Space–time fractional nonlinear partial differential system: Exact solution and conservation laws

https://doi.org/10.1007/s12043-021-02205-x

Space–time fractional nonlinear partial differential system:

Exact solution and conservation laws

BALJINDER KOUR^1,2

1Department of Mathematics, Akal University, Talwandi Sabo, Bathinda 151 302, India

2Department of Mathematics and Statistics, Central University of Punjab, Bathinda 151 401, India E-mail: baljindervirk172@gmail.com

MS received 24 November 2020; revised 11 May 2021; accepted 18 June 2021

Abstract. The objective of this manuscript is to analyse space–time fractional generalised Hirota–Satsuma coupled Korteweg–de Vries (HSCKdV) system with time-dependent variable coefficients for exact solution using power series method corresponding to Lie symmetry reduction of HSCKdV system. The exact solution obtained in power series form is further analysed for convergence. Conservation laws of the HSCKdV system are constructed by using the new conservation theorem and generalised fractional Noether’s operator.

Keywords. Fractional differential equations with time-dependent variable coefficients; power series solution;

conservation laws.

PACS Nos 02.30. Jr; 45.10.Hj; 02.30.Lt; 11.30.-j

1. Introduction

In numerous research and development applications, the fractional partial differential equations (FPDEs) appear in certain fields including physics, biology, hydrody- namics, viscoelasticity, control theory, electrochemistry etc. [1–12]. They have recently gained great attention, and there has been relevant scientific progress that has taken place in this field. It is very well-known that the key instrument for characterising the nonlinear physical phenomena is to study the nonlinear fractional partial differential equations (NLFPDEs).

Several powerful methods for finding exact solu- tions of NLFPDEs have been developed in the literature recently. Some of them are: the method of Lie symme- try, the method of exp function, the method of fractional sub-equation, the method of exact power series solu- tions, residual power series, (D^α_t G/G)-method, etc.

For a deeper understanding of nonlinear phenomena, researchers are increasingly interested in space–time NLFPDEs rather than in integer-order NLPDEs. There are many techniques to find solution of fractional partial differential equations. In some publications, the inves- tigation of conservation laws has been defined [13–22].

The conservation laws can be investigated for NLFPDEs which are very important mechanisms for the study of

mathematical along with physical differential equations [23,24]. Noether’s theorem [25] involves a structured methodology for constructing FPDE’s conservation laws, using symmetries associated with Noether’s oper- ator. However, there are several other methods to obtain the conservation laws of PDEs [26–28].

After finding symmetry transformations by classi- cal method, the conserved vectors can be found using symmetry transformations. For this, the considered problem must be investigated for the nonlinear self- adjointness. Fractional-order Noether’s operators have been generalised to find conservation laws using a new conservation theorem [22].

In certain FPDEs, the power series approach is being used to discover exact solution in the form of convergent power series. This type of solution generally assumes a power series having coefficients to be determined, and the substitution that series into the FPDEs provides a recurrence relation between coefficients. Power series solutions are very important solutions of FPDEs. The power series method (PSM) [29,30] illustrates how to find the exact series solution of FPDEs. The system of space–time fractional generalised Hirota–Satsuma coupled Korteweg–de Vries (HSCKdV) equations is studied for explicit power series solution, and conser- vation laws in this work.

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The main aim of this article is to find exact solutions of the governing system. For this, the Lie symmetry reduction of the governing system obtained in our pre- vious work [31] is used. The PSM is imposed on the reduced form of the governing system, involving frac- tional Erdèlyi–Kober differential operators. The power series solutions are obtained for reduced form and then using symmetry transformations can be obtained for the original equation. The conservation laws and symme- tries of FPDEs have one to one correspondence. The new conservation theorem [22] and generalised Noether’s operators are imposed to construct conservation laws.

