# Journal of Nonlinear Mathematical Physics

Volume 27, Issue 4, September 2020, Pages 581 - 591

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

Authors
Yarong Xia
School of Computer Science, Shaanxi Normal University, Xi’an, Shaanxi, 710119, People’s Republic of China
School of Information and Engineering, Shaanxi Joint Laboratory for Artificial Intelligence, Xi’an University, Xi’an, Shaanxi, 710065, People’s Republic of China,xiayarong2014@126.com
Ruoxia Yao*
School of Computer Science, Shaanxi Normal University, Xi’an, Shaanxi, 710119, People’s Republic of China,rxyao2@hotmail.com
Xiangpeng Xin
School of Mathematical Sciences, Liaocheng University, Liaocheng, Shandong, 252059, People’s Republic of China,xinxiangpeng2012@gmail.com
*Corresponding author.
Corresponding Author
Ruoxia Yao
Received 18 July 2019, Accepted 3 January 2020, Available Online 4 September 2020.
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.

Open Access

## 1. Introduction

Symmetry study is always a powerful method in physics and mathematics fields  . 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 [27 ]. 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 [1012 ]. For these reasons, there are many other interesting applications of the symmetry study [1317 ]. 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  , Galas presented the nonlocal Lie-Bäcklund symmetries by using the Pseudo-potentials, in , , 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  . Recently, based on the auxiliary system (lax pair), Xin and Chen obtained the nonlocal symmetries for the modified Korteweg-de Vries equation  , Miao, Xin and Chen obtained the nonlocal symmetries for the Ablowitz-Kaup-Newell-Segur system  . 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  . Besides, considering the results obtained from nonlocal symmetry reduction, many references  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

pt=12α(t)pxxα(t)p2q+2β(t)p,qt=12α(t)qxx+α(t)q2p2β(t)q,(2.1)
where 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, a(t)=12α(t), b(t) = −α(t) = −2β(t), the system (2.1) reduces to the following variable-coefficient Newell-Whitehead system,
ut=a(t)uxx+b(t)(u3u).

The system (2.1) can efficiently describe many nonlinear phenomena which appear in fluid mechanics, plasma physics, thermonuclear reactions, and population proliferation. The literature  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 

ϕx=Uϕ,ϕt=Vϕ,(2.2)
where
ϕ=(ϕ1ϕ2),U=(λpqλ),V=(ABCA)(2.3)
and
A=αλ212αpq+β,B=αpλ+12αpx,C=αqλ12αqx.

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

pp+ɛσ1,qq+ɛσ2,αα+ɛσ3,ββ+ɛσ4,
that is to say, σi (i = 1,...,4), are symmetries of p, q, α and β, which should satisfy the following linearized equations
σt112σ3pxx12ασxx1+σ3p2q+2ασ1pq+αp2σ22σ4p2βσ1=0,σt2+12σ3qxx12ασxx2σ3q2p2ασ2qpαq2σ1+2σ4q+2βσ2=0,(2.4)
and the symmetries σ1, σ2, σ3 and σ4 can be written as
σ1=X¯px+T¯ptU¯,σ2=X¯qx+T¯qtV¯,σ3=T¯αtA1,σ4=T¯βtB1,(2.5)
where X¯, T¯, U¯, V¯, A1, B1 are functions of x, t, p, q, α, β, ϕ1, ϕ2. Substituting Eq. (2.5) into Eq. (2.4), eliminating pt, qt, and ϕ1x, ϕ1t, ϕ2x, ϕ2t in terms of the lax pair (2.2), we derive a system of determining equations for the functions X¯, T¯, U¯, V¯, A1, B1, which on solving by Maple gives
X¯=F1(x),T¯=F2(t),U¯=c1¯ϕ2pF3(t)+12pddxF1(x)+c2¯p,V¯=12qdF1(x)dx+c1¯ϕ22+F3(t)q,A1=α(dF2(t)dt+2dF1(x)dx),B1=32pqxdF1(x)dx+12c2¯αqpβdF2(t)dt18αd3F1(x)dx312dF3(t)dt,(2.6)
where c1¯, c2¯ are arbitrary constants, F1(x) arbitrary function of x, and F2(t), F3(t) arbitrary functions of t.

