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

Abstract

Using the extended F-expansion method 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.

1. 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 Schrödinger’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 Schrödinger-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 Porubov et al. [27–29] 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 well-known 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+1)-dimensional 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 𝑢_𝑡𝑡+𝑢_{𝑥𝑥𝑥𝑡}+3𝑢_𝑥𝑢_𝑡+𝑢𝑢_𝑥𝑡+3𝑢_𝑥𝑥^𝑥_∞𝑢_𝑡𝑑𝑥^′𝑢=0,_𝑡𝑡+𝑢_{𝑥𝑥𝑥𝑡}+32𝑢_𝑥𝑢_𝑡+𝑢𝑢_𝑥𝑡+3𝑢_𝑥𝑥^𝑥_∞𝑢_𝑡𝑑𝑥^′+𝛼𝑢_𝑦𝑡+𝛽𝑢_𝑥𝑡=0. (1.1)(1.2) Also Sawada-Kotera-Ito (SK-Ito) seventh-order equation is the special case of the generalized seventh-order KdV equation as 𝑢_𝑡+252𝑢²𝑢_𝑥+63𝑢_3𝑥+378𝑢𝑢_𝑥𝑢_2𝑥+126𝑢²𝑢_3𝑥+63𝑢_2𝑥𝑢_3𝑥+42𝑢_𝑥𝑢_4𝑥+21𝑢𝑢_5𝑥+𝑢_7𝑥=0. (1.3) 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]. Moreover, the Ito-type coupled KdV (ItcKdV) equation [44], written in the following form: 𝑢_𝑡+𝛼𝑢𝑢_𝑥+𝛽𝑣𝑣_𝑥+𝛾𝑢_𝑥𝑥𝑥=0;𝑣_𝑡+𝛽(𝑢𝑣)_𝑥=0, (1.4) if we take the special values 𝛼=−6,  𝛽=−2, and 𝛾=−1. Equation (1.4) describes the interaction process of two internal long waves which has infinitely many conserved quantities [45, 46].

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.

2. Extended F-Expansion Method

In this section, we introduce a simple description of the EFE method, for a given partial differential equation as 𝐺𝑢,𝑢_𝑥,𝑢_𝑦,𝑢_𝑧,𝑢_𝑥𝑦,…=0. (2.1) We like to know whether travelling waves (or stationary waves) are solutions of (2.1). The first step is to unite the independent variables 𝑥,  𝑦, and 𝑡 into one particular variable through the new variable as 𝜁=𝑥+𝑦−𝜈𝑡,𝑢𝑥,𝑦,𝑡=𝑈(𝜁), (2.2) where 𝜈 is wave speed, and reduce (2.1) to an ordinary differential equation (ODE) as 𝐺𝑈,𝑈^′,𝑈^′′,𝑈^′′′,…=0. (2.3) Our main goal is to derive exact or at least approximate solutions, if possible, for this ODE. For this purpose, let us simply use 𝑈 as the expansion in the form 𝑢𝑥,𝑦,𝑡=𝑈(𝜁)=^𝑁_𝑖=0𝑎_𝑖𝐹^𝑖+^𝑁_𝑖=1𝑎_−𝑖𝐹^−𝑖, (2.4) where 𝐹^′=𝐴+𝐵𝐹²+𝐶𝐹⁴, (2.5) the highest degree of (𝑑^𝑝𝑈/𝑑𝜁^𝑝), is taken as 𝑂𝑑^𝑝𝑈𝑑𝜁^𝑝𝑂𝑈=𝑁+𝑝,𝑝=1,2,3,…,_𝑞𝑑^𝑝𝑈𝑑𝜁^𝑝=𝑞+1𝑁+𝑝,𝑞=0,1,2,…,𝑝=1,2,3,…, (2.6)(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,𝑎₀,𝑎₁,…. We will solve the over-determined system of nonlinear algebraic equations by use of Mathematica. The relations between values of 𝐴,  𝐵,  𝐶, and corresponding JEF solution 𝐹(𝜁) of (2.4) are given in Table 1. Substituting the values of 𝐴,  𝐵,  𝐶, and the corresponding JEF solution 𝐹(𝜁) chosen from Table 1 into the general form of solution, then an ideal periodic wave solution expressed by JEF can be obtained.

Table 1 

Relation between values of (𝐴,𝐵,𝐶) and corresponding 𝐹.

sn(𝜁), cn(𝜁), and dn(𝜁) are the JE sine function, JE cosine function, and the JEF of the third kind, respectively. And cn²(𝜁)=1−sn²(𝜁),dn²(𝜁)=1−𝑚²sn²(𝜁), (2.8) with the modulus 𝑚(0<𝑚<1).

