The first integral method and traveling wave solutions to Davey–Stewartson equation

The first integral method was first proposed for solving Burger-KdV equation [1] which is based on the ring theory of commutative algebra. This method was further developed by the same author [2, 3] and some other mathematicians [2, 4, 5]. The present paper investigates, for the first time, the applicability and effectiveness of the first integral method on the Davey–Stewartson equations. We consider the Davey–Stewartson (DS) equations [6, 7]:


Introduction
The first integral method was first proposed for solving Burger-KdV equation [1] which is based on the ring theory of commutative algebra. This method was further developed by the same author [2,3] and some other mathematicians [2,4,5]. The present paper investigates, for the first time, the applicability and effectiveness of the first integral method on the Davey-Stewartson equations. We consider the Davey-Stewartson (DS) equations [6,7]: The case σ = 1 is called the DS-I equation, while σ = i is the DS-II equation. The parameter λ characterizes the focusing or defocusing case. The Davey-Stewartson I and II are two well-known examples of integrable equations in two space dimensions, which arise as higher dimensional generalizations of the nonlinear Schrodinger (NLS) equation [8].
They appear in many applications, for example in the description of gravity-capillarity surface wave packets in the limit of the shallow water. Davey and Stewartson first derived their model in the context of water waves, from purely physical considerations. In the context, q(x, y, t) is the amplitude of a surface wave packet, while φ(x, y) reperesents the velocity potential of the mean flow interacting with the surface wave [8].
The remaining portion of this article is organized as follows: Section 2 is a brief introduction to the first integral method. In Section 3, the first integral method will be implemented and some new exact solutions for Davey-Stewartson equation will be reported. Additionally, in this section, the traveling wave solution will also be obtained to retrieve soliton solution. A conclusion and future directions for research will be summarized in the last section.

The first integral method
Consider a general nonlinear PDE in the form P (u, u t , u x , u y , u xx , u tt , u yy , u xt , u xy , u yt , u xxx , . . .) = 0.
Initially, we consider case σ = 1 (DS-I). So using the wave variable η = x − 2αy + αt leads into the following ordinary differential equation (ODE): where prime denotes the derivative with respect to the same variable η. For case σ = i (DS-II), we use the wave variable η = x + 2αy − αt into Eq. (2). Next, we introduce new independent variables x = u, y = u η which change to a dynamical system of the following type: According to the qualitative theory of differential equations [1,9], if one can find two first integrals to Eqs. (3) under the same conditions, then analytic solutions to Eqs. (3) can be solved directly. However, in general, it is difficult to realize this even for a single first integral, because for a given autonomous system in two spatial dimensions, there does not exist any general theory that allows us to extract its first integrals in a systematic way.
A key idea of our approach here, to find first integral, is to utilize the Division theorem. For convenience, first let us recall the Division theorem for two variables in the complex domain C [10].
Division theorem. (See [11].) Suppose P (x, y) and Q(x, y) are polynomials of two variables x and y in C[x, y] and P (x, y) is irreducible in C[x, y]. If Q(x, y) vanishes at all zero points of P (x, y), then there exists a polynomial G(x, y) in C[x, y] such that Q(x, y) = P (x, y)G(x, y).

Exact solutions of Davey-Stewartson equation
In order to seek exact solutions of Eqs. (1), we first consider case σ = 1. So Eqs. (1) reduces to Now, in order to seek exact solutions of Eqs. (4), we assume q(x, y, t) = u(x, y, t) exp i(αx + βy + kt + l) , where α, β and k are constants to be determined later, l is an arbitrary constant. We assume β = 1. Substitute Eq. (5) into Eqs. (4) to yield Using the transformation where α is a constant, Eqs. (6) further reduces to where prime denotes the differential with respect to η. Integrating the second part of Eq. (7) with respect to η and taking the integration constant as zero yields Substituting Eq. (8) into the first part of (7) yields

Application of Division theorem
In order to apply the Division theorem, we introduce new independent variables x = u, y = u η which change Eq. (9) to the dynamical system given by www.mii.lt/NA Now, we are going to apply the Division theorem to seek the first integral to (10). Suppose that x = x(η) and y = y(η) are the nontrivial solutions to (10), and where a i (x) (i = 0, 1, . . . , m) are polynomials in x and all relatively prime in C[x, y], a m (x) = 0. Equation (11) is also called the first integral to (10). We start our study by assuming m = 1 in (11). Note that dP dη is a polynomial in x and y, and P [x(η), y(η)] = 0 implies dP dη = 0. By the Division theorem, there exists a polynomial H( where prime denotes differentiating with respect to the variable x. On equating the coefficients of y i (i = 2, 1, 0) on both sides of (12), we have Since, a 1 (x) is a polynomial in x, from (13) we conclude that a 1 (x) is a constant and g(x) = 0. For simplicity, we take a 1 (x) = 1, and balancing the degrees of h(x) and a 0 (x) we conclude that deg h(x) = 1. Now suppose that h(x) = Ax+B, then from (14), we find a 0 (x) = 1 2 where D is an arbitrary integration constant. Substituting a 0 (x), a 1 (x) and h(x) in (15) and setting all the coefficients of powers x to be zero, we obtain a system of nonlinear algebraic equations and by solving it, we obtain and Using (16) and (17) in (11), we obtain respectively, where λ = 0. Combining this equations with (10), where we obtain the exact solutions of Eq. (10) as follows: where λ = 0 and c 1 is an arbitrary constant. Therefore, the exact solutions to (10) can be written as where λ = 0. Then exact solutions for Eqs. (4) are where λ = 0.

