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ORIGINAL RESEARCH article

Front. Phys., 28 March 2023

Sec. Mathematical Physics

Volume 11 - 2023 | https://doi.org/10.3389/fphy.2023.1160391

This article is part of the Research TopicSymmetry and Exact Solutions of Nonlinear Mathematical Physics EquationsView all 20 articles

Applications of the invariant subspace method on searching explicit solutions to certain special-type non-linear evolution equations

Gaizhu Qu
Gaizhu Qu1*Mengmeng WangMengmeng Wang2Shoufeng Shen
Shoufeng Shen3*
  • 1School of Mathematics and Physics, Weinan Normal University, Weinan, China
  • 2Department of Mathematics, Hangzhou Zhongce Vocational School Qiantang, Hangzhou, China
  • 3Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, China

We extend the invariant subspace method (ISM) to a class of Hamilton–Jacobi equations (HJEs) and a family of third-order time-fractional dispersive PDEs with the Caputo fractional derivative in this letter. More precisely, the complete classification is presented for such HJEs that admit invariant subspaces governed by solutions of the second-order and third-order linear ordinary differential equations (ODEs). Meanwhile, some concrete equations are derived for the construction of new exact solutions u(x,t)=ni=1Ci(t)fi(x). Then a set of invariant subspaces of the considered third-order time-fractional non-linear dispersive equations are obtained. Based on the Laplace transform method (LTM) and applying several properties of the well known Mitta-Leffer (ML) function, the different types of explicit solutions of a family of third-order time-fractional dispersive PDEs are finally derived.

1 Introduction

One of the recently invented methods to derive the explicit solution of NPDE is ISM, which was initiated by Galaktionov and Svirshchevskii in [1] and many researchers have illustrated its applicability in Refs. [26]. Specifically, Refs. [2, 3, 5, 6] have addressed the basic question of the dimension of invariant subspaces, which in addition to ISM is also relevant to Lie-B¨acklund symmetry (LBS) and the conditional Lie-B¨acklund symmetry (CLBS) [714]. Very recently, Refs. [1523] generalized this method to resolve fractional non-linear partial differential equations (fNPDEs). It is verified that by applying ISM, a fNPDE can be reduced to a system of fractional non-linear ordinary differential equations (fNODEs), which can be solved by known analytical approaches.

In this paper, we analyze the following two families of special-type non-linear evolution equations.

1.1 Hamilton–Jacobi equations

Hamilton–Jacobi equations (HJEs) can be regarded as models for various processes in theoretical physics, quantum mechanics and contemporary problems of control, etc. In Refs. [2428], the authors analyzed HJEs in different directions. References [2932] have also indicated that these equations can be used to depict several properties including blow up behavior and the long time action of non-linear diffusion equations. We will consider the following HJEs

ut=um+2x+p(x)B(u)um+1x+Q(x,u),tR+,xR,(1.1)

where u = u(t, x) and p(x), B(u), Q(x, u) are sufficiently smooth functions of indicated variables. Here we suppose that m ≠ − 1, −2. This assumption means that Eq. 1.1 is a fully non-linear HJE. In Ref. [7], Qu showed that Eq. 1.1 preserves the second-order CLBS with η=uxx+H(u)u2x+G(u)ux+F(u) and classified the solutions for Eq. 1.1.

1.2 Third-order time-fractional dispersive PDEs

The concept of fractional order derivative was initiated with the half-order derivative as considered by Leibniz and L’Hopital and many authors have generalized it to an arbitrary order derivative. Different concepts of fractional derivatives were proposed in [3336]. Now fNPDEs have gained much attention because they can be utilized to represent a large number of physical processes. Some techniques have been employed to solve fNPDEs, but the study of fNPDEs has been still handicapped due to the limitations on dealing with more complex fNODEs.