Consider the space–time fractional generalised Hirota–Satsuma coupled Korteweg–de Vries equations [5,32] with time-dependent variable coefficients as

∂^γu

∂t^γ = A1(t)u∂^βu

∂x^β +A2(t)v∂^βv

∂x^β +A3(t)∂^βw

∂x^β +A4(t)∂³u

∂x³,

∂^γv

∂t^γ = A₅(t)u∂^βv

∂x^β +A₆(t)∂³v

∂x³,

∂^γw

∂t^γ = A7(t)u∂^βw

∂x^β +A8(t)∂³w

∂x³. (1)

The generalised HSCKdV equations with integer order and fractional order have been studied, by many researchers, using different methods. The generalised HSCKdV equations has been studied in [5] using vari- ational iteration method. In [11], soliton solutions have been obtained. Numerical solutions have been obtained in [33]. Lie symmetry reduction of fractional HSCKdV has been obtained in [34]. We have studied the govern- ing system using Lie symmetry method for symmetry reduction in [31].

The sections are organised as follows: Section2con- tains the preliminaries. Power series solution is provided in §3. Section4 presents the convergence of the exact solution from §3. In §5, conservation laws are con- structed. In §6, conclusion of the study is drawn.

2. Preliminaries

The governing system (1) is studied for symmetry reduc- tion via Lie symmetry method in our previous work [31].

The present section contains the important results from [31] for further study of system (1) in this article.

Under one-parameter Lie group of transformations, the space–time fractional generalised HSCKdV with time-dependent variable coefficients (1) with A4(t) = K₄t^m, A₆(t) = K₆t^s and A₈(t) = K₈t^r (where K4,K6,K8 are arbitrary constants andm,s,r are real numbers) is invariant. Then, the following values are

obtained in [31]:

A1(t)=K1t^{1−23γ+β(γ3+m)},

A2(t)=K2A1(t), A3(t)=K3A1(t), A5(t)=K5A1(t), A7(t)=K7A1(t),

whereK1,K2,K3,K5,K7 are arbitrary constants.

Further infinitesimals of system (1) has been obtained in [31]

ξ = c1x

3 , τ = c1t

γ +m, η= γ −1 2(γ +m)c1u, φ = γ −1

2(γ +m)c1v, ψ = γ −1

2(γ +m)c1w, (2) for some arbitrary constantc₁.

Also, symmetry variable and similarity transforma- tions have been obtained as

y =xt^−γ+m3 (3)

u(x,t)=t^γ−21g(y), v(x,t)=t^γ−21h(y),

w(x,t)=t^γ−12 f(y). (4) The symmetry reductions of system (1) have been obtained in [31] as

T ^1−γ23 ^,γ

(γ+m) g

(y)=y^−β

M1g(y)

R^−β,β₁ g (y)

+(M₂h(y)

R^−β,β₁ h (y)

+(M3

R^−β,β₁ f (y))

+K4g(y),

T ^1−γ23 ^,γ

(γ+m) h

(y)=K6h(y)+M4y^−βg(y)

×

R^−β,β₁ h (y),

T ^1−γ23 ^,γ (γ+m) f

(y)=K8f(y)+M5y^−βg(y)

×

R^−β,β₁ f

(y), (5)

where

T ^1−γ23 ^,γ (γ+m)

is the left fractional Erdèlyi–Kober differential operator and(R^−β,β₁ )is the right fractional Erdèlyi–Kober differential operator [31] and M₂ = K2M1, M3 = K3M1, M4 = K5M1, M5 = K7M1 are arbitrary constants.

3. Power series solutions

Here, we examine the analytic explicit solution using power series technique [29,30,35,36] of the reduced sys- tem of NLFODEs (5) of considered system (1).

Let us take the solution in the form of the following series:

g(y)= ∞ n=0

anyⁿ,

h(y)= ∞ n=0

bnyⁿ,

f(y)= ∞ n=0

c_nyⁿ, (6)

wherean,bnandcnforn=0,1,2, . . .are coefficients.

