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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.

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 [

The first integral method was first proposed by Feng in [

Consider a general nonlinear PDE in the form

Raslan in [

Using a wave variable

Assume that the solution of ODE (

We introduced new independent variables

According to the qualitative theory of ODEs [

We will apply the division theorem to obtain one first integral to (

An exact solution to (

Let us now recall the division theorem for two variables in the complex domain

Suppose that

The division theorem follows immediately from the Hilbert-Nullstellensatz Theorem [

Let

Every ideal

Let

For an ideal

For a polynomial

In this section, we have investigated three NPDEs using the first integral method for the first time.

Let us consider the (2+1)-dimensional hyperbolic nonlinear Schrodinger (HNLS) equation [

By considering the transformations

Using (

Equation (

Suppose that

Balancing the degrees of

Substituting

Combining (

Assume that

In this case, let us assume that

Substituting

Let us Consider the generalized nonlinear Schrodinger (GNLS) equation with a source [

The GNLS equation (

Furthermore, the GNLS equation enjoys many remarkable properties (e.g., bright and dark soliton solutions, Lax pair, Liouvile integrability, inverse scattering transformation, conservation laws, Backlund transformation, etc.).

We have considered a plane wave transformation in the form

Equation (

Suppose that

Balancing the degrees of

Substituting

Combining (

Suppose that

In this case, it was assumed that

Substituting

Equations (

The higher-order nonlinear Schrodinger equation describing propagation of ultrashort pulses in nonlinear optical fibers [

where

To seek traveling wave solutions of (

Equation (

Suppose that

Balancing the degrees of

Substituting

Combining (

Suppose that

In this case, it was assumed that

Substituting

Similarly, in the case of (

Comparing these results with Liu’s results [

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.

The author would like to thank the referees for their useful comments which led to some improvements of the current paper.