### Remark 2.1.

From the results in (2.6), we find that the system (2.5) denotes the local symmetries of the coupled variable-coefficient Newell-Whitehead system when c1¯=0, and when c1¯0 they are nonlocal symmetries.

## 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 c1¯=1, c2¯=0, F1(x) = 0, F2(t) = 0, F3(t) = 0 in (2.6), we obtain the following nonlocal symmetries for (2.1)

σ1=ϕ12,σ2=ϕ22,σ3=0,σ4=0.(3.1)

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

σx5λσ5σ1ϕ2pσ6=0,σx6+λσ6σ2ϕ2qσ5=0,12σ4pxϕ212ασx1ϕ212αpxσ6λ2σ4ϕ1αλ2σ5βσ5+σt5+12ασ1q1ϕ1+12αpσ2ϕ1+12αpqσ5λσ4pϕ2λασ1ϕ2αpλσ6+12σ4pqϕ1=0,λ2σ4ϕ2+αλ2σ6+12σ4qxϕ1+12ασx2ϕ1+12αqxσ5+βσ6+σt612σ4pqϕ212ασ1qϕ212αpσ2ϕ212αpqσ6λσ4qϕ1λασ2ϕ1αqλσ5=0,(3.2)
with σ1, σ2, σ3, σ4 given by (3.1), and σ5, σ6 satisfying the following transformations
ϕ1ϕ1+ɛσ5,ϕ2ϕ2+ɛσ6,ff+ɛσ7.(3.3)

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

σ5=ϕ1f,σ6=ϕ2f,(3.4)
with f satisfying
fx=ϕ1ϕ2,ft=12α(pϕ22+4λϕ1ϕ2),(3.5)
whose corresponding symmetries are
σ7=σf=f2.(3.6)

### 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

α3Sx16α2Cxλ+12αCCx4αC4+4Cα4=0,C=ftfx,S=fxxxfx32(fxxxfx)2,(3.7)
which is just the Schwartz form of the coupled variable-coefficient Newell-Whitehead system (2.1). This may provide a new method to seek for the Schwartz form for PDEs, especially for the discrete integrable models, without using the Painlevé analysis.

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

σ1=ϕ12σ2=ϕ22σ3=0σ4=0σ5=ϕ1f,σ6=ϕ2f,σ7=f2,(3.8)
in the extended space x, t, p, q, α, β, ϕ1, ϕ2, f, and the prolonged equations of (2.1), (2.2) and (3.5) have the following the Lie point symmetry vector:
V=ϕ12pϕ22q+0α+0β+ϕ1fϕ1+ϕ2fϕ2+f2f.(3.9)

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:

dp^(ɛ)dɛ=ϕ12^(ɛ),p^(0)=p,dq^(ɛ)dɛ=ϕ22^(ɛ),q^(0)=q,dα^(ɛ)dɛ=0,α^(0)=α,dβ^(ɛ)dɛ=0,β^(0)=β,dϕ1^(ɛ)dɛ=ϕ1^(ɛ)f^(ɛ),ϕ1^(0)=ϕ1,dϕ2^(ɛ)dɛ=ϕ2^(ɛ)f^(ɛ),ϕ2^(0)=ϕ2,df^(ɛ)dɛ=f2^(ɛ),f^(0)=f,(3.10)
will easily yield the symmetry group transformation theorem as follows.