Therefore, from (5) with (6) we get the following sys- tem:

∞ n=0

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠anyⁿ−M1y^−β

× ∞ n=0

n k=0

akan−k

(1+k) (1+k−β)

yⁿ

−M2y^−β ∞ n=0

n k=0

(1+k) (1+k−β)

bkbn−kyⁿ

−M3y^−β ∞ n=0

(1+n) (1+n−β)

cnyⁿ

−K4

∞ n=0

(n+1)(n+2)(n+3)an+3yⁿ =0, ∞

n=0

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠bnyⁿ

−M4y^−β ∞ n=0

n k=0

(1+k) (1+k−β)

×bkan−kyⁿ−K6

∞ n=0

(n+1)(n+2)

×(n+3)bn+3yⁿ =0, ∞

n=0

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠c_nyⁿ

−

M5y^−β ∞ n=0

n k=0

(1+k) (1+k−β)

×ckan−kyⁿ−K8

∞ n=0

(n+1)(n+2)

×(n+3)cn+3yⁿ =0. (7) Comparing the system of eqs (7) forn ≥0, we have the coefficients as follows:

an+3= 1

K4(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠an

−M₁y^−β n k=0

(1+k) (1+k−β)

a_ka_n−k

−M2y^−β n k=0

(1+k) (1+k−β)

bkbn−k

−M3y^−β

(1+n) (1+n−β)

cn

⎫⎬

⎭, (8)

bn+3= 1

K6(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠bn

−M₄y^−β n k=0

(1+k) (1+k−β)

b_ka_n−k

⎫⎬

⎭,

cn+3= 1

K8(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠cn

−M5y^−β n k=0

(1+k) (1+k−β)

ckan−k

⎫⎬

⎭. (9) With arbitrary constants ai, bi, ci (i = 0,1,2), all coefficientsan(n≥3),bn(n≥3)andcn(n ≥3)can be determined for the power series (6).

Thus, the power series (6) can be written as

g(y)=a0+a1y+a2y²+ 1 6K₄

⎧⎨

⎩

⎛

⎝

1+γ 2 )

1−γ 2 )

⎞

⎠a0

−M1y^−β

1 (1−β)

a₀²

−M₂y^−β

1 (1−β)

y³ +

∞ n=1

1

K4(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠an

−M₁y^−β n k=0

(1+k) (1+k−β)

a_ka_n−k

−M2y^−β n k=0

(1+k) (1+k−β)

bkbn−k

−M3y^−β

(1+n) (1+n−β)

cn

⎫⎬

⎭yⁿ⁺³,

h(y)=b0+b1y+b2y²+ 1 6K6

⎧⎨

⎩

⎛

⎝

1+γ 2 )

1−γ 2 )

⎞

⎠b0

−M₄y^−β

1 (1−β)

b₀a₀

⎫⎬

⎭y³ +

∞ n=1

1

K6(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠bn

−M4y^−β n k=0

(1+k) (1+k−β)

bkan−k

⎫⎬

⎭

×yⁿ⁺³,

f(y)=c₀+c₁y+c₂y² + 1

6K8

⎧⎨

⎩

⎛

⎝

1+γ 2 )

1−γ 2 )

⎞

⎠c0

−M₅y^−β

1 (1−β)

c₀a₀

⎫⎬

⎭y³ +

∞ n=1

1

K8(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠c_n

−M5y^−β n k=0

(1+k) (1+k−β)

ckan−k

⎫⎬

⎭

×yⁿ⁺³. (10) Hence,

u(x,t)=a0t^γ−21 +a1xt^{γ−21−(γ+3m)} +a2x²t^{γ−21−2(γ+3m)} +

∞ n=0

1

K4(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+m₃ )

1−γ

2 −n(^γ+₃^m)

⎞

⎠an

−x^−βt^{γ−21−−β(γ3+m)} n k=0

(1+k) (1+k−β)

×

M1akan−k+M2bkbn−k

−M3x^−βt^{γ−21−−β(γ3+m)}

×

(1+n) (1+n−β)

c_n

⎫⎬

⎭

×xⁿ⁺³t^{γ−12 −(n+3)(γ3 +m)},

v(x,t)=b0t^γ−12 +b1xt^{γ−12 −(γ+m)3} +b2x²t^{γ−12 −2(γ3+m)} +

∞ n=0

1

K6(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠bn

−M₄y^−β n k=0

(1+k) (1+k−β)

b_ka_n−k

⎫⎬

⎭

×xⁿ⁺³t^{γ−12 −(n+3)(γ3 +m)},

w(x,t)=c0t^γ−12 +c1xt^{γ−12 −(γ+m)3} +c2x²t^{γ−12 −2(γ+m)3} +

∞ n=0

1

K8(n+1)(n+2)(n+3)

×

⎧⎨

⎩

⎛

⎝

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m)

⎞

⎠cn

−M5y^−β n k=0

(1+k) (1+k−β)

ckan−k

⎫⎬

⎭

×xⁿ⁺³t^{γ−12 −(n+3)(γ3 +m)} (11) are the required solutions of (1) in the form of power series.