When 𝑚→1. the Jacobi functions degenerate to the hyperbolic functions, that is, sn𝜁→tanh𝜁,cn𝜁→sech𝜁,dn𝜁→sech𝜁, (2.9) when 𝑚→0, the Jacobi functions degenerate to the triangular functions, that is, sn𝜁→sin𝜁,cn𝜁→cos𝜁,dn→1. (2.10)

3. Generalized (1+1)-Dimensional Ito Equation

We first consider the generalized (1+1)-dimensional Ito equation (1.1) as follows: 𝑢_𝑡𝑡+𝑢_{𝑥𝑥𝑥𝑡}+3𝑢_𝑥𝑢_𝑡+𝑢𝑢_𝑥𝑡+3𝑢_𝑥𝑥^𝑥_∞𝑢_𝑡𝑑𝑥^′=0, (3.1) if we use the transformation 𝑢=𝑣_𝑥, it carries (3.1) into 𝑣_𝑥𝑡𝑡+𝑣_{𝑥𝑥𝑥𝑥𝑡}+3𝑣_𝑥𝑥𝑣_𝑥𝑡+𝑣_𝑥𝑣_𝑥𝑥𝑡+3𝑣_𝑥𝑥𝑥𝑣_𝑡=0, (3.2) if we use 𝜁=𝑥−𝜈𝑡 transforms (3.2) into the ODE, we have −𝜈𝑉^′′′+𝑉^(𝑣)−3𝜈𝑉^′′𝑉^′′+𝑉^′𝑉^′′′−3𝜈𝑉^′′′𝑉^′=0, (3.3) where by integrating once we obtain, upon setting the constant of integration to zero, 𝑉^′′′+3𝑉^′2−𝜈𝑉^′=0, (3.4) if we use the transformation 𝑊=𝑉^′, then (3.4) can be written as follows: 𝑊^′′+3𝑊²−𝜈𝑊=0. (3.5) Balancing the term 𝑊^′′ with the term 𝑊² we obtain 𝑁=2 then 𝑊(𝜁)=𝑎₀+𝑎₁𝜓+𝑎₋₁𝜓⁻¹+𝑎₂𝜓²+𝑎₋₂𝜓⁻²,𝜓^′=𝐴+𝐵𝜓²+𝐶𝜓⁴. (3.6)

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) 𝑎₋₁=𝑎₋₁=0,𝑎₀=−2𝐵3,𝑎₂𝑎=−2𝐶,₋₂2𝐵=−2𝐴,𝜈=−²+12𝐴𝐶𝐵, (3.7)

(2) 𝑎₋₁=𝑎₂=𝑎₋₁=0,𝑎₀=−2𝐵3,𝑎₋₂2𝐵=−2𝐴,𝜈=−²−3𝐴𝐶𝐵, (3.8)