We will study a family of third-order time-fractional dispersive PDEs

αtα[uδ22ux2]+σux+γ3ux3=F[u]=x[b1u2+b2(ux)2+b3u2ux2],(1.2)

where u = u(t, x), 0 < α ≤ 1, t > 0, and αutα is the Caputo fractional derivative of u with respect to t. The ordinary case α = 1 of Eq. 1.2 was first introduced in [37] and has been discussed in depth by many researchers [38, 39]. In fact, when α = 1, δ = b2 = b3 = 0, Eq. 1.2 becomes the KdV equation. If we take α=δ2=b3=1,b1=32,b2=12, Eq. 1.2 becomes the Camassa–Holm equation [40]:

ut+σux+γuxxxuxxt+3uux=2uxuxx+uuxxx.(1.3)

If α=δ2=b2=b3=b12=1,σ=γ=0, Eq. 1.2 is the Degasperis–Procesi equation [41, 42]:

utuxxt+4uux=3uxuxx+uuxxx.(1.4)

If α = δ2 = 2b2 = b3 = 1, σ = γ = b1 = 0, Eq. 1.2 becomes the Hunter-Saxton equation [1]:

utuxxt=2uxuxx+uuxxx.(1.5)

These equations arise as asymptotic models in the theory of shallow water waves. Many authors have concentrated on studying the above special cases of Eq. 1.2.

The major contents of this paper are as follows. We recall the method of the invariant subspace, and also introduce several definitions and fundamental theorems on fractional derivatives and integrals in Section 2. In Section 3 we obtain the complete invariant subspace classification of Eq. 1.1 and derive the reductions and explicit solutions of several examples by utilizing ISM. In Section 4, combined with LTM and inspired by several properties of the well known ML function, we investigate exact solutions of different cases for Eq. 1.2. In the last section, we make some concluding remarks.

2 Preliminaries

First, we introduce ISM. Then, we give several definitions and properties.

2.1 Invariant subspace method

Now, we will present brief details of ISM for a kth-order NPDE

ut=F(x,u,ux,,ukx)F[u],(2.1)

where ujx=juxj(j=1,,k).

In [15], Gazizov and Kasatkin demonstrated that ISM can be used to reduce a fNPDE to a system of fNODEs.

We focus on the fNPDE of the form

αutα=F(x,u,ux,,ukx)F[u],(2.2)

where αtα is the time-fractional Caputo derivative. Let f1(x), f2(x), …, fn(x) be linearly independent functions and their linear span over R be Wn, namely,

Wn=L{f1(x),f2(x),,fn(x)}{ni=1Cifi(x),CiR}.

Definition 2.1. If differential operator F satisfies F[Wn] ⊆ Wn, the subspace Wn is invariant under F.Let us suppose Eq. 2.2 preserves the subspace Wn, then

F[ni=1Cifi(x)]=ni=1Ψi(C1,C2,,Cn)fi(x)

(C1,C2,,Cn)Rn. Thus Eq. 2.2 has the solution

u(x,t)=ni=1Ci(t)fi(x),

{Ci(t), (i = 1, 2, …, n)} satisfy the n-dimensional dynamical system

αCi(t)tα=Ψ(C1(t),C2(t),,Cn(t)),i=1,2,,n.

Observing that the subspace Wn is determined by a basic solution set of a linear nth-order ODE,

L[y]y(n)+an1(x)y(n1)++a1(x)y+a0(x)y=0.(2.3)

Therefore, the invariant condition F is

L[F[u]]|[H]=0.(2.4)

2.2 Some results on fractional calculus

Definition 2.2. The Riemann–Liouville fractional integral operator of order α > 0 is represented as the following expression:

Iαa+f(t)=1Γ(α)ta(tτ)α1f(τ)dτ,t>a.(2.5)

Where Γ(p)=0exxp1dx is the Euler Gamma function. Note that I0a+f(t)=f(t).

Definition 2.3. The Caputo fractional differential operator of order α > 0 is represented as the following expression:

Dαa+f(t)=Inαa+Dnf(t)={1Γ(nα)ta(tτ)nα1f(n)(τ)dτ,α(n1,n),nN,f(n)(t),α=nN.(2.6)

When α=0,Dαa+f(t)=f(t).We can replace operators Dα0+f(t) and Iα0+f(t) by Dαf(t) and Iαf(t) respectively. The following properties are true for fractional integral and derivative:

Dα[f(t)+g(t)]=Dαf(t)+Dαg(t),DαIαf(t)=f(t),IαDαf(t)=f(t)n1k=0f(k)(0)k!tk,α>0,t>0,Iαtβ=Γ(β+1)Γ(β+α+1)tβ+α,α>0,t>0,β>1,Dαtβ=Γ(β+1)Γ(βα+1)tβα,β>0.