### Theorem 3.1.

If {p, q, α, β, ϕ1, ϕ2, f } is a solution to the prolonged system (2.1), (2.2) and (3.5), then so is the {p^, q^, α^, β^, ϕ1^, ϕ2^, f^} with

p^=p+ɛϕ121+ɛf,q^=q+ɛϕ221+ɛf,α^=α,ϕ1^=ϕ11+ɛf,ϕ2^=ϕ21+ɛf,β^=β.(3.11)

### 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^ from a given 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

V=Xx+Tt+Pt+Qq+Aα+Bβ+P1ϕ1+P2ϕ2+Ff,(3.12)
where X, T, P, Q, A, B, P1, P2, F are functions of x, t, p, q, α, β, ϕ1, ϕ2, f, which means that the closed system is invariant under the following infinitesimal transformations
(x,t,p,q,α,β,ϕ1,ϕ2,f)(x+ɛX,t+ɛT,p+ɛP,q+ɛQ,α+ɛA,β+ɛB,ϕ1+ɛP1,ϕ2+ɛP2,x+ɛF)
with
σ1=Xpx+TptP,σ2=Xqx+TqtQ,σ3=TαtA,σ4=TβtB,σ5=Xϕ1x+Tϕ1tP1,σ6=Xϕ2x+Tϕ2tP2,σ7=Xfx+TftF,(3.13)
the symmetries σi (i = 1,...,7) are defined as the solution of the linearized equations of the prolonged systems, i.e., (2.4), (3.2) and
σx7+σ5ϕ2+ϕ1σ6=0,σt7+2σ3λϕ2ϕ1+2αλσ5ϕ2+12σ3pϕ2212σ3qϕ12+12ασ1ϕ2212αϕ12σ2+2αλσ6ϕ1+αpσ6ϕ2ασ5ϕ1q=0(3.14)

Substituting Eq. (3.13) into (2.4), (3.2) and (3.14), eliminating the pt, qt, ϕ1x, ϕ1t, ϕ2x, ϕ2t, fx, ft, and collecting the coefficients of the independent partial derivatives of dependent variables p, q, α, β, we obtain a system of overdetermined linear equations for the infinitesimals X, T, P, Q, A, B, P1, P2, F, and by solving these determining equations, we can obtain the following results

X=c1,T=F4(t),P=c2ϕ12+F5(t)p,Q=c2ϕ22+F5(t)q,P1=12ϕ1(2c2f+F5(t)+c3),P2=12ϕ2(2c2f+F5(t)c3),A=ddtF4(t)α,B=ddtF4(t)β+12ddtF5(t),F=c2f2+c3f+c4,(3.15)
where ci (i = 1,...,4) are arbitrary constants, F4(t), F5(t) are functions of t. Specially, when setting c1 = c3 = F3(t) = F4(t) = F5(t) = 0, and c2 = −1, the above symmetries (3.15) are just the same ones with that of (3.8). If setting c2 = c3 = 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 σi = 0 (i = 1,...,7) defined by (3.13) with (3.15), which is equivalent to solving the following characteristic equation:

dxX=dtT=dpP=dqQ=dαA=dβB=dϕ1P1=dϕ2P2=dfF.(4.1)

Solving the characteristic equation (4.1) under the condition of c2 ≠ 0, we obtain two nontrivial similar reductions and several substantial invariant solutions listed in the follows.

## Case 1: c4 ≠ 0

Without the loss of generality, we assume c1 = 1, c2 = 1, c3 = 0, c4 = k1, F4(t) = k2, F5(t) = k3, with k1, k2, k3 are arbitrary constants, so the similarity solutions can be obtained after solving out the characteristic equations (4.1)

p=ek3tk2tanh(Θ)F22(ξ)k1F4(ξ)k1,q=ek3tk2tanh(Θ)F32(ξ)k1F5(ξ)k1,ϕ1=F2(ξ)tanh2(Θ1)ek3t2k2,ϕ2=F3(ξ)tanh2(Θ1)ek3t2k2,α=C1,β=C2,f=k1tanh(Θ),(4.2)
where Θ=k1F1(ξ)+tk2, ξ=k2xtk2. Substituting the expression (4.2) into the prolonged system, we show that F2(ξ), F3(ξ), F4(ξ), and F5(ξ) are of the forms:
F2(ξ)=C2exp((λ+F1ξξ2F1ξ1C1k2+1C1F1ξ)dξ),F3(ξ)=k1F1ξk2F2,F4(ξ)=4k2C1λF22F1ξ+k2C1F22F1ξξ2F22F1ξ+2k2F222k1C1F1ξ2,F5(ξ)=4k1k2C1λF1ξ+k1k2C1F1ξξ+2k1F1ξ2k1k22C1k22F22,(4.3)
where C1, C2 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=k1tanh(Θ) into (3.7), we find that F1 satisfies the following reduction equation
4k1C12F4Fξk22C12F2Fξξξ+4C12k22FFξFξξ3C12k22Fξ316C1k22λFFξ8k2FFξ+12k22Fξ=0,(4.4)
where F(ξ) = F = F1ξ.