(3) 𝑎₋₁=𝑎₋₂=𝑎₋₁=0,𝑎₀=−2𝐵3,𝑎₂2𝐵=−2𝐶,𝜈=−²−3𝐴𝐶𝐵. (3.9) The solutions of (3.1) are 𝑢₁2=−1+𝑚²3−2𝑚²sn²2𝑥−1+𝑚²²+12𝑚²1+𝑚²𝑡−2ns²2𝑥−1+𝑚²²+12𝑚²1+𝑚²𝑡,𝑢₂2=−1+𝑚²3−2𝑚²cd²2𝑥−1+𝑚²²+12𝑚²1+𝑚²𝑡−2dc²2𝑥−1+𝑚²²+12𝑚²1+𝑚²𝑡,𝑢₃2=−2𝑚²−13+2𝑚²cn²2𝑥+12𝑚²𝑚²−1+2𝑚²−1²2𝑚²𝑡−1−21−𝑚²nc²2𝑥+12𝑚²𝑚²−1+2𝑚²−1²2𝑚²𝑡,𝑢−1₄2=−2−𝑚²3+2dn²2𝑥−2−𝑚²²−12−1+𝑚²2−𝑚²𝑡−2𝑚²−1nd²2𝑥−2−𝑚²²−12−1+𝑚²2−𝑚²𝑡,𝑢₅2=−1+𝑚²3−2ns²2𝑥−1+𝑚²²−3𝑚²1+𝑚²𝑡,𝑢₆2=−2−𝑚²3−21−𝑚²sc²2𝑥+121−𝑚²+2−𝑚²²2−𝑚²𝑡−2cs²2𝑥+121−𝑚²+2−𝑚²²2−𝑚²𝑡,𝑢₇=22𝑚²−13+2𝑚²1+𝑚²sd²2𝑥+−12𝑚²1−𝑚²+−1+2𝑚²²2𝑚²𝑡−1−2ds²2𝑥+−12𝑚²1−𝑚²+−1+2𝑚²²2𝑚²𝑡,𝑢−1₈=21−2𝑚²6−122ns𝑥+0.75+0.5−𝑚²²0.5−𝑚²𝑡2+cs𝑥+0.75+0.5−𝑚²²0.5−𝑚²𝑡²−122ns𝑥+0.75+0.5−𝑚²²0.5−𝑚²𝑡2+cs𝑥+0.75+0.5−𝑚²²0.5−𝑚²𝑡⁻²,𝑢₉=−1+𝑚²3+1−𝑚²2×2nc𝑥+120.5−0.5𝑚²0.25−0.25𝑚²+0.5+0.5𝑚²²0.5𝑚²𝑡2+0.5+sc𝑥+120.5−0.5𝑚²0.25−0.25𝑚²+0.5+0.5𝑚²²0.5𝑚²𝑡+0.5²+1−𝑚²22nc𝑥+120.5−0.5𝑚²0.25−0.25𝑚²+0.5+0.5𝑚²²0.5𝑚²𝑡2+0.5+sc𝑥+120.5−0.5𝑚²0.25−0.25𝑚²+0.5+0.5𝑚²²0.5𝑚²𝑡+0.5⁻²,𝑢₁₀𝑚=−²−23−𝑚²22ns𝑥+0.75𝑚²+−1+0.5𝑚²²0.5𝑚²𝑡2−1+ds𝑥+0.75𝑚²+−1+0.5𝑚²²0.5𝑚²𝑡−1²−122ns𝑥+0.75𝑚²+−1+0.5𝑚²²0.5𝑚²𝑡2−1+ds𝑥+0.75𝑚²+−1+0.5𝑚²²0.5𝑚²𝑡−1⁻²,𝑢₁₁=𝑚²−23−𝑚²22sn𝑥+0.75𝑚⁴+−1+0.5𝑚²²0.5𝑚²𝑡2−1+𝑖cs𝑥+0.75𝑚⁴+−1+0.5𝑚²²0.5𝑚²𝑡−1²+𝑚²22sn𝑥+0.75𝑚⁴+−1+0.5𝑚²²0.5𝑚²𝑡2−1+𝑖cs𝑥+0.75𝑚⁴+−1+0.5𝑚²²0.5𝑚²𝑡−1⁻²,𝑢₁₂2=−1+𝑚²3−2dc²2𝑥−1+𝑚²²−3𝑚²1+𝑚²𝑡,𝑢₁₃2=−2𝑚²−13−21−𝑚²nc²2𝑥+−3𝑚²𝑚²−1+2𝑚²−1²2𝑚²𝑡,𝑢−1₁₄2=−2−𝑚²3−2𝑚²−1nd²2𝑥−2−𝑚²²+3−1+𝑚²2−𝑚²𝑡,𝑢₁₅2=−2−𝑚²3−2cs²2𝑥+2−𝑚²²−31−𝑚²2−𝑚²𝑡,𝑢₁₆=22𝑚²−13−2ds²2𝑥+3𝑚²1−𝑚²+−1+2𝑚²²2𝑚²𝑡,𝑢−1₁₇=21−2𝑚²6−122ns𝑥+0.5−𝑚²²−3/160.5−𝑚²𝑡2+cs𝑥+0.5−𝑚²²−(3/16)0.5−𝑚²𝑡⁻²,𝑢₁₈=−1+𝑚²3+1−𝑚²2×2nc𝑥+0.5+0.5𝑚²²−30.5−0.5𝑚²0.25−0.25𝑚²0.5𝑚²𝑡2+0.5+sc𝑥+0.5+0.5𝑚²²−30.5−0.5𝑚²0.25−0.25𝑚²0.5𝑚²𝑡+0.5⁻²,𝑢₁₉𝑚=−²−23−122ns𝑥+0.5𝑚²−1²−3/160.5𝑚²𝑡2−1+ds𝑥+0.5𝑚²−1²−3/160.5𝑚²𝑡−1⁻²,𝑢₂₀=𝑚²−23+𝑚²22sn𝑥+0.5𝑚²−1²−𝑚⁴/160.5𝑚²𝑡2−1+𝑖cs𝑥+0.5𝑚²−1²−𝑚⁴/160.5𝑚²𝑡−1⁻²,𝑢₂₁2=−1+𝑚²3−2𝑚²sn²2𝑥−1+𝑚²²−3𝑚²1+𝑚²𝑡,𝑢₂₂2=−1+𝑚²3−2𝑚²cd²2𝑥−1+𝑚²²−3𝑚²1+𝑚²𝑡,𝑢₂₃2=−2𝑚²−13+2𝑚²cn²2𝑥+2𝑚²−1²−3𝑚²𝑚²−12𝑚²𝑡,𝑢−1₂₄2=−2−𝑚²3+2dn²2𝑥−2−𝑚²²+3−1+𝑚²2−𝑚²𝑡,𝑢₂₅2=−2−𝑚²3−21−𝑚²sc²2𝑥+2−𝑚²²−31−𝑚²2−𝑚²𝑡,𝑢₂₆=22𝑚²−13+2𝑚²1+𝑚²sd²2𝑥+3𝑚²1−𝑚²+−1+2𝑚²²2𝑚²𝑡,𝑢−1₂₇=21−2𝑚²6−122ns𝑥+0.5−𝑚²²−3/160.5−𝑚²𝑡2+cs𝑥+0.5−𝑚²²−3/160.5−𝑚²𝑡²,𝑢₂₈=−1+𝑚²3+1−𝑚²2×2nc𝑥+0.5+0.5𝑚²²−30.5−0.5𝑚²0.25−0.25𝑚²0.5𝑚²𝑡2+0.5+sc𝑥+0.5+0.5𝑚²²−30.5−0.5𝑚²0.25−0.25𝑚²0.5𝑚²𝑡+0.5²,𝑢₂₉𝑚=−²−23−𝑚²22ns𝑥+0.5𝑚²−1²−𝑚²/160.5𝑚²𝑡2−1+ds𝑥+0.5𝑚²−1²−𝑚²/160.5𝑚²𝑡−1²,𝑢₃₀=𝑚²−23−𝑚²22sn𝑥+0.5𝑚²−1²−𝑚⁴/160.5𝑚²𝑡2−1+𝑖cs𝑥+0.5𝑚²−1²−𝑚⁴/160.5𝑚²𝑡−1². (3.10)(3.11)(3.12)(3.13)(3.14)(3.15)(3.16)(3.17)(3.18)(3.19)(3.20)(3.21)(3.22)(3.23)(3.24)(3.25)(3.26)(3.27)(3.28)(3.29)(3.30)(3.31)(3.32)(3.33)(3.34)(3.35)(3.36)(3.37)(3.38)(3.39)