When α ∈ (0, 1], the LT of Caputo fractional derivative has the following expression

L{dαf(t)dtα}=sαˉf(s)sα1f(0),

where ˉf(s)=0estf(t)dt.

Definition 2.4. A ML function is

Eα,β(z)=k=0zkΓ(αk+β),Re(α)>0,Re(β)>0.

Also, Eα,1(z) = Eα(z).We can see the γth order Caputo derivatives of the ML function are:

Dγ[tβ1Eα,β(atα)]=tβγ1Eα,βγ(atα),Dγ[Eα(atα)]=aEα(atα),

aR,γ>0,α>0, and the following presentation gives the LT of function tαk+β1E(k)α,β(±atα), that is

L{tαk+β1E(k)α,β(±atα)}=0tαk+β1estE(k)α,β(±atα)dt=k!sαβ(sαa)k+1,Re(s)>|a|1α.

3 Exact solutions of HJEs

3.1 Invariant subspace classification of Eq. 1.1

For Eq. 1.1, we write it in the form ut=F[u]=um+2x+p(x)B(u)um+1x+Q(x,u). By the maximal dimension n ≤ 2k + 1, we consider the following cases for n = 2, 3.

We investigate n = 2 first. After a straightforward calculation, we obtain that

J1um+3x+J2um+2x+J3um+1x+J4umx+J5um1x+J6u2x+J7ux+J8=0,(3.1)

where Ji(i = 1, 2, …, 8) have the following expressions:

J1=pB,J2=(m+1)(m+2)a21(m+1)a0(m+2)a1+2pB2(m+1)pa1B,J3=pB(2m+3)pa0Bu(2m+1)a1pB+[m(m+1)a21(m+1)a1ma0]pB+2(m+1)(m+2)a1a0u(m+2)a0u,J4=(m+1)[(m+2)a20u+(2ma1a0a0)pB2a0pB]u,J5=m(m+1)pa20u2B,J6=Quu,J7=2Qxu,J8=a0Q+a1Qxa0uQu+Qxx.(3.2)

Observing the above expression Eq. 3.1, we shall discuss four possibilities: m = −3, 1, 2 and m ≠ − 3, 1, 2. For the case of m = −3, we derive the following system

2a0+2a21+a1+2(p+2a1p)B=0,pB+5a1pB+(3a0+6a21+2a1)pB+(3a0pB+4a0a1+a0)u=0,a20u+(a0+6a0a1)pB+2a0pB=0,pa20B=0,Qxu=0,Quu=0,pB+a0Q+a1Qxa0uQu+Qxx=0.(3.3)

From the first equation of Eq. 3.3, it is apparent that B(u) = b1u + b2. By solving the fifth and sixth equations of Eq. 3.3, we obtain Q(x, u) = q1u + Q1(x), where b1, b2 and q1 are arbitrary constants and Q1(x) is a function of x. Inserting B(u) = b1u + b2 and Q(x, u) = q1u + Q1(x) into system Eq. 3.3, we have

2a21+4b1a1p+a1+2a0+2b1p=0,6b1a21p+(4a0+5b1p)a1+2b1a1p+6b1a0p+a0+b1p=0,6b2a21p+5b2a1p+2b2a1p+3b2a0p+b2p=0,6b1a0a1p+a20+2b1a0p+b1a0p=0,6b2a0a1p+2b2a0p+b2a0p=0,b1a20p=0,b2a20p=0.a1Q1+a0Q1+Q1=0.(3.4)

Taking into account the assumption p(x) ≠ 0 and solving the system (3.4), the corresponding classifying equations and two-dimensional invariant subspaces are listed as the first three lines in Table 1 with the case m = −3. The cases of m = 1, 2 and m ≠ − 3, 1, 2 can be dealt in a similar way; therefore, we obtain the invariant subspace classification results, which are presented in Table 1.

TABLE 1
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TABLE 1. Classifications of W2 governed by linear ODEs (2.3) of Eq. 1.1.

