International Journal of Differential Equations
Volume 2025, Issue 1 6540374
Research Article
Open Access

The Nonlinear Schrödinger Equation Derived From the Fifth-Order Korteweg–de Vries Equation Using Multiple Scales Method

Murat Koparan, 

Corresponding Author

Murat Koparan

Department of Mathematics and Science Education , Anadolu University Faculty of Education , Mathematics Education Division , Yunusemre Campus , Eskişehi , 26470 , Türkiye

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First published: 14 April 2025
https://doi.org/10.1155/ijde/6540374
Academic Editor: Salim A. Messaoudi

Abstract

The mathematical models of problems that arise in many branches of science are nonlinear equations of evolution (NLEE). For this reason, NLEE have served as a language in formulating many engineering and scientific problems. Although the origin of nonlinear evolution equations dates back to ancient times, significant developments have been made in these equations up to the present day. The main reason for this situation is that NLEE involve the problem of nonlinear wave propagation. Therefore, many different and effective techniques have been developed regarding nonlinear evolution equations and solution methods. Studies conducted in recent years show that evolution equations are becoming increasingly important in applied mathematics. This study is about the multiple scales methods, known as the perturbation method, for NLEE. In this report, the multiple scales method was applied for the analysis of (1 + 1) dimensional fifth-order nonlinear Korteweg–de Vries (KdV) equation, and nonlinear Schrödinger (NLS) type equations were obtained.

1. Introduction

Very diverse physical, chemical, and biological phenomena are depicted by nonlinear evolution of equations (NLEE). Recently, NLEEs have become an important field of study in applied mathematics. Also, nonlinear equations that model these scientific phenomena in other branches of science have long been a major concern for research studies. Since data of their exact solutions facilitates the confirmation of numerical solvers and supports in decisiveness analysis of solutions, analytical study of these NLEEs is significant. This not only helps to better understand the solutions but also helps us to understand the phenomenon they describe.

Known for its distinguished role in the development of nonlinear physics, the Korteweg–de Vries (KdV) equation has been derived in a variety of physical contexts. For example, it can be thought of as modeling the unilateral propagation of long wavelength gravitational waves of small amplitude in a shallow channel. In any case, the KdV equation is obtained with a certain degree of approximation and, therefore cannot be considered to represent physical reality with perfect accuracy. Therefore, an important question arises on what would happen to the solutions of KdV equation when perturbations from neglected terms in the derivation come into play. For instance, would the perturbed KdV equation still have a well-localized single-wave solution? The answer, sure, would depend on the nature and physical origin of the perturbation. Currently, studies are underway on the nonlinear fifth-order KdV-type equations as they can describe real properties in various scientific applications and engineering fields; and have practical and physical significance [1–5].

The nonlinear Schrödinger (NLS) equation is an example of a universal nonlinear model because it describes a wide variety of physical systems. As a result, the equation may be used to describe a wide range of nonlinear physical events [6–8]. It is renowned that a multiscale analysis of the KdV equation gives rise to the NLS equation for modulated amplitude [9–13]. In [9] Zakharov and Kuznetsov demonstrated a much deeper correspondence between these integrable equations, not only at the equation level but also at the linear spectral problem level, by showing that multiscale analysis of the Schrödinger spectral problem yields the Zakharov-Shabat problem for the NLS equation. Özer and Dağ demonstrated a similar link between the NLS and integrable fifth-order nonlinear evolution equations [14]. Additionally, similar results were obtained using multiscale analysis in different equations [15–19].

In this paper, we apply a multiple scale method following Zakharov and Kuznetsov [9] related to the connection of the KdV and NLS equations. This is an important derivation because the KdV flow equations follow from the NLS equation. The strength of this method is that for each degree of coefficients in epsilon, the equations contain no secular terms. Therefore, there is no freedom in choosing the coefficients and the expansion is uniquely determined. The derivation of this hierarchy is not a simple case of algorithms, but basically relies on two facts. First, different time flows must commute; that is: . Second, In each order in epsilon, the coefficient equations contain secular terms. Eliminating the secular terms requires to have a certain high symmetry (flow) of the hierarchy, and this way all coefficients of expansion are fixed and no arbitrariness is left.