3.1. Soliton Solutions

Some solitary wave solutions can be obtained, if the modulus 𝑚 approaches to 1 in (3.10)–(3.39) as follows: 𝑢₃₁4=−3−2tanh²(𝑥−16𝑡)−2coth²(),𝑢𝑥−16𝑡₃₂2=−3+2sech²(),𝑢𝑥+2𝑡₃₃2=−3+2sech²(),𝑢𝑥−2𝑡₃₄4=−3−2coth²(),𝑢𝑥−𝑡₃₅2=−3−2csch²(),𝑢𝑥+2𝑡₃₆=23+4sinh²(𝑥+2𝑡)−2csch²(),𝑢𝑥+2𝑡₃₇=−13−12(coth(𝑥−4𝑡)+csch(𝑥−4𝑡))²−12(coth(𝑥−4𝑡)+csch(𝑥−4𝑡))⁻²,𝑢₃₈=−13−12(tanh(𝑥−4𝑡)+𝑖csch(𝑥−4𝑡))²+12(tanh(𝑥−4𝑡)+𝑖csch(𝑥−4𝑡))⁻²,𝑢₃₉=−13−121coth𝑥−4𝑡1+csch𝑥−4𝑡⁻²,𝑢₄₀=−13+121tanh𝑥−4𝑡1+𝑖csch𝑥−4𝑡⁻²,𝑢₄₁4=−3−2tanh²(),𝑢𝑥−𝑡₄₂=23+4sinh²(),𝑢𝑥+2𝑡₄₃=−13−121coth𝑥−4𝑡1+csch𝑥−4𝑡²,𝑢₄₄=13−12(coth(𝑥+2𝑡)+csch(𝑥+2𝑡))²,𝑢₄₅=−13−123tanh𝑥−4𝑡3+𝑖csch𝑥−4𝑡². (3.40)

3.2. Triangular Periodic Solutions

Some trigonometric function solutions can be obtained, if the modulus 𝑚 approaches to zero in (3.10)–(3.39) as follwos: 𝑢₄₆2=−3−2csc²(),𝑢𝑥−2𝑡₄₇2=−3−2sec²(),𝑢𝑥−2𝑡₄₈=23−2sec²(),𝑢𝑥−2𝑡₄₉4=−3−2tan²(𝑥+16𝑡)−2cot²(),𝑢𝑥+16𝑡₅₀=13−12(csc(𝑥+4𝑡)+cot(𝑥+4𝑡))²−12(csc(𝑥+4𝑡)+cot(𝑥+4𝑡))⁻²,𝑢₅₁1=−3+12(sec(𝑥+7𝑡)+tan(𝑥+7𝑡))²+12(sec(𝑥+7𝑡)+tan(𝑥+7𝑡))⁻²,𝑢₅₂=23−sin²(),𝑢𝑥−2𝑡₅₃4=−3−2cot²(),𝑢𝑥+𝑡₅₄=13−121csc𝑥+4𝑡1+cot𝑥+4𝑡⁻²,𝑢₅₅1=−3+121sec𝑥−2𝑡1+tan𝑥−2𝑡⁻²,𝑢₅₆=23−12sin²3𝑥+8𝑡,𝑢₅₇4=−3−2tan²(),𝑢𝑥+𝑡₅₈=13−121csc𝑥+4𝑡1+cot𝑥+4𝑡²,𝑢₅₉1=−3+121sec𝑥−2𝑡1+tan𝑥−2𝑡². (3.41)

The modulus of solitary wave solutions 𝑢₁,  𝑢₂,  𝑢₂₁, and 𝑢₂₃ is displayed in Figures 1, 2, 3, and 4, respectively, with values of parameters listed in their captions.