When n = 3, we find there is only one case: m = 0, and the corresponding results are listed in Table 2.

TABLE 2
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TABLE 2. Classifications of W3 governed by linear ODEs (2.3) of Eq. 1.1.

3.2 Applications

In this section, we provide a further discussion for addressing with the explicit solutions using the above classification results.

Example 1: The equation

ut=u3x+94xuu2x2716x3u3+q1u(3.5)

admits the two-dimensional invariant subspace W{x32,x32} generated by ODE

y+1xy94x2y=0.

As a result, we derive that

u(x,t)=C1(t)x32+C2(t)x32,

Substituting the above solution into Eq. 3.5, we obtain

C1=q1C1+274C31,C2=814C21C2+q1C2,

For q1 = 0, we can see that

C1=24c154t,C2=c2(27t2c1)32.

For q1 ≠ 0, we have

C1=24c1q1e2q1t27,C2=c2(4c1q1e2q1t27)32e4q1t.

The corresponding solution shown in Figure 1

FIGURE 1
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FIGURE 1. Solution profile of Eq. 3.5.

Example 2: The equation

ut=u4x+q1u(3.6)

admits the invariant subspace W{1,(x13a1)43} governed by ODE

y13xa1y=0.

Then, we arrive at

u(x,t)=C1(t)+C2(t)(x13a1)43,

Inserting the above solution into Eq. 3.6, we obtain

C1=q1C1,C2=25681C42+q1C2,

For q1 = 0, we obtain

C1=c1,C2=3327c2256t.

For q1 ≠ 0, we have

C1=c1eq1t,C2=333q181c2q1e3q1t256.

The corresponding solution shown in Figure 2

FIGURE 2
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FIGURE 2. Solution profile of Eq. 3.6.

Example 3: The equation

ut=um+2xm+2mxuum+1x(3.7)

admits the two-dimensional invariant subspace W{1,xm+2m} governed by ODE

y2mxy=0.

Then we arrive at

u(x,t)=C1(t)+C2(t)xm+2m,

Inserting the above solution into Eq. 3.7, we obtain

C1=0,C2=(m+2m)m+2C1Cm+12,

we can see that

C1=c1,C2=1mm(m+2m)m+2c1t+c2.

The corresponding solution shown in Figure 3

FIGURE 3
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FIGURE 3. Solution profile of Eq. 3.7 with m = 2, c1 = c2 = 1.

Example 4: The equation

ut=u2x+43a2uux+49a22u2+q2u(3.8)

admits the three-dimensional trigonometric invariant subspace W{1,e13a2x,e23a2x} governed by ODE

y+a2y+29a22y=0.

Then we arrive at

u(x,t)=C1(t)+C2(t)e13a2x+C3(t)e23a2x,

Inserting the above solution into Eq. 3.8, we obtain

C1=49a22C21+q2C1,C2=49a22C1C2+q2C2,C3=19a22C22+q2C3,

For q2 = 0, we can see that

C1=99c14a22t,C2=c29c14a22t,C3=c2236(9c14a22t)+c3.

For q2 ≠ 0, we have

C1=9q29c1q2eq2t4a22,C2=c29c1q2eq2t4a22,C3=[a22c2281c1q22(9c1q2eq2t4a22)+c3]eq2t.

The corresponding solution shown in Figure 4

FIGURE 4
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FIGURE 4. Solution profile of Eq. 3.8.

4 Exact solutions of a family of third-order time-fractional dispersive PDEs

Now, we will investigate the different invariant subspaces of non-linear differential operator F[u] and discuss explicit solutions of Eq. 1.2, see the following discussions.

Case 1. Let us consider the following equation

αutα+γ3ux3δ22x2(αutα)=F[u]=x[b1u2+b2(ux)2+b3u2ux2].(4.1)

Here α(0,1){12}, Eq. 4.1 admits the invariant subspace W2=L{1,x}, the reason is that

F[C1+C2x]=2b1C1C2+2b1C22xW2.