It is easy to verify that the equation (4.4) is equivalent to the following elliptic equation

Fξ=1k2C1L0+L1F+L2F2+L3F3+L4F4(4.5)
with
L0=4k22,L1=16C1k22λ8k2,L2=2C12C3k22,L3=2C12C2k2,L4=4C12k1,(4.6)
where C1, C2, C3 are arbitrary constants.

It is obvious that once the F1 is solved with Eq. (4.4), F2, F3, F4, F5 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

F=b0+b1JacobiSN(ξ,n).(4.7)

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

b0=0,b1=b1,k1=k1,k2=14b1λ,k3=2b1,n=8k1λ,(4.8)
with k1, b1, λR, 0 ⩽ n ⩽ 1. Substituting Eq. (4.7), (4.8) and F1ξ = 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 {k1 = 1, k2 = 10000, b1 = 1000, k3 = 10, C1 = 0.10, C2 = 0.1, λ = 0.001}.

## Case 2: c4 = 0

Assume c1 = k4, c2 = k5, c3 = 0, c4 = 0, F1(t) = 1, F4(t) = 1, with k4, k5 as arbitrary constants. Solving the following characteristic equations

dxk4=dt1=dpk5ϕ12p=dqk5ϕ22+q=dα0=dβ0=dϕ112ϕ1(2k5f1)=dϕ212ϕ2(2k5f+1)=dfk5f2.(4.9)
p=et(F22(ζ)k5t+F1(ζ)+F4(ζ)),q=et(F32(ζ)k5t+F1(ζ)+F5(ζ)),ϕ1=e12tF2(ζ)k5t+F1(ζ),ϕ2=e12tF3(ζ)k5t+F1(ζ),α=K1,β=K2,f=1k5t+F1(ζ),(4.10)
where ζ = −k4t + x, K1, K2 are arbitrary constants.

Substituting Eq. (4.10) into the prolonged system yields

F2(ζ)=C2exp((λk4C1¯+F1ζζ2F1ζ+k5C1¯F1ζ)dζ),F3(ζ)=F1ζF2,F4(ζ)=(4λC1¯F1ζ+2k4F1ζC1¯F1ζζ2k5)2C1¯F1ζ2F22,F5(ξ)=4λC1¯F1ζ+2k4F1ζ+C1¯F1ζζ2k52C1¯F22(4.11)
where C1¯ is an arbitrary constant and F = F(ζ) = F1ζ meets the reduced equation as follows
C12¯F2Fζζζ4C12¯FFζFζζ+16k5λC1¯FFζ+3C12¯Fζ3+8k4k5FFζ12k52Fζ=0,(4.12)
which is equivalent to the following elliptic equation
dF(ζ)dζ=2C12¯C2¯F3(ζ)+2C12¯C3¯F2(ζ)8(2k5λC1¯+k4k5)F(ζ)+4k52C1¯.(4.13)

Then, another type of soliton-cnodial wave solutions can be obtained via taking

F(ζ)=1L0+L1JacobiSN(ζ,m)(4.14)
into the reduced Eq. (4.12), which yields the following eight sets of solutions
{L0=±L1m,L1=L1,k4=(C1+2C1λ),k5=C1m2L1},{L0=±L1m,L1=L1,k4=(C12C1λ),k5=±C1m2L1},{L0=±L1,L1=L1,k4=(C1m+2C1λ),k5=C1m2L1},{L0=±L1,L1=L1,k4=(C1m+2C1λ),k5=±C1m2L1}.(4.15)

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).