(a)

(b)

Figure 1 

The modulus of solitary wave solution 𝑢1 (3.10) where 𝑚=0.5.

(a)

(b)

Figure 2 

The modulus of solitary wave solution 𝑢2 (3.11) where 𝑚=0.5.

(a)

(b)

Figure 3 

The modulus of solitary wave solution 𝑢21 (3.30) where 𝑚=0.5.

(a)

(b)

Figure 4 

The modulus of solitary wave solution 𝑢23 (3.32) where 𝑚=0.5.

4. Generalized (2+1)-Dimensional Ito Equation

In this section we consider the generalized (2+1)-dimensional Ito equation (1.2) as follows: 𝑢_𝑡𝑡+𝑢_{𝑥𝑥𝑥𝑡}+32𝑢_𝑥𝑢_𝑡+𝑢𝑢_𝑥𝑡+3𝑢_𝑥𝑥^𝑥_∞𝑢_𝑡𝑑𝑥^′+𝛼𝑢_𝑦𝑡+𝛽𝑢_𝑥𝑡=0, (4.1) if we use the transformation 𝑢=𝑣_𝑥, it carries (4.1) into 𝑣_𝑥𝑡𝑡+𝑣_{𝑥𝑥𝑥𝑥𝑡}+32𝑣_𝑥𝑥𝑣_𝑥𝑡+𝑣_𝑥𝑣_𝑥𝑥𝑡+3𝑣_𝑥𝑥𝑥𝑣_𝑡+𝛼𝑣_𝑥𝑦𝑡+𝛽𝑣_𝑥𝑥𝑡=0, (4.2) if we use 𝜁=𝑥+𝑦−𝜈𝑡 carries (4.2) into the ODE, we have 𝜈−𝛼−𝛽𝑉^′′′−𝑉^(𝑣)−3𝑉^′2′′=0, (4.3) where by integrating twice we obtain, upon setting the constant of integration to zero, 𝜈−𝛼−𝛽𝑉^′−𝑉^′′′−3𝑉^′2=0, (4.4) if we use the transformation 𝑊=𝑉^′, it carries (4.4) into 𝜈−𝛼−𝛽𝑊−𝑊^′′−3𝑊²=0. (4.5) Balancing the term 𝑊^′′ with the term 𝑊², we obtain 𝑁=2, then 𝑊(𝜁)=𝑎₀+𝑎₁𝜓+𝑎₋₁𝜓⁻¹+𝑎₂𝜓²+𝑎₋₂𝜓⁻²,𝜓^′=𝐴+𝐵𝜓²+𝐶𝜓⁴. (4.6)

Proceeding as in the previous case, we obtain

(1) 𝑎₋₁=𝑎₋₁=0,𝑎₀=−2𝐵3,𝑎₂𝑎=−2𝐶,₋₂2𝐵=−2𝐴,𝜈=𝛼+𝛽−²+12𝐴𝐶𝐵, (4.7)

(2) 𝑎₋₁=𝑎₂=𝑎₋₁=0,𝑎₀=−2𝐵3,𝑎₋₂2𝐵=−2𝐴,𝜈=𝛼+𝛽−²−3𝐴𝐶𝐵, (4.8)