This means that Eq. 4.1 has the following explicit solution:

u(x,t)=C1(t)+C2(t)x,

Substituting the solution into Eq. 4.1, we have

dαC1(t)dtα=2b1C1(t)C2(t),(4.2)
dαC2(t)dtα=2b1C22(t).(4.3)

Eqs 4.2, 4.3 provide

C2(t)=12b1Γ(1α)Γ(12α)tα,

and

C1(t)=tα.

Then

u(x,t)=tα+12b1Γ(1α)Γ(12α)tαx.

The corresponding solution shown in Figure 5

FIGURE 5
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FIGURE 5. Solution profile of Eq. 4.1 with α = 1/3, b1 = 2.

Case 2. We consider the equation

αutα+σux+γ3ux3δ22x2(αutα)=F[u]=x[a21(b2+b3)u2+b2(ux)2+b3u2ux2],(4.4)

α ∈ (0, 1], Eq. 4.4 preserves invariant subspace W2=L{1,ea1x}, since

F[C1+C2ea1x]=a31(2b2+b3)C1C2ea1xW2,

which means that Eq. 4.4 has the solution

u(x,t)=C1(t)+C2(t)ea1x.

Plugging the solution into Eq. 4.4, we find

dαC1(t)dtα=0,(4.5)
(1a21δ2)dαC2(t)dtα=a1(σ+γa21)C2(t)+a31(2b2+b3)C1(t)C2(t).(4.6)

Solving Eq. 4.5, C1(t) = c1, c1 is an arbitrary constant, and when a21δ21, letting

μ=a1[σ+γa21+a21(2b2+b3)c1]1a21δ2.

Therefore, Eq. 4.6 becomes

dαC2(t)dtα=μC2(t).(4.7)

Applying the LT to Eq. 4.7, we have

sαL{C2(t)}sα1C2(0)=μL{C2(t)},

namely,

ˉC2(s)=L{C2(t)}=asα1sαμ.

Here C2(0) = a, its inverse LT is

C2(t)=aEα,1(μtα),α(0,1].

where Eα,1(.) is the ML function

Eα,1(μtα)=k=0(μtα)kΓ(αk+1).

Hence, we derive that

u(x,t)=c1+aEα,1(μtα)ea1x.

In the case of α = 1, it is a traveling wave solution

u(x,t)=c1+aeμta1x.

The corresponding solution shown in Figure 6

FIGURE 6
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FIGURE 6. Solution profile of Eq. 4.4.

Case 3. We consider the equation

αutα+σux+γ3ux3δ22x2(αutα)=F[u]=x[a0(b2+b3)u2+b2(ux)2+b3u2ux2],(4.8)

α ∈ (0, 1], Eq. 4.8 admits the two-dimensional invariant subspace W2=L{cos(a0x),sin(a0x)}, since

F[C1cos(a0x)+C2sin(a0x)]=0W2.

This indicates that Eq. 4.8 has the solution

u(x,t)=C1cos(a0x)+C2sin(a0x).

Substituting the solution into Eq. 4.8, we have

dαC1(t)dtα=λC2(t),(4.9)
dαC2(t)dtα=λC1(t).(4.10)

Here, λ=a0(σa0γ)1+a0δ2. By applying the time-fractional derivative dαdtα to Eq. 4.9, we derive that

dαdtαdαC1(t)dtα=λ2C1(t).

Now we discuss the following Cauchy problem:

{dαdtαdαC1(t)dtα=λ2C1(t),C1(0)=a,dαC1(t)dtα|t=0=0.(4.11)

Then, define g(t)=dαC1(t)dtα, and utilizing the LT to this equation, we can see

ˉg(s)=sαˉC1(s)asα1.(4.12)

At the same time, applying LT to the first equation of Eq. 4.11, we obtain

L{dαdtαdαC1(t)dtα}=L{dαg(t)dtα}=sαˉg(s)sα1g(0),(4.13)

Inserting Eq. 4.12 into Eq. 4.13, we find

ˉC1(s)=as2α1s2α+λ2.

whose inverse LT is

C1(t)=aE2α,1(λ2t2α),α(0,1].(4.14)

where E2α,1(.) is the ML function

E2α,1(λ2t2α)=k=0(1)kλ2kt2αkΓ(2αk+1).

