The First Integral Method to the Nonlinear Schrodinger Equations in Higher Dimensions

The first integral method introduced by Feng is adopted for solving some important nonlinear partial differential equations, including the (2 + 1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers. This method provides polynomial first integrals for autonomous planar systems. Through the established first integrals, exact traveling wave solutions are formally derived in a concise manner. Published in : Hindawi Publishing Corporation Abstract and Applied Analysis References [1] A.-L. Guo and J. Lin, “Exact solutions of (2 + 1)-dimensional HNLSequation,”Communications inTheoretical Physics, vol. 54, no. 3, pp. 401–406, 2010. 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El-Ganaini 1 Mathematics Department ,Faculty of Science , Minia University, Egypt 2 Mathematics Department , Faculty of Science , Damanhour University, Egypt Mathematics Department, Faculty of Science at Dawadmi Shaqra University, Saudi Arabia *e-mail Address: ganaini5533@hotmail.com


Introduction
It is well known that nonlinear complex physical phenomena are related to nonlinear partial differential equations (NLPDEs) which are involved in many fields from physics to biology, chemistry, mechanics, etc.As mathematical models of the phenomena, the investigation of exact solutions of NLPDEs will help us to understand these phenomena better.Many effective methods for obtaining exact solutions of NLPDEs have been established and developed, such as the Lie point symmetries method [1], the exp-function method [2,3], the sine-cosine method [4,5], the extended tanh-coth method [6,7], the projective Riccati equation method [8,9], and so on.
The first integral method was first proposed by Feng in [10] in solving Burgers-KdV equation which is based on the ring theory of commutative algebra.Recently, this useful method has been widely used by many such as in [11][12][13][14][15][16][17][18][19][20][21] and by the references therein.In Section 2, we have described this method for finding exact travelling wave solutions of nonlinear evolution equations.In Section 3, we have illustrated this method in detail with the (2+1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, the generalized nonlinear Schrodinger (GNLS) equation with a source, and the higher-order nonlinear Schrodinger equation in nonlinear optical fibers.In Section 4, we have given some conclusions.

The First Integral Method
Consider a general nonlinear PDE in the form  (,   ,   ,   ,   ,   ,   , . ..) = 0, where  is a polynomial in its arguments.
Raslan in [22] has summarized the first integral method in the following steps.
Step 2. Assume that the solution of ODE (2) can be written as  (, ) =  (). (3) Step 3. We introduced new independent variables  () =  () , =  () , which leads a system of nonlinear ODEs () =  ( () ,  ()) . (5b) Step 4. According to the qualitative theory of ODEs [23], if we can find the integrals to (5a) and (5b) under the same conditions, then the general solution to (5a) and (5b) can be found directly.However, in general, it is really difficult to realize this even for one first integral, because for a given plane autonomous system, there is no systematic theory that can tell us how to find its first integrals, nor is there a logical way for telling us what these first integrals are.We will apply the division theorem to obtain one first integral to (5a) and (5b) which reduces (2) to a first-order integrable ODE.
An exact solution to (1) is then obtained by solving this equation.
Let us now recall the division theorem for two variables in the complex domain (, ).

Applications
In this section, we have investigated three NPDEs using the first integral method for the first time.
Balancing the degrees of ℎ() and  0 (), we have concluded that deg(ℎ()) = 1 only.Suppose that ℎ() =  +  and  ̸ = 0, we find  0 () where  is an arbitrary integration constant.Substituting  0 (),  1 (), and ℎ() for (12c) and setting all the coefficients of powers  to be zero, we have obtained a system of nonlinear algebraic equations and by solving it, we have obtained Using the conditions ( 14) in (10), we obtain respectively.
Substituting  0 (),  1 (),  2 (), and ℎ() for (17d) and setting all the coefficients of powers  to be zero, then we have obtained a system of nonlinear algebraic equations and by solving it, we get Using the conditions (19a) and (19b) in (10), we have obtained Combining ( 20) with (9a) we have obtained the exact solutions to (9a) and (9b) and hence the exact traveling wave solutions to the (2+1)-dimensional hyperbolic nonlinear Schrodinger equation ( 7) can be written as Comparing these results with the results obtained in [1], it can be seen that the solutions here are new.