4. Convergence analysis

In this segment, we analyse the convergence [37] of solution (11).

From (8), we have

|an+3| ≤J

|an| + n k=0

|ak||an−k| +

n k=0

|bk||bn−k| + |cn|

, (12)

where J =max

⎧⎨

⎩

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+₃^m) , M1y^−β

(1+k) (1+k−β)

, M2y^−β

(1+k) (1+k−β)

,

M3y^−β

(1+n) (1+n−β)

⎫⎬

⎭, for allk =0,1,2, . . . ,n,

|b_n+3| ≤K

|b_n| + n k=0

(|b_k||a_n−k|)

, (13)

where K =max

⎧⎨

⎩

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+m₃ ) , M1y^−β

(1+k) (1+k−β)

⎫⎬

⎭, for allk =0,1,2, . . . ,n,and

|c_n+3| ≤L

|c_n| + n k=0

(|c_k||a_n−k|)

, (14)

where L =max

⎧⎨

⎩

1+γ

2 −n(^γ+₃^m)

1−γ

2 −n(^γ+m₃ ) ,

M5y^−β

(1+k) (1+k−β)

⎫⎬

⎭, for allk =0,1,2, . . . ,n.

Let us choose following power series:

P = P(y)= ∞ n=0

pnyⁿ,

Q= Q(y)= ∞ n=0

qnyⁿ

and

R =R(y)= ∞ n=0

r_nyⁿ,

having the coefficients with particular condition pi = |ai|, qi = |bi|, ri = |ci|,for i =0,1,2,3 and

p_n+3= J

p_n+ n k=0

p_kp_n−k+ n k=0

q_kq_n−k+r_n

,

q_n+3= K

q_n+ n k=0

(q_kp_n−k)

,

r_n+3= L

r_n+ n k=0

(r_kp_n−k)

, (15)

wheren =0,1,2,3, . . . .Therefore,

|an| ≤ pn, |bn| ≤qn,

|c_n| ≤r_n, n =0,1,2, . . . . (16) Therefore, we have the series

P = P(y)= ∞ n=0

p_nyⁿ,

Q= Q(y)= ∞ n=0

qnyⁿ

and

R =R(y)= ∞ n=0

rnyⁿ,

which are the majorant series of the series g(y), h(y) and f(y)in (6) respectively. Now, we prove the conver- gence of the seriesg(y),h(y)and f(y)by comparison test using the convergence of the majorant power series P(y), Q(y)andR(y).

Therefore, we prove the convergence of the power series P(y), Q(y)andR(y)by using implicit function theorem [38].

P = p0+p1y+ p2y²+p3y³+ ∞ n=1

pn+3yⁿ⁺³

= p0+p1y+ p2y²+p3y³ +

∞ n=1

J

pn+ n k=0

pkpn−k

+ n k=0

q_kq_n−k+r_n

yⁿ⁺³

= p0+p1y+ p2y²+p3y³+J y³((P−p0) +(P²−(p0)²)+(Q²−(q0)²)+(R−r0)),

(17) Q = q₀+q₁y+q₂y²+q₃y³+

∞ n=1

q_n+3yⁿ⁺³

=q0+q1y+q2y²+q3y³ +

∞ n=1

K

qn+ n k=0

(qkpn−k)

yⁿ⁺³

=q0+q1y+q2y²+q3y³+K y³[(Q−q0) +(P Q− p₀q₀)] (18) and

R= r0+r1y+r2y²+r3y³+ ∞ n=1

rn+3yⁿ⁺³

=r0+r1y+r2y²+r3y³ +

∞ n=1

L

r_n+ n k=0

(r_kp_n−k)

yⁿ⁺³

=r0+r1y+r2y²+r3y³+L y³[(R−r0)