Substituting Eq. 4.14 in Eq. 4.10, we get

dαC2(t)dtα=λaE2α,1(λ2t2α).(4.15)

By applying Iα on both sides of Eq. 4.15, we obtain

C2(t)=aλtαE2α,α+1(λ2t2α).

For the sake of simplicity, we set the integration constant to zero. Assuming a = 1, the solution of Eq. 4.8 is

u(x,t)=E2α,1(λ2t2α)cos(a0x)λtαE2α,α+1(λ2t2α)sin(a0x).

Note that for α = 1,

E2,1(λ2t2)=k=0(λ2t2)kΓ(2k+1)=cos(λt),λtE2,2(λ2t2)=λtk=0(λ2t2)kΓ(2k+2)=sin(λt),

and the solution becomes

u(x,t)=cos(λt)cos(a0x)sin(λt)sin(a0x)=cos(λt+a0x).

The corresponding solution shown in Figure 7

FIGURE 7
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FIGURE 7. Solution profile of Eq. 4.8 with a0 = 100, σ = γ = 1, δ = 2.

Case 4. We consider the equation

αutα49γa21ux+γ3ux3δ22x2(αutα)=F[u]=x[19a21u234(ux)2+u2ux2],(4.16)

α ∈ (0, 1], Eq. 4.16 admits the two-dimensional invariant subspace W2=L{e13a1x,e23a1x}, since

F[C1e13a1x+C2e23a1x]=118a31C21e23a1xW2.

This means that the explicit solution has the following form

u(x,t)=C1(t)e13a1x+C2(t)e23a1x.

Substituting the solution into Eq. 4.16, we have

dαC1(t)dtα=λ1C1(t),(4.17)
dαC2(t)dtα=λ2[C1(t)]2,(4.18)

where λ1=a31γa21δ29,λ2=a31188a21δ2. Setting C1(0) = 1 and employing the LT of both sides of Eq. 4.17, we have

ˉC1(s)=sα1sαλ1.

Its inverse LT is

C1(t)=Eα,1(λ1tα),α(0,1].

Utilizing C1(t) in Eq. 4.18, we obtain

dαC2(t)dtα=λ2(Eα,1(λ1tα))2.

However, while the ML function does not fulfill the following composition property

Eα(x)Eα(y)Eα(x+y),

it should be noted that

Eα(xα)=k=0xαkΓ(αk+1)

which satisfies the composition property, that is,

Eα(xα)Eα(yα)=Eα((x+y)α),α>0.

Thus, we find

dαC2(t)dtα=λ2Eα,1(λ1(2t)α)).(4.19)

Taking Iα on Eq. 4.19 and applying the integration of the ML function relation, we derive the following result:

C2(t)=λ2(2t)αEα,α+1(λ1(2t)α)).

Here, we set C2(0) = 0. Hence, the exact solution of Eq. 4.16 associated with W2=L{e13a1x,e23a1x} reads

u(x,t)=Eα,1(λ1tα)e13a1x+λ2(2t)αEα,α+1(λ1(2t)α))e23a1x.

Note that for α = 1,

E1,1(λ1t)=k=0(λ1t)kΓ(k+1)=eλ1t,E1,2(λ1(2t))=k=0(2λ1t)kΓ(k+2)=e2λ1t12λ1t,
u(x,t)=eλ1t13a1x+λ2λ1(e2λ1t1)e23a1x.

The corresponding solution shown in Figure 8

FIGURE 8
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FIGURE 8. Solution profile of Eq. 4.16 with a1 = 1, λ1 = 1, λ2 = 2, δ = 2.

Case 5. We consider the equation

αutα+σux+γ3ux3δ22x2(αutα)=F[u]=x[(b2+b3)u2+b2(ux)2+b3u2ux2],(4.20)

α ∈ (0, 1], Eq. 4.20 admits the three-dimensional invariant subspace W3=L{1,cosx,sinx}, since

F[C1+C2cosx+C3sinx]=(2b2+b3)C1C3cosx(2b2+b3)C1C2sinxW3.

This means that the exact solution has the following form:

u(x,t)=C1(t)+C2(t)cosx+C3(t)sinx.

