# Nonlocal symmetries and group invariant solutions for the coupled variable-coefficient Newell-Whitehead system

^{*}

^{*}Corresponding author.

- DOI
- 10.1080/14029251.2020.1819601How to use a DOI?
- Keywords
- Nonlocal symmetry; Group invariant solution; Lie point symmetry; Symmetry reduction; Variable-coefficient Newell-Whitehead system
- Abstract
Starting from the Lax pairs, the nonlocal symmetries of the coupled variable-coefficient Newell-Whitehead system are obtained. By introducing an appropriate auxiliary dependent variable, the nonlocal symmetries are localized to Lie point symmetries and the coupled variable-coefficient Newell-Whitehead system is extended to an enlarged system with the auxiliary variable. Then the finite symmetry transformation for the prolonged system is found by solving the initial-value problems. Furthermore, by applying symmetry reduction method to the enlarged system, two kinds of the group invariant solutions are given.

- Copyright
- © 2020 The Authors. Published by Atlantis Press and Taylor & Francis
- Open Access
- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

## 1. Introduction

Symmetry study is always a powerful method in physics and mathematics fields [1]
. Especially in soliton theory, the study of symmetries is more important because of the following three important applications. The first is using symmetry to obtain new solutions from old ones [2–7
]. The second is symmetry can be used to reduce dimensions of a partial differential equation(PDE) [8, 9]. The third one is that any flow equation corresponding to a symmetry of integrable models is an integrable one, that is to say, through symmetry a new soliton equation can be acquired [10–12
]. For these reasons, there are many other interesting applications of the symmetry study [13–17
]. Nonlocal symmetry as a generalization of the symmetry was first researched by Vinogradov and Krasil’shchik in 1980, which enlarges the class of symmetries and connected with integrable models. Compared with the local symmetry, the nonlocal symmetries of PDE are not easy to find and its similarity reductions cannot be calculated directly. However, the importance of the nonlocal symmetry has attracted many mathematicians to make great efforts to explore it and achieve fruitful results, such as in reference [4]
, Galas presented the nonlocal Lie-Bäcklund symmetries by using the Pseudo-potentials, in [8], [18–20], Bluman, Euler *et al.* constructed nonlocal symmetries of PDES by using potential system, Lou, Hu and Chen derived the nonlocal symmetries from the Bäcklund transformation [21]
. Recently, based on the auxiliary system (lax pair), Xin and Chen obtained the nonlocal symmetries for the modified Korteweg-de Vries equation [23]
, Miao, Xin and Chen obtained the nonlocal symmetries for the Ablowitz-Kaup-Newell-Segur system [24]
. Gao, Lou and Tang found that Painlevé analysis can be applied to obtain nonlocal symmetries corresponding to the residues with respect to the singular manifold of the truncated Painlevé expansion, which is also called residual symmetries [25]
. Besides, considering the results obtained from nonlocal symmetry reduction, many references [26–32] have shown that nonlocal symmetry method is one of the effective tools to find nonlinear waves interacting with each other.

In this paper, further generalizing the method based on lax pair to construct nonlocal symmetries for the variables coefficient system, and then employing this method to the coupled variable-coefficient Newell-Whitehead system, we achieve the nonlocal symmetries of system (2.1), and use the obtained nonlocal symmetries to construct group invariant solution for it. Because of the variables coefficient equation contain the corresponding constant coefficient equation, this study is very meaningful.

This paper is arranged as follows: in Section 2, nonlocal symmetries of coupled variable-coefficient Newell-Whitehead system are obtained by using the Lax pair. In Section 3, by extending the original system, nonlocal symmetries are transformed into the Lie point symmetry, and the corresponding finite transformation group are given with Lie’s first theorem. In Section 4, some symmetry reductions and group invariant solutions of system (2.1) are obtained with lie point symmetry of prolonged system. A short conclusion and discussion is included in the last Section.