(3) 𝑎₋₁=𝑎₋₂=𝑎₋₁=0,𝑎₀=−2𝐵3,𝑎₂2𝐵=−2𝐶,𝜈=𝛼+𝛽−²−3𝐴𝐶𝐵. (4.9) The solutions of (4.1) are 𝑢₁2=−1+𝑚²3−2𝑚²sn²2𝑥+𝛼+𝛽−1+𝑚²²+12𝑚²1+𝑚²𝑡−2ns²2𝑥+𝑦+𝛼+𝛽−1+𝑚²²+12𝑚²1+𝑚²𝑡,𝑢₂2=−1+𝑚²3−2𝑚²cd²2𝑥+𝑦+𝛼+𝛽−1+𝑚²²+12𝑚²1+𝑚²𝑡−2dc²2𝑥+𝑦+𝛼+𝛽−1+𝑚²²+12𝑚²1+𝑚²𝑡,𝑢₃2=−2𝑚²−13+2𝑚²cn²2𝑥+𝑦+𝛼+𝛽+12𝑚²𝑚²−1+2𝑚²−1²2𝑚²𝑡−1−21−𝑚²nc²2𝑥+𝑦+𝛼+𝛽+12𝑚²𝑚²−1+2𝑚²−1²2𝑚²𝑡,𝑢−1₄2=−2−𝑚²3+2dn²2𝑥+𝑦+𝛼+𝛽−2−𝑚²²−12−1+𝑚²2−𝑚²𝑡−2𝑚²−1nd²2𝑥+𝑦+𝛼+𝛽−2−𝑚²²−12−1+𝑚²2−𝑚²𝑡,𝑢₅2=−1+𝑚²3−2ns²2𝑥+𝑦+𝛼+𝛽−1+𝑚²²−3𝑚²1+𝑚²𝑡,𝑢₆2=−2−𝑚²3−21−𝑚²sc²2𝑥+𝑦+𝛼+𝛽+121−𝑚²+2−𝑚²²2−𝑚²𝑡−2cs²2𝑥+𝑦+𝛼+𝛽+121−𝑚²+2−𝑚²²2−𝑚²𝑡,𝑢₇=22𝑚²−13+2𝑚²1+𝑚²sd²2𝑥+𝑦+𝛼+𝛽+−12𝑚²1−𝑚²+−1+2𝑚²²2𝑚²𝑡−1−2ds²2𝑥+𝑦+𝛼+𝛽+−12𝑚²1−𝑚²+−1+2𝑚²²2𝑚²𝑡,𝑢−1₈=21−2𝑚²6−122ns𝑥+𝑦+𝛼+𝛽+0.75+0.5−𝑚²²0.5−𝑚²𝑡2+cs𝑥+𝑦+𝛼+𝛽+0.75+0.5−𝑚²²0.5−𝑚²𝑡²−122ns𝑥+𝑦+𝛼+𝛽+0.75+0.5−𝑚²²0.5−𝑚²𝑡2+cs𝑥+𝑦+𝛼+𝛽+0.75+0.5−𝑚²²0.5−𝑚²𝑡⁻²,𝑢₉=−1+𝑚²3+1−𝑚²2×2nc𝑥+𝑦+𝛼+𝛽+120.5−0.5𝑚²0.25−0.25𝑚²+0.5+0.5𝑚²²0.5𝑚²𝑡2+0.5+sc𝑥+𝑦+𝛼+𝛽+120.5−0.5𝑚²0.25−0.25𝑚²+0.5+0.5𝑚²²0.5𝑚²𝑡+0.5²+1−𝑚²2×2nc𝑥+𝑦+𝛼+𝛽+120.5−0.5𝑚²0.25−0.25𝑚²+0.5+0.5𝑚²²0.5𝑚²𝑡2+0.5+sc𝑥+𝑦+𝛼+𝛽+120.5−0.5𝑚²0.25−0.25𝑚²+0.5+0.5𝑚²²0.5𝑚²𝑡+0.5⁻²,𝑢₁₀𝑚=−²−23−𝑚²22ns𝑥+𝑦+𝛼+𝛽+0.75𝑚²+−1+0.5𝑚²²0.5𝑚²𝑡2−1+ds𝑥+𝑦+𝛼+𝛽+0.75𝑚²+−1+0.5𝑚²²0.5𝑚²𝑡−1²−122ns𝑥+𝑦+𝛼+𝛽+0.75𝑚²+−1+0.5𝑚²²0.5𝑚²𝑡2−1+ds𝑥+𝑦+𝛼+𝛽+0.75𝑚²+−1+0.5𝑚²²0.5𝑚²𝑡−1⁻²,𝑢₁₁=𝑚²−23−𝑚²22sn𝑥+𝑦+𝛼+𝛽+0.75𝑚⁴+−1+0.5𝑚²²0.5𝑚²𝑡2−1+𝑖cs𝑥+𝑦+𝛼+𝛽+0.75𝑚⁴+−1+0.5𝑚²²0.5𝑚²𝑡−1²+𝑚²22sn𝑥+𝑦+𝛼+𝛽+0.75𝑚⁴+−1+0.5𝑚²²0.5𝑚²𝑡2−1+𝑖cs𝑥+𝑦+𝛼+