New Soliton Solutions for the Higher-Dimensional Non-Local Ito Equation


 In this article, (2+1)-dimensional Ito equation that models waves motion on shallow water surfaces is analyzed for exact analytic solutions. Two reliable techniques involving the simplest equation and modified simplest equation algorithms are utilized to find exact solutions of the considered equation involving bright solitons, singular periodic solitons, and singular bright solitons. These solutions are also described graphically while taking suitable values of free parameters. The applied algorithms are effective and convenient in handling the solution process for Ito equation that appears in many phenomena.

This article is prepared as follows: The algorithms of SEM and MSEM are presented in section 2. By applying the proposed techniques, the Logarithmic form of obtained exact solutions for Eq. (1.1) is studied in section 3. Results' discussion including simulations of some obtained solutions is provided in section 4.

Mathematical Analysis
In this section, SEM and MSEM achieve the exact analytic solutions of constant-coe cients (2+1)-D PDE of the following form: where P is an assumed polynomial in v = v (x, y, t) and its derivatives involving the highest derivative and the higher power of linear terms. The solution process is also valid for the time-dependent coe cients and NPDEs' systems.
To investigate Eq. (2.1), we transform it into an ODE under the transformation: where ξ = x + y − η t is the wave variable and η is the wave frequency. In case of time-dependent, one can use 3) several times as possible as we can and setting the integration's constants to . The transformed equation is reduced and kept the solution process very simple.
From the above schemes, positive integer m is determined by balancing the highest order derivative and the nonlinear term of the highest order in the completely integrated form of Eq. (2.3). As a result, each method's solution process is discussed.

. The Method of Simplest Equation
The Eq. (2.3) solution via the SEM [26,27] is expressed as: The solution function ϕ (ξ ) is expressed as: I. If λ = , we have the rational form: II. If λ > and µ < , we have the rational-exponential form: III. If λ < and µ > , we have the rational-exponential form: Whereas in the case Riccati equation 9) and the solutions of Eq. (2.9) can be expressed in the following forms: I. If λ µ < , we have the hyperbolic form: II. If λ µ > , we have the periodic form: 12) or,

. Modi ed Simple Equation Method
The MSEM [28] considers the Eq. (2.1) solution as follows: where A i (i = , , . . . , m) are parameters to be determined later. Positive integer m is found by the homogeneous balance principle. ϕ (ξ ) is an unspeci ed function to be determined later. Once Eq. (2.14) is substituted into Eq. (2.3), an algebraic equations' system, which can be an algebraic-di erential system, is resulted. When the constructed system's numerator is forced to be vanished and substituted our results into Eq. (2.14), the exact solution is determined for the studied problem.

The Ito Equations' Applications
The (2+1)-D non-local Ito equation is investigated via the Kudryashov SEM, which was discussed in the previous section. With the application of the transformation given in Eq. (2.2), Eq. (1.2) will be carried into the following ODE form: In further compact form, Eq. (3.1) can be expressed as: By the twice integrartion Eq. (3.2) w.r.t ξ and let the integration's constants to be , we have: Let w (ξ ) = v (ξ ) to get: The balance is made between w and w in Eq. (3.4) which m = is obtained.

. Application of SEM
As a result, Eq. (3.5) has a solution as: By the substitution of Eq. (3.5) into Eq. (3.4), and from the Bernoulli Eq. (2.5), as well as, let us set up the coe cients of ϕ i , i = , , . . . , , to be zero,the following system is constructed in terms of A , A , A , λ, µ and η:

Application of MSEM
The MSEM is applied for Eq. (1.2) with m = , Eq. (3.4) constructs a solution in the form: It is simple to nd that   (3.38) where C and C are arbitrary constants. A is nonzero arbitrary constant.        1). The proposed techniques can be applied to various related NPDEs via Mathematica symbolic computation package 11. The considered can be investigated by using di erent de nitions of fractional derivatives. Also, solitonic, super nonlinear, periodic, quasi-periodic, and chaotic waves can be done in future works.