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Notice that integrating of Eq. (8) with respect to η and taking the integration constant as zero yields Now we assume that m = 2 in (11 On equating the coefficients of y i (i = 3, 2, 1, 0) from both sides of (18), we have Since, a 2 (x) is a polynomial of x, from (19) we conclude that a 2 (x) is a constant and g(x) = 0. For simplicity, we take a 2 (x) = 1, and balancing the degrees of h(x), a 0 (x) and a 1 (x) we conclude that deg h(x) = 1 or 0, therefore we have two cases: Case 1. Suppose that deg h(x) = 1 and h(x) = Ax + B, then from (20) we find where D is an arbitrary integration constant. From (21) we find where E is an arbitrary integration constant. Substituting a 0 (x), a 1 (x) and h(x) in (22) and setting all the coefficients of powers x to be zero, we obtain a system of nonlinear algebraic equations and by solving it, we obtain: and Using (23) and (24) in (11), we obtain respectively, where λ = 0. Combining this equations with (10), we obtain two exact solutions to Eq. (10) which was obtained in case m = 1.

Case 2.
In this case suppose that deg h(x) = 0 and h(x) = A, then from (20) we find a 1 (x) = Ax + B, where B is an arbitrary integration constant. From (21) we find where D is an arbitrary integration constant. Substituting a 0 (x), a 1 (x) and h(x) in (22) and setting all the coefficients of powers x to be zero, we obtain a system of nonlinear algebraic equations and by solving it, we obtain: Using (25) in (11), we obtain Combining this equations with (10), we obtain the exact solutions to Eq. (10) as follows: and λ = 0 while c 1 is an arbitrary constant. Then the exact solutions to (10) can be written as: and Then solutions of Eqs. (4) are where λ = 0. Now, we consider the case when σ = i, where Eqs. (1) transforms the following: We will solve equation (26) similarly as in the case where σ = 1, with the only difference being η = x + 2αy − αt, φ = 2λ 1+4α 2 u 2 and By applying the Division theorem as in the case where σ = 1, and by assuming m = 1, the exact solutions of Eqs. (10) as follows: and λ = 0.
Then the exact solutions of (26) are where λ = 0 and for case m = 2 in Case 1, we obtain two exact solutions to Eqs. (10) which was obtained in case m = 1.

Traveling wave solutions
The traveling wave hypothesis will be used to obtain the 1-soliton solution to the Davey-Stewartson equation (1). It needs to be noted that this equation was already studied by traveling wave hypothesis in 2011 [12] where the power law nonlinearity was considered. Additionally, the ansatz method was used to extract the exact 1-soliton solution to the Davey-Stewartson equation in 2011 and that too was studied with power law nonlinearity [13].
In this subsection, the starting point is going to be Eq. (9) which can now be re-written as Now, multiplying both sides of (27) by u and integrating while choosing the integration constant to be zero, since the search is for soliton solutions, yields where a = λ 4α 2 − 1 and b = α 2 + 2k + 1 4α 2 + 1 .
Separating variables in (28) and integrating gives that yields the 1-soliton solution as where the amplitude A of the soliton is given by and this leads to the constraints α 2 + 2k + 1 > 0 and λ 4α 2 − 1 α 2 + 2k + 1 > 0.

Conclusions
We described the first integral method for finding some new exact solutions for the Davey-Stewartson equation. We have obtained four exact solutions to the Davey-Stewartson equation. The solutions obtained are expressed in terms of trigonometric and exponential functions. In addition, the traveling wave hypothesis is used to obtain the 1-soliton solution of the equation where a topological and non-topological soliton pair is retrieved. These new solutions may be important for the explanation of some practical problems. One context where this equation is studied from a practical standpoint is in the study of water waves with finite depth which moves in one direction [14]. However, in a twodimensional scenario, this equation models both short waves and long waves. Many explicit solutions are given in this reference which are all applicable to the study of finite depth water waves. This paper also gives a substantial set of solution set that is also meaningful. Addditionally, this equation can also be generalized to a fractional derivative case that has been recently touched (15).