## 2. Nonlocal symmetries of the coupled variable-coefficient Newell-Whitehead system

The coupled Newell-Whitehead system is given by

*α*(

*t*) and

*β*(

*t*) are real functions of

*t*, and

*p*=

*p*(

*x,t*),

*q*=

*q*(

*x,t*) real functions of

*x*and

*t*. When

*u*=

*p*=

*q*,

*b*(

*t*) = −

*α*(

*t*) = −2

*β*(

*t*), the system (2.1) reduces to the following variable-coefficient Newell-Whitehead system,

The system (2.1) can efficiently describe many nonlinear phenomena which appear in fluid mechanics, plasma physics, thermonuclear reactions, and population proliferation. The literature [33] has studied the Darboux transformation and multi-soliton solutions for system (2.1), so our main objective in this paper is to obtain nonlocal symmetries for the coupled variable-coefficient Newell-Whitehead system from its Lax pair, and further to localize the nonlocal symmetries by introducing potential variables, then by the method of symmetry reduction to derive the interaction solution of system (2.1). To the best of our knowledge, the nonlocal symmetries and interaction solution to the system (2.1) have not been discussed.

The lax pair of system (2.1) reads [33]

For system (2.1), we consider the invariant property under

*σ*(

^{i}*i*= 1

*,...,*4), are symmetries of

*p*,

*q*,

*α*and

*β*, which should satisfy the following linearized equations

*σ*

^{1},

*σ*

^{2},

*σ*

^{3}and

*σ*

^{4}can be written as

*A*

_{1},

*B*

_{1}are functions of

*x*,

*t*,

*p*,

*q*,

*α*,

*β*,

*ϕ*

_{1},

*ϕ*

_{2}. Substituting Eq. (2.5) into Eq. (2.4), eliminating

*p*,

_{t}*q*, and

_{t}*ϕ*

_{1}

*,*

_{x}*ϕ*

_{1}

*,*

_{t}*ϕ*

_{2}

*,*

_{x}*ϕ*

_{2}

*in terms of the lax pair (2.2), we derive a system of determining equations for the functions*

_{t}*A*

_{1},

*B*

_{1}, which on solving by

*Maple*gives

*F*

_{1}(

*x*) arbitrary function of

*x*, and

*F*

_{2}(

*t*),

*F*

_{3}(

*t*) arbitrary functions of

*t*.

## 3. Localization of the nonlocal symmetry

We know that Lie point symmetry can be applied to construct explicit solutions for PDEs, however, it is invalid for nonlocal symmetry. So we need to transform the nonlocal symmetries into local ones, especially into Lie point symmetries. Following this idea, we extend the original system to a closed prolonged system by introducing some auxiliary variables, and construct the Lie point symmetries for the prolonged system which contains the nonlocal symmetries (2.6) of the original system.

For simplicity, letting *F*_{1}(*x*) = 0, *F*_{2}(*t*) = 0, *F*_{3}(*t*) = 0 in (2.6), we obtain the following nonlocal symmetries for (2.1)

To localize the nonlocal symmetries (3.1), we have to solve the following linearized equations

*σ*

^{1},

*σ*

^{2},

*σ*

^{3},

*σ*

^{4}given by (3.1), and

*σ*

^{5},

*σ*

^{6}satisfying the following transformations

It is easy to verify that the solutions of (3.2) have the following forms

*f*satisfying

### Remark 3.1.

What is more interesting here is that the symmetry *σ* * ^{f}* shown in (3.6) indicates that the auxiliary-dependent variable

*f*satisfying

From the expression (3.4), one can see that the nonlocal symmetries (3.1) in the original space *x*, *t*, *p*, *q*, *α*, *β* have been successfully localized to a lie point symmetries

*x*,

*t*,

*p*,

*q*,

*α*,

*β*,

*ϕ*

_{1},

*ϕ*

_{2},

*f*, and the prolonged equations of (2.1), (2.2) and (3.5) have the following the Lie point symmetry vector:

To proceed, we study the finite symmetry transformation of the Lie point symmetry (3.9). According to Lie’s first theorem, solving the following initial value problem:

### Theorem 3.1.

*If* {*p, q, α, β, ϕ _{1}*

*, ϕ*

_{2}*, f*}

*is a solution to the prolonged system*(2.1)

*,*(2.2)

*and*(3.5)

*, then so is the*{

*with*

### Remark 3.2.

From the Theorem 3.1. one can see that the finite symmetry transformation (3.11) can give a new solution *p, q* for the system (2.1). It should be mentioned that the last expression in (3.10) is nothing but the Möbious transformation.