The Generalized Nonlinear Schrodinger (GNLS) Equation
with a Source.Let us Consider the generalized nonlinear Schrodinger (GNLS) equation with a source [30,31], in the form where  = ( − ) is a real function and , , , , , , , and  are all real.
The GNLS equation ( 23) plays an important role in many nonlinear sciences.It arises as an asymptotic limit for a slowly varying dispersive wave envelope in a nonlinear medium.For example, its significant application in optical soliton communication plasma physics has been proved.
We have considered a plane wave transformation in the form where () is a real function.For convenience, let  = + 0 where  and  0 are real constants and  = ( − ) + .Then by replacing (23) and its appropriate derivatives in (22) and separating the real and imaginary parts of the result, we have obtained the following two ordinary differential equations: Integrating ( 25) once, with respect to , we have where  is an arbitrary integration constant.Since the same function () satisfies ( 26) and ( 27), we have obtained the following constraint condition: Using ( 4) and (5a) and (5b), we can get According to the first integral method, we suppose that () and () are nontrivial solutions of (29a) and (29b) and (, ) = ∑  =0   ()  is an irreducible polynomial in the complex domain [, ] such that where   (), ( = 0, 1, 2, . . ., ) are polynomials of  and   () ̸ = 0. Equation ( 30) is called the first integral to (29a) and (29b).Due to the division theorem, there exists a polynomial ℎ()+ () in the complex domain [, ] such that In this example, we have taken two different cases, assuming that  = 1 and  = 2 in (30).
Balancing the degrees of ℎ() and  0 (), it can be concluded that deg(ℎ()) = 1 only.Suppose that ℎ() =  + , and  ̸ = 0, then we find where  is an arbitrary integration constant.Substituting  0 (),  1 (), and ℎ() in (32c) and setting all the coefficients of powers  to be zero, we have obtained a system of nonlinear algebraic equations and by solving it, we obtain Using the conditions (34a) and (34b) in (30), we obtain respectively.
Combining (35) with (29a), the exact solutions to (29a) and (29b) were obtained and then the exact traveling wave solutions to the generalized nonlinear Schrodinger (GNLS) equation with a source (23) can be written as where  0 is an arbitrary integration constant.
In this case, it was assumed that ℎ() = + and  ̸ = 0; then we find  1 () and  0 () as follows: where , , , and  are arbitrary integration constants.Substituting  0 (),  1 (),  2 (), and ℎ() for (38d) and setting all the coefficients of powers  to be zero, a system of nonlinear algebraic equations was obtained and by solving it, we got Using the conditions (40a) and (40b) in (30), we obtain respectively.Combining ( 41) with (29a) we have obtained the exact solutions to (29a) and (29b) and thus the exact traveling wave solutions to the generalized nonlinear Schrodinger (GNLS) equation with a source ( 23) can be written as respectively, where  0 is an arbitrary integration constant.Equations ( 36)-( 37) and ( 42)-( 43) are new types of exact traveling wave solutions to the generalized nonlinear Schrodinger (GNLS) equation with a source (23).It could not be obtained by the methods presented in [32].

The Higher-Order Nonlinear Schrodinger Equation in
Nonlinear Optical Fibers.The higher-order nonlinear Schrodinger equation describing propagation of ultrashort pulses in nonlinear optical fibers [33][34][35][36][37][38][39] reads where  is slowly varying envelope of the electric field, the subscripts  and  are the spatial and temporal partial derivative in retard time coordinates, and  1 ,  2 ,  3 ,  4 ,  5 are the real parameters related to the group velocity dispersion (GVD), self-phase modulation (SPM), third-order dispersion (TOD), and self-steepening and self-frequency shift arising from simulated Raman scattering, respectively.Some properties of the equation, as well as many versions of it have been studied [33][34][35][36][37][38][39].Up to now, the bright, dark and the combined bright and dark solitary waves and periodic waves were found of (43) and its special case.
To seek traveling wave solutions of (44), we make the gauge transformation where , , , ,  are constants.Substituting (45) into (44) yields a complex ODE of (), the real and imaginary parts of which, respectively, It is easy to see that (46) becomes an equation under the constraint conditions Abstract and Applied Analysis 7 Using ( 4) and (5a) and (5b), we can get According to the first integral method, we suppose that () and () are nontrivial solutions of (49a) and (49b) and (, ) = ∑  =0   ()  is an irreducible polynomial in the complex domain [, ] such that where   (), ( = 0, 1, 2, . . ., ) are polynomials of  and   () ̸ = 0. Equation ( 50) is called the first integral to (49a) and (49b) due to the division theorem, there exists a polynomial ℎ() + () in the complex domain [, ] such that In this example, we take two different cases, assuming that  = 1 and  = 2 in (50).
Balancing the degrees of ℎ() and  0 (), it was concluded that deg(ℎ()) = 1 only.Suppose that ℎ() =  +  and  ̸ = 0, then we find where  is an arbitrary integration constant.Substituting  0 (),  1 (), and ℎ() in (52c) and setting all the coefficients of powers  to be zero, then we have obtained a system of nonlinear algebraic equations and by solving it, we obtain Using the conditions (54) in (50), we obtain respectively.Combining (55) with (49a), the exact solutions to (49a) and (49b) were obtained and then the exact traveling wave solutions to the higher-order nonlinear Schrodinger equation in nonlinear optical fibers can be written as respectively, where and  0 is an arbitrary integration constant.
In this case, it was assumed that ℎ() = + and  ̸ = 0; then we find  1 () and  0 () as follows: where , , , and  are arbitrary integration constants.Substituting  0 (),  1 (),  2 (), and ℎ() into (58d) and setting all the coefficients of powers  to be zero, a system of nonlinear algebraic equations was obtained and by solving it, we get Using the conditions (60a) in (50), we obtain respectively.Combining (61) with (49a), the exact solutions to (49a) and (49b) were obtained and thus the exact traveling wave solutions to the higher-order nonlinear Schrodinger equation in nonlinear optical fibers (44) can be written as Similarly, in the case of (60b), from (50), we get respectively.Combining (63) with (49a), the exact solutions to (49a) and (49b) were obtained and thus the exact traveling wave solutions to the higher-order nonlinear Schrodinger equation in nonlinear optical fibers (44) can be written as respectively, where  is as defined in (57) and  0 is an arbitrary integration constant.Comparing these results with Liu's results [39], it can be seen that the solutions here are new.

Conclusion
Searching for first integrals of nonlinear ODEs is one of the most important problems since they permit us to solve a nonlinear differential equation by quadratures.Applying the first integral method, which is based on the ring theory of commutative algebra, some new exact traveling wave solutions to the (2+1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation, generalized nonlinear Schrodinger (GNLS) equation with a source and higher-order nonlinear Schrodinger equation in nonlinear optical fibers were established.
These solutions may be important for the explanation of some practical physical problems.
The first integral method described herein is not only efficient but also has the merit of being widely applicable.
Therefore, this method can be applied to other nonlinear evolution equations and this will be done elsewhere.