+(P R−p0r0)]. (19) Let us assume the implicit functional system

(y,P,Q,R)= P−p0−p1y− p2y²−p3y³

−J y³((P−p₀)+(P²−(p₀)²) +(Q²−(q0)²)+(R−r0)),

(y,P,Q,R)= Q−q0−q1y−q2y²−q3y³

−K y³[(Q−q0)+(P Q− p0q0)],

(y,P,Q,R)= R−r0−r1y−r2y²−r3y³

−L y³[(R−r0)+(P R−p0r0)]. (20) It can be easily seen that the functions , , are analytic in the neighbourhood of (0,p0,q0,r0). Also, (0,p0,q0,r0) = 0, (0,p0,q0,r0) = 0 and

(0,p0,q0,r0)= 0, with non-zero Jacobian determi- nant

∂(, , )

∂(P,Q,R)

(0,p₀,q₀,r₀)=1.

Therefore, the series P = P(y), Q = Q(y) and R = R(y) are analytic in a neighbourhood of the point (0,p0,q0,r0), using implicit function theorem [38]. This proves the convergence of power seriesP = P(y), Q = Q(y) and R = R(y) about the point (0,p0,q0,r0). Therefore, by comparison test the series (6) are convergent with positive radius of convergence.

5. Conservation laws

The present paper contains the conservation laws of fractional generalised HSCKdV (1), by using new con- servation theorem and fractional Noether’s operator [22,39,40].

The continuation equation Dt(C^t)+Dx(C^x)=0,

defines conservation laws. The vectors C^t(x,t,u, v) andC^x(x,t,u, v)define the conserved vectors of (1).

So we have to calculate these vectors by using fractional Noether’s operators. These vector fields spans vector field. In order to construct conserved vectors, the nec- essary definitions are given below:

DEFINITION 1

Euler–Lagrange operator for the FPDEs system Euler–Lagrange operators denoted byδ/δu^jis defined as follows:

δ

δu^j = ∂

∂u^j +(D_t^γ)^∗ ∂

∂(D_t^γu^j) +(D^β_x)^∗ ∂

∂(D^β_xu^j) +

∞ k=1

(−1)^kDi₁Di₂, . . . ,Di_k ∂

∂(u^j)i1,i2,...,ik

, (21) where Di_k is thei_k^th variable total derivative operator.

Also(D^γ_t )^∗and(D^βx)^∗are the adjoint operators of frac- tional derivativesD_t^γ andD_x^β , respectively.

DEFINITION 2

Lagrangian for the FPDEs system(1) The operator defined by

L= p ∂^γu

∂t^γ −A1(t)u∂^βu

∂x^β−A2(t)v∂^βv

∂x^β−A3(t)∂^βw

∂x^β

−A₄(t)∂³u

∂x³

+q ∂^γv

∂t^γ −A₅(t)u∂^βv

∂x^β

−A6(t)∂³v

∂x³

+r ∂^γw

∂t^γ −A7(t)u∂^βw

∂x^β −A8(t)∂³w

∂x³

, (22) where p,qandr are new dependent variables, is called Lagrangian [40] for the FPDEs system (1).

DEFINITION 3

Adjoint equation for the FPDEs system

The adjoint equations for FPDEs system are given by F^∗_j ≡ δL

δu^j =0, j =1,2,3. (23) Now the adjoint equations for (1), are obtained as δL

δu = F₁^∗=(D_t^γ)^∗p−A1(t)(D_x^β)^∗(pu)−A1(t)p∂x^βu

−A5(t)q∂x^βv−A7(t)r∂x^βw+A4(t)D³_x(p)=0, δL

δv = F₂^∗=(D_t^γ)^∗q−A2(t)(D_x^β)^∗(pv)

−A5(t)(D^β_x)^∗(qu)

−A2(t)p∂_x^βv+ A6(t)D³_x(q)=0, δL

δw =F₃^∗=(D_t^γ)^∗r −A3(t)(D^β_x)^∗(p)

−A7(t)(D^β_x)^∗(r u)+A8(t)D³_x(r)=0. (24) The non-linearity of system (1) depends on the condition that (24) satisfies, where new dependent variables

p=ψ(x,t,u, v), q =ϕ(x,t,u, v),

r =μ(x,t,u, v) (25)

are not simultaneously zero.