After the introduction, in chapter two, we briefly expressed the fifth-order KdV (fKdV) flow equations and the multiple scales method, respectively. In Chapter three, we applied the method given in Chapter two to the (1 + 1) dimensional fKdV equation. The last part consists of the conclusion part.

2. Background Materials

In this section, we present some background material on the best-known fKdV equations and the multiscale method.

2.1. The fKdV Flow Equations

The best-known fKdV equations look like this
()
where α, β, γ, and ω are arbitrary nonzero and real parameters, and u = u(x, t) is a smooth enough function. Since the parameters α, β, γ, and ω are arbitrary and take different values, this will greatly change the properties of the fKdV equation (1). Changing the actual values of the parameters allows you to generate many different variations of the fKdV equation. The fKdV equations, which are widely used in nonlinear optics and quantum physics, are an important mathematical model. Characteristic examples are commonly utilized in many domains, including plasma physics, quantum field theory, solid-state physics, and fluid physics [20, 21].

Some important special cases of the (1) are:

Kaup–Kupershmidt (KK) equation [22–26].
()
Sawada–Kotera (SK) equation [27, 28].
()
Caudrey–Dodd–Gibbon (CDG) equation.
()
Lax equation [20].
()
Ito equation [29, 30].
()

As the values of α, β, and γ vary, the characteristics of Equation (1) change drastically. For example, the KK equation with α = 10, β = 25, and γ = 20, which is known to be integrable [24], and has bilinear representations [24, 26], but the obvious form of the N-soliton solutions is not known. Another example is the SK equation where α = β = γ = 5, and the Lax equation with α = 10, β = 20, and γ = 30, are both fully integrable. These two equations have N-soliton solutions and an endless set of conserved densities. One last equation in this class is the Ito equation, with α = 3, β = 6, and γ = 2, which cannot be fully integrated but has a limited number of special conserved densities [30].

2.2. The Multiple Scales Method

The multiple scales method is a perturbation method. In the multiple scales method first recommended by Zakharov and Kuznetzov [9], Zakharov and Kuznetzov used this method to decrease the KdV equation to the NLS equation and apply it to a class of nonlinear evolution equations. Using this method, they showed that integrable systems can be decreased to integrable systems. If the system we have taken at the beginning is not an integrable system, it has been seen that the reduced system as a result of the application of the method is either integrable or nonintegrable. However, if the method is applied to a suitable integrable system, it is seen that the system obtained as a result of the analysis is always an integrable system. This is the master purpose of applying the multiscale expansion method to integrable systems.

In this section, multiple scales method of nonlinear evolution equations is discussed. By applying the Zakharov and Kuznetsov [9] technique, the steps of the multiscale method in obtaining NLS type equations from KdV equations are shown in order.

Let consider the general evolution equation in the following form
()
where K[u] is a function of u and its derivatives with respect to the x-spatial variables. The well known of this type equations is KdV equation. L[∂x, ∂y]u is the linear part of K[u]. So, using K[u] we can reach the dispersion relation for equation (7). Substituting the wave solution space, we obtain the following equation:
()
Into the linear part of (7)
()
We get dispersion relation
()
Then, dispersion relation (10) is substituted in (7). We assume the following series expansions for the solution of the (7):
()
Based on this solution, we also define slow spaces ξ and multiple time variable τ with respect to the scaling parameter ϵ > 0 respectively as follows:
()
A nonlinear equation modulates the amplitude of this plane wave solution in such a way that may consider it depend upon the slow variables. If we choose the slow variables different forms, we can derive higher order NLS equations. The multiple scales analysis starts with the assumption:
()
and solution of U is in the form
()

In this case, considering the transformation (13) and solution (14), using (10) and (12), the terms included derivative in (7) are obtained. Substituting these terms with (13) and (14) into the (7), we get a polynomial in ε. We obtain a series of algebraic equations by equalizing each coefficient of this polynomial to zero. Using wave solution space (8) and dispersion relation (10), these equations may be solved by iteration and Reduce. Thus, we can obtain NLS type equations from (7). Furthermore, this approach allows us to obtain numerical solutions to KdV-type problems.