Next, we study the Lie point symmetries of the prolonged system instead of the single Eq. (2.1). According to the classical Lie point symmetry method, we assume that the vector of the symmetries has the following form

*X*,

*T*,

*P*,

*Q*,

*A*,

*B*,

*P*

_{1},

*P*

_{2},

*F*are functions of

*x*,

*t*,

*p*,

*q*,

*α*,

*β*,

*ϕ*

_{1},

*ϕ*

_{2},

*f*, which means that the closed system is invariant under the following infinitesimal transformations

*σ*(

^{i}*i*= 1

*,...,*7) are defined as the solution of the linearized equations of the prolonged systems, i.e., (2.4), (3.2) and

Substituting Eq. (3.13) into (2.4), (3.2) and (3.14), eliminating the *p _{t}*,

*q*,

_{t}*ϕ*

_{1}

*,*

_{x}*ϕ*

_{1}

*,*

_{t}*ϕ*

_{2}

*,*

_{x}*ϕ*

_{2}

*,*

_{t}*f*,

_{x}*f*, and collecting the coefficients of the independent partial derivatives of dependent variables

_{t}*p*,

*q*,

*α*,

*β*, we obtain a system of overdetermined linear equations for the infinitesimals

*X*,

*T*,

*P*,

*Q*,

*A*,

*B*,

*P*

_{1},

*P*

_{2},

*F*, and by solving these determining equations, we can obtain the following results

*c*(

_{i}*i*= 1

*,...,*4) are arbitrary constants,

*F*

_{4}(

*t*),

*F*

_{5}(

*t*) are functions of

*t*. Specially, when setting

*c*

_{1}=

*c*

_{3}=

*F*

_{3}(

*t*) =

*F*

_{4}(

*t*) =

*F*

_{5}(

*t*) = 0, and

*c*

_{2}= −1, the above symmetries (3.15) are just the same ones with that of (3.8). If setting

*c*

_{2}=

*c*

_{3}= 0, and then the classical Lie point symmetry for system (2.1) can be given.

## 4. Symmetry reduction and group invariant solutions to the coupled variable-coefficient Newell-Whitehead system

In order to derive the group invariant solutions in this Section, we solve the symmetry constraint conditions *σ ^{i}* = 0 (

*i*= 1

*,...,*7) defined by (3.13) with (3.15), which is equivalent to solving the following characteristic equation:

Solving the characteristic equation (4.1) under the condition of *c*_{2} ≠ 0, we obtain two nontrivial similar reductions and several substantial invariant solutions listed in the follows.

## Case 1: *c*_{4} ≠ 0

Without the loss of generality, we assume *c*_{1} = 1, *c*_{2} = 1, *c*_{3} = 0, *c*_{4} = *k*_{1}, *F*_{4}(*t*) = *k*_{2}, *F*_{5}(*t*) = *k*_{3}, with *k*_{1}, *k*_{2}, *k*_{3} are arbitrary constants, so the similarity solutions can be obtained after solving out the characteristic equations (4.1)

*F*

_{2}(

*ξ*),

*F*

_{3}(

*ξ*),

*F*

_{4}(

*ξ*), and

*F*

_{5}(

*ξ*) are of the forms:

*C*

_{1},

*C*

_{2}are arbitrary constants. Since the auxiliary dependent variable

*f*satisfies the Schwartzian form (3.7) of the coupled variable-coefficient Newell-Whitehead system, substituting

*F*

_{1}satisfies the following reduction equation

*F*(

*ξ*) =

*F*=

*F*

_{1}

*.*

_{ξ}It is easy to verify that the equation (4.4) is equivalent to the following elliptic equation

*C*

_{1},

*C*

_{2},

*C*

_{3}are arbitrary constants.