𝛽+0.75𝑚⁴+−1+0.5𝑚²²0.5𝑚²𝑡−1⁻²,𝑢₁₂2=−1+𝑚²3−2dc²2𝑥+𝑦+𝛼+𝛽−1+𝑚²²−3𝑚²1+𝑚²𝑡,𝑢₁₃2=−2𝑚²−13−21−𝑚²nc²2𝑥+𝑦+𝛼+𝛽+−3𝑚²𝑚²−1+2𝑚²−1²2𝑚²𝑡,𝑢−1₁₄2=−2−𝑚²3−2𝑚²−1nd²2𝑥+𝑦+𝛼+𝛽−2−𝑚²²+3−1+𝑚²2−𝑚²𝑡,𝑢₁₅2=−2−𝑚²3−2cs²2𝑥+𝑦+𝛼+𝛽+2−𝑚²²−31−𝑚²2−𝑚²𝑡,𝑢₁₆=22𝑚²−13−2ds²2𝑥+𝑦+𝛼+𝛽+3𝑚²1−𝑚²+−1+2𝑚²²2𝑚²𝑡,𝑢−1₁₇=21−2𝑚²6−122ns𝑥+𝑦+𝛼+𝛽+0.5−𝑚²²−3/160.5−𝑚²𝑡2+cs𝑥+𝑦+𝛼+𝛽+0.5−𝑚²²−3/160.5−𝑚²𝑡⁻²,𝑢₁₈=−1+𝑚²3+1−𝑚²2×2nc𝑥+𝑦+𝛼+𝛽+0.5+0.5𝑚²²−30.5−0.5𝑚²0.25−0.25𝑚²0.5𝑚²𝑡2+0.5+sc𝑥+𝑦+𝛼+𝛽+0.5+0.5𝑚²²−30.5−0.5𝑚²0.25−0.25𝑚²0.5𝑚²𝑡+0.5⁻²,𝑢₁₉𝑚=−²−23−122ns𝑥+𝑦+𝛼+𝛽+0.5𝑚²−1²−3/160.5𝑚²𝑡2−1+ds𝑥+𝑦+𝛼+𝛽+0.5𝑚²−1²−3/160.5𝑚²𝑡−1⁻²,𝑢₂₀=𝑚²−23+𝑚²2×2sn𝑥+𝑦+𝛼+𝛽+0.5𝑚²−1²−𝑚⁴/160.5𝑚²𝑡2−1+𝑖cs𝑥+𝑦+𝛼+𝛽+0.5𝑚²−1²−𝑚⁴/160.5𝑚²𝑡−1⁻²,𝑢₂₁2=−1+𝑚²3−2𝑚²sn²2𝑥+𝑦+𝛼+𝛽−1+𝑚²²−3𝑚²1+𝑚²𝑡,𝑢₂₂2=−1+𝑚²3−2𝑚²cd²2𝑥+𝑦+𝛼+𝛽−1+𝑚²²−3𝑚²1+𝑚²𝑡,𝑢₂₃2=−2𝑚²−13+2𝑚²cn²2𝑥+𝑦+𝛼+𝛽+2𝑚²−1²−3𝑚²𝑚²−12𝑚²𝑡,𝑢−1₂₄2=−2−𝑚²3+2dn²2𝑥+𝑦+𝛼+𝛽−2−𝑚²²+3−1+𝑚²2−𝑚²𝑡,𝑢₂₅2=−2−𝑚²3−21−𝑚²sc²2𝑥+𝑦+𝛼+𝛽+2−𝑚²²−31−𝑚²2−𝑚²𝑡,𝑢₂₆=22𝑚²−13+2𝑚²1+𝑚²sd²2𝑥+𝑦+𝛼+𝛽+3𝑚²1−𝑚²+−1+2𝑚²²2𝑚²𝑡,𝑢−1₂₇=21−2𝑚²6−122ns𝑥+𝑦+𝛼+𝛽+0.5−𝑚²²−3/160.5−𝑚²𝑡2+cs𝑥+𝑦+𝛼+𝛽+0.5−𝑚²²−3/160.5−𝑚²𝑡²,𝑢₂₈=−1+𝑚²3+1−𝑚²2×2nc𝑥+𝑦+𝛼+𝛽+0.5+0.5𝑚²²−30.5−0.5𝑚²0.25−0.25𝑚²0.5𝑚²𝑡2+0.5+sc𝑥+𝑦+𝛼+𝛽+0.5+0.5𝑚²²−30.5−0.5𝑚²0.25−0.25𝑚²0.5𝑚²𝑡+0.5²,𝑢₂₉𝑚=−²−23−𝑚²2×2ns𝑥+𝑦+𝛼+𝛽+0.5𝑚²−1²−𝑚²/160.5𝑚²𝑡2−1+ds𝑥+𝑦+𝛼+𝛽+0.5𝑚²−1²−𝑚²/160.5𝑚²𝑡−1²,𝑢₃₀=𝑚²−23−𝑚²2×2sn𝑥+𝑦+𝛼+𝛽+0.5𝑚²−1²−𝑚⁴/160.5𝑚²𝑡2−1+𝑖cs𝑥+𝑦+𝛼+𝛽+0.5𝑚²−1²−𝑚⁴/160.5𝑚²𝑡−1². (4.10)