3. Applications

Following the method given in Section 2.2, we use a multiple scales method to derive the NLS equations from the fifth-order fKdV equation (2). To find dispersion relation for (2), we consider the linear part of (2) in the form
()
and linear differential equation (14) satisfies the solution
()
Substituting the solution (16) into the linear differential equation (14), we get and from this we reach the
()
Dispersion relation. Thus the solution of linear differential equation (14) is as follows:
()
Let the solution of (2) is in the form
()
ϵ scale parameter and slow variables
()
Then, we assume the following series expansions for solutions:
()
In this case, considering the transformation and solution in (19), using (17) and (20), the terms included derivative in (2) are obtained. Substituting these terms and (19) into the (2), we get a polynomial in ϵ. Equalizing each coefficient of this polynomial to zero, we get a set of algebraic equations. If we let ε⟶0 we find the following:
()
()
()
()
()
()
Then, we can find the solution of (22) as follows:
()
where c.c. is complex conjugate of v1. Substituting the solution (28) into (23), the solution of (23) is in the form
()
where f0 is integration constant. Thus, we get
()
where v−1 is the complex conjugate of v1 and v−2 is the complex conjugate of v2. Substituting solutions (27), (28) and (29) into the (24), we find the solution of (24) in the form
()
where f1 is integration constant and v−3 is the complex conjugate of v3. Then, we get
()
and
()
Describing as v1 = kq and v−1 = kp, from (33) we get the NLS type equations
()
Substituting solutions into (24) are obtained as
()
where f2 is integration constant and v−4 is the complex conjugate of v4. Thus, we get
()
and
()
Describing as v1 = kq and v−1 = kp, from (37) we get the following equation:
()
Similarly, substituting the solutions into (25) are obtained as
()
where f3 is integration constant and v−5 is the complex conjugate of v5. Thus, we get
()
and
()
Describing as v1 = kq and v−1 = kp, from (41) we get the following equation:
()
Finally, similarly substituting the solutions into (26) are obtained as
()
where f4 is integration constant and v−6 is the complex conjugate of v6. Thus, we get
()
and
()
Describing as v1 = kq and v−1 = kp, from (41) we get the following equation:
()

4. Conclusion

The multiple scales method, which is a perturbation method, is used to find approximate solutions of nonlinear evolution equations. This method allows us to find the solution of the given nonlinear equation depending on the solution of the linear part by using the multiple scales, which is defined as the slow variables and depends on a parameter, in time and space variables. First, Zakharov and Kuznetzov showed that integrable systems can be reduced to other integrable systems using this method. If the initially taken system is not integrable, it is seen that the reduced system is either integrable or nonintegrable as a result of the application of the method. However, if the method is applied to an integrable system, it is seen that the system obtained as a result of the analysis is always integrable. This is the main purpose of applying the multiple scales methods to integrable systems. In this study, we examined how only NLS-type equations are derived from fKdV-type equations by increasing the slow variables depending on a parameter using the multiple scales methods. We hope this conclusion serves as a valuable resource for researchers, scientists, and engineers interested in nonlinear wave theory, inspiring a deeper desire to explore further and reveal the mysteries still held within the intricate language of the NLS equation. At the same time, we think that the obtained results will form the basis of numerical calculations. Also, the method can be applied to many different NLEE equations.

Conflicts of Interest

The author declares no conflicts of interest.

Funding

The author received no financial support for this manuscript.

Data Availability Statement

No data were used to support the findings of this study.

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