just as solutions.

Figure 1 :
Figure 1: The modulus of solitary wave solution u 1 3.10 where m 0.5.

2 0
5 0.5m 2 2 − 3 0.5 − 0.5m 2 0.25 − 0.25m Moreover, the Ito-type coupled KdV ItcKdV equation 44 , written in the following form: SK-Ito equation is characterized by the presence of three dispersive terms u x , u 3x , and u 7x , respectively.SK-Ito seventh-order equation is completely integrable and admits of conservation laws 43 .

Table 1 :
Relation between values of A, B, C and corresponding F.
Some solitary wave solutions can be obtained, if the modulus m approaches to 1 in 4.10 as follows: 2.