The third derivative of p=ψ(x,t,u, v)with respect toxis

p_{x x x} =ψx x x+6ψxuvu_xvx+6ψxuwu_xwx

+6ψxvwvxwx+3ψuuvu²_xvx

+3ψuuwu²_xwx+3ψvvwv²xwx

+3ψuwwuxw²x+3ψvwwvxwx²+3ψuuuxux x

+3ψ_vvvxvx x +3ψ_wwwxwx x+3ψuvvuxv_x² +3ψxuux x+3ψxvvx x+3ψxwwx x+ψvvx x x

+3ψuv(uxvx x+vxux x)+3ψuw(uxwx x+wxux x) +3ψvw(vxwx x+wxvx x)+3ψx xvvx +3ψx xuux

+3ψx xwwx+ψuux x x+ψwwx x x+3ψxuuu²_x +3ψxvvv²x+3ψxwwwx²+ψuuuu³_x

+ψvvvv³x+ψwwww³x+2ψuxux x

+2ψvxvx x+2ψwxwx x+6ψuvwuxvxwx. (26) Similarly, we can find the other required derivativesqx x x

andr_{x x x}. Therefore, the nonlinear self-adjointness con- ditions are obtained as follows:

δL

δu =λ1(∂t^γu−A₁(t)u∂_x^βu−A₂(t)v∂^βvx

−A₃(t)∂_x^βw−A₄(t)∂_x³u)

+λ2(∂t^γv−A5(t)u∂x^βv−A6(t)∂x³v) +λ3(∂t^γw−A7(t)u∂x^βw−A8(t)∂x³w), δL

δv =λ4(∂t^γu−A1(t)u∂x^βu−A2(t)v∂^βvx

−A3(t)∂x^βw−A4(t)∂x³u)

+λ5(∂t^γv−A₅(t)u∂^βxv− A₆(t)∂x³v) +λ6(∂t^γw−A7(t)u∂x^βw−A8(t)∂x³w), δL

δw =λ7(∂t^γu− A1(t)u∂x^βu−A2(t)v∂^βvx

−A3(t)∂x^βw−A4(t)∂x³u)

+λ8(∂t^γv−A5(t)u∂^β_xv− A6(t)∂_x³v)

+λ9(∂t^γw−A7(t)u∂x^βw−A8(t)∂x³w), (27) for some undetermined coefficientsλi(i =1,2,3,4,5, 6,7,8,9).

Thus,

(Dt^γ)^∗ψ−A1(t)(D_x^β)^∗(ψu)−A1(t)ψ∂_x^βu

−A5(t)ϕ∂^βxv− A7(t)μ∂x^βw

+A₄(t)(ψx x x+6ψxuvu_xvx+6ψxuwu_xwx

+6ψxvwvxwx+3ψuuvu²_xvx+3ψuuwu²_xwx

+3ψvvwv²xwx+3ψuwwuxw²x

+3ψvwwvxw²x+3ψuuuxux x+3ψvvvxvx x

+3ψwwwxwx x+3ψuvvu_xv_x²+3ψxuu_{x x} +3ψxvvx x+3ψxwwx x

+3ψuv(uxvx x+vxux x) +3ψuw(uxwx x+wxux x)

+3ψvw(vxwx x+wxvx x)+3ψx xvvx

+3ψx xuux+3ψx xwwx+ψuux x x

+ψvvx x x+ψwwx x x+3ψxuuu²_x

+3ψxvvvx²+3ψxwww²x+ψuuuu³_x+ψvvvv³x

+ψwwww³x+2ψuxux x

+2ψvxvx x+2ψwxwx x

+6ψuvwuxvxwx)

=λ1(∂t^γu−A1(t)u∂x^βu−A2(t)v∂^βvx−A3(t)∂x^βw

−A4(t)∂x³u)+λ2(∂t^γv−A5(t)u∂x^βv−A6(t)∂x³v)