4.1. Soliton Solutions

Some solitary wave solutions can be obtained, if the modulus 𝑚 approaches to 1 in (4.10) as follows: 𝑢₃₁4=−3−2tanh²𝑥+𝑦+𝛼+𝛽−16𝑡−2coth²𝑡,𝑢𝑥+𝑦+𝛼+𝛽−16₃₂2=−3+2sech²𝑡,𝑢𝑥+𝑦+𝛼+𝛽+2₃₃2=−3+2sech²𝑡,𝑢𝑥+𝑦+𝛼+𝛽−2₃₄4=−3−2coth²𝑡,𝑢𝑥+𝑦+𝛼+𝛽−1₃₅2=−3−2csch²𝑡,𝑢𝑥+𝑦+𝛼+𝛽+2₃₆=23+4sinh²𝑥+𝑦+𝛼+𝛽+2𝑡−2csch²𝑡,𝑢𝑥+𝑦+𝛼+𝛽+2₃₇=−13−12coth𝑥+𝑦+𝛼+𝛽−4𝑡+csch𝑥+𝑦+𝛼+𝛽−4𝑡²−12coth𝑥+𝑦+𝛼+𝛽−4𝑡+csch𝑥+𝑦+𝛼+𝛽−4𝑡⁻²,𝑢₃₈=−13−12tanh𝑥+𝑦+𝛼+𝛽−4𝑡+𝑖csch𝑥+𝑦+𝛼+𝛽−4𝑡²+12tanh𝑥+𝑦+𝛼+𝛽−4𝑡+𝑖csch𝑥+𝑦+𝛼+𝛽−4𝑡⁻²,𝑢₃₉=−13−121coth𝑥+𝑦+𝛼+𝛽−4𝑡1+csch𝑥+𝑦+𝛼+𝛽−4𝑡⁻²,𝑢₄₀=−13+121tanh𝑥+𝑦+𝛼+𝛽−4𝑡1+𝑖csch𝑥+𝑦+𝛼+𝛽−4𝑡⁻²,𝑢₄₁4=−3−2tanh²𝑡,𝑢𝑥+𝑦+𝛼+𝛽−1₄₂=23+4sinh²𝑡,𝑢𝑥+𝑦+𝛼+𝛽+2₄₃=−13−121coth𝑥+𝑦+𝛼+𝛽−4𝑡1+csch𝑥+𝑦+𝛼+𝛽−4𝑡²,𝑢₄₄=13−12coth𝑥+𝑦+𝛼+𝛽+2𝑡+csch𝑥+𝑦+𝛼+𝛽+2𝑡²,𝑢₄₅=−13−123tanh𝑥+𝑦+𝛼+𝛽−4𝑡3+𝑖csch𝑥+𝑦+𝛼+𝛽−4𝑡². (4.11)

4.2. Triangular Periodic Solutions

Some trigonometric function solutions can be obtained, if the modulus 𝑚 approaches to zero in (4.10) as follows: 𝑢₄₆2=−3−2csc²𝑡,𝑢𝑥+𝑦+𝛼+𝛽−2₄₇2=−3−2sec²𝑡,𝑢𝑥+𝑦+𝛼+𝛽−2₄₈=23−2sec²𝑡,𝑢𝑥+𝑦+𝛼+𝛽−2₄₉4=−3−2tan²𝑥+𝑦+𝛼+𝛽+16𝑡−2cot²𝑡,𝑢𝑥+𝑦+𝛼+𝛽+16₅₀=13−12csc𝑥+𝑦+𝛼+𝛽+4𝑡+cot𝑥+𝑦+𝛼+𝛽+4𝑡²−12csc𝑥+𝑦+𝛼+𝛽+4𝑡+cot𝑥+𝑦+𝛼+𝛽+4𝑡⁻²,𝑢₅₁1=−3+12sec𝑥+𝑦+𝛼+𝛽+7𝑡+tan𝑥+𝑦+𝛼+𝛽+7𝑡²+12sec𝑥+𝑦+𝛼+𝛽+7𝑡+tan𝑥+𝑦+𝛼+𝛽+7𝑡⁻²,𝑢₅₂=23−sin²𝑡,𝑢𝑥+𝑦+𝛼+𝛽−2₅₃4=−3−2cot²𝑡,𝑢𝑥+𝑦+𝛼+𝛽+1₅₄=13−121csc𝑥+𝑦+𝛼+𝛽+4𝑡1+cot𝑥+𝑦+𝛼+𝛽+4𝑡⁻²,𝑢₅₅1=−3+121sec𝑥+𝑦+𝛼+𝛽−2𝑡1+tan𝑥+𝑦+𝛼+𝛽−2𝑡⁻²,𝑢₅₆=23−12sin²3𝑥+𝑦+𝛼+𝛽+8𝑡,𝑢₅₇4=−3−2tan²𝑡,𝑢𝑥+𝑦+𝛼+𝛽+1₅₈=13−121csc𝑥+𝑦+𝛼+𝛽+4𝑡1+cot𝑥+𝑦+𝛼+𝛽+4𝑡²,𝑢₅₉1=−3+121sec𝑥+𝑦+𝛼+𝛽−2𝑡1+tan𝑥+𝑦+𝛼+𝛽−2𝑡². (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.

5. 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 system-mathematica, 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.

Acknowledgments

This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks DSR technical and financial support. Also, The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the paper.