It is obvious that once the *F*_{1} is solved with Eq. (4.4), *F*_{2}, *F*_{3}, *F*_{4}, *F*_{5} can be obtained directly from Eq. (4.3). From Eq. (4.5), we know that *F* can be written in terms of Jacobi Elliptic functions, with *F* bearing the following form

Substituting Eq. (4.7) into Eq. (4.5) and solving the over-determined equations with *Maple* will yield

*k*

_{1},

*b*

_{1},

*λ*∈

*R*, 0 ⩽

*n*⩽ 1. Substituting Eq. (4.7), (4.8) and

*F*

_{1}

*=*

_{ξ}*F*into Eq. (4.3) leads to the explicit solution of

*p, q*. The result is omitted here because of its prolixity, but the corresponding images for

*p*and

*q*are as follows

In Fig. 1, we plot the interaction solutions between solitary waves and elliptic function waves expressed by (4.2) with parameters {*k*_{1} = 1, *k*_{2} = 10000, *b*_{1} = 1000, *k*_{3} = 10, *C*_{1} = 0*.*10, *C*_{2} = 0*.*1, *λ* = 0*.*001}.

## Case 2: *c*_{4} = 0

Assume *c*_{1} = *k*_{4}, *c*_{2} = *k*_{5}, *c*_{3} = 0, *c*_{4} = 0, *F*_{1}(*t*) = 1, *F*_{4}(*t*) = 1, with *k*_{4}, *k*_{5} as arbitrary constants. Solving the following characteristic equations

*ζ*= −

*k*

_{4}

*t*+

*x*,

*K*

_{1},

*K*

_{2}are arbitrary constants.

Substituting Eq. (4.10) into the prolonged system yields

*F*=

*F*(

*ζ*) =

*F*

_{1}

*meets the reduced equation as follows*

_{ζ}Then, another type of soliton-cnodial wave solutions can be obtained via taking

Substituting the expressions (4.15), (4.14) and (4.11) into (4.10) comes the exact solutions for the variable coefficient Newell-Whitehead system (2.1), which proves that the solutions *p, q* are rational functions.

## 5. Summary and Discussion

On the basis of the classical Lie point method and Lax pair, by making some changes of the assumption of the symmetry, we not only derive the local symmetries, but also the nonlocal ones to the variable coefficient Newell-Whitehead system. The latter is the prime object of this paper. By introducing the auxiliary variable, the nonlocal symmetries are readily localized to Lie point symmetry via extending the original system to a large system. Meanwhile, in the process of localization, the corresponding Schwartz form of the system (2.1) is given from the Lax pair. Then exploring the Lie group theorem to these local symmetries, the group invariant solutions of system (2.1) are derived. Moreover, by using standard Lie point symmetry approach to study the similarity reductions of the prolonged system, two types of group invariant solution are presented in this paper, including the special interaction solution between the soliton and the cnodial periodic wave, and the soliton with rational function solution. This kind of solution can be easily applicable to the analysis of physically interesting processes.

The method presented in this paper could be applied to other variable coefficient integrable models. One can also consider another interesting subject, the relationship between nonlocal symmetry obtained from the Lax pair and other methods, such as the truncated Painlevé expansion and Darboux transformation method. Furthermore, though the localization is the important step to expand application of the nonlocal symmetry, there is no universal way to estimate which type of nonlocal symmetry can be localized to the point symmetry, which will be discussed in our future work.

## Acknowledgments

The project is supported by the National Natural Science Foundation of China (Grant Nos. 11471004, 11775047, 11505090), The Chinese Post doctoral Science Foundation (No. 2020M673332), the Natural Science Foundation of Shaanxi Province (No. 2018JQ-1045), Research Award Foundation for Outstanding Young Scientists of Shandong Province (No. BS2015SF009) and the Science and Technology Innovation Foundation of Xi’an (2017CGWL06), the Scientific research project of Shaanxi Provincial Department of Education (No. 19JK0737).

## References

### Cite this article

TY - JOUR AU - Yarong Xia AU - Ruoxia Yao AU - Xiangpeng Xin PY - 2020 DA - 2020/09/04 TI - Nonlocal symmetries and group invariant solutions for the coupled variable-coefficient Newell-Whitehead system JO - Journal of Nonlinear Mathematical Physics SP - 581 EP - 591 VL - 27 IS - 4 SN - 1776-0852 UR - https://doi.org/10.1080/14029251.2020.1819601 DO - 10.1080/14029251.2020.1819601 ID - Xia2020 ER -