Substituting the solution into Eq. 4.20, we obtain

dαC1(t)dtα=0,(4.21)
(1+δ2)dαC2(t)dtα=(γσ)C3(t)+(2b2+b3)C1(t)C3(t),(4.22)
(1+δ2)dαC3(t)dtα=(σγ)C2(t)(2b2+b3)C1(t)C2(t).(4.23)

Solving Eq. 4.21, we obtain C1(t) = c1, inserting it into Eq. 4.22 and Eq. 4.23, we find

dαC2(t)dtα=λC3(t),dαC3(t)dtα=λC2(t),

where λ=γδ+c1(2b2+b3)1+δ2, Following the procedure described in case 3, we obtain the exact solution

u(x,t)=c1+E2α,1(λ2t2α)cosxλtαE2α,α+1(λ2t2α)sinx.

Note that for α = 1,

E2,1(λ2t2)=k=0(1)k(λt)2kΓ(2k+1)=cos(λt),λtE2,2(λ2t2)=λtk=0(1)k(λt)2k+1Γ(2k+1)=sin(λt),

and the solution is

u(x,t)=c1+cos(λt)cosxsin(λt)sinx=c1+cos(λt+x),

which is a compacton solution.The corresponding solution shown in Figure 9

FIGURE 9
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FIGURE 9. Solution profile of Eq. 4.20 with α = γ = b2 = b3 = c1 = 1, δ = 10.

Case 6. We consider the equation

αutαδ22x2(αutα)=F[u]=x[b2(ux)2+b3u2ux2],(4.24)

α(0,1){12}, Eq. 4.24 admits the four-dimensional invariant subspace W4=L{1,x,x2,x3}, since

F[C1+C2x+C3x2+C4x3]=6b3C1C4+(4b2+2b3)C2C3+[(8b2+4b3)C23+12(b2+b3)C2C4]x+12(3b2+2b3)C3C4x2+12(3b2+2b3)C24x3W4.

This means that the exact solution has the following form

u(x,t)=C1(t)+C2(t)x+C3(t)x2+C4(t)x3.

Substituting the solution into (4.24), we have

dαC1(t)dtα2δ2dαC3(t)dtα=6b3C1(t)C4(t)+(4b2+2b3)C2(t)C3(t),dαC2(t)dtα6δ2dαC4(t)dtα=(8b2+4b3)C23(t)+12(b2+b3)C2(t)C4(t),dαC3(t)dtα=12(3b2+2b3)C3(t)C4(t),dαC4(t)dtα=12(3b2+2b3)C24(t).

Solving this system, we derive that

C1(t)=2(3b2+2b3)2b2+b3δ2tα+163(3b2+2b3)2[Γ(12α)Γ(1α)]2tα,C2(t)=[δ22(2b2+b3)Γ(1α)Γ(12α)+4(3b2+2b3)Γ(12α)Γ(1α)]tα,C3(t)=tα,C4(t)=112(3b2+2b3)Γ(1α)Γ(12α)tα.

Thus, Eq. 4.24 has the solution

u(x,t)=(3b2+2b3)[22b2+b3δ2+163(3b2+2b3)η2]tα+[4(3b2+2b3)η+12(2b2+b3)ηδ2]tαx+tαx2+112(3b2+2b3)ηtαx3.

where η=Γ(12α)γ(1α).The corresponding solution shown in Figure 10

FIGURE 10
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FIGURE 10. Solution profile of Eq. 4.24 with α = 1/3, b2 = b3 = 1, δ = 10.

5 Conclusion

In this work, a class of HJEs (1.1) and a family of third-order time-fractional dispersive PDEs (1.2) are investigated by utilizing ISM. All invariant subspaces for the considered HJEs are derived and displayed in Table 1 and Table 2. Meanwhile, some exact solutions to the equations are obtained due to the corresponding symmetry reductions. For the third-order time-fractional dispersive PDEs, the right-hand side of Eq. 1.2 is the derivative of a quadratic differential polynomial, therefore they preserve more than one invariant subspace, each of which generates a solution. Then, by employing the LT method and applying several properties of the well known ML function, the different kinds of explicit solutions of Eq. 1.2 are derived.

There are still some important problems to be considered. For instance, how does one use ISM to resolve initial value problems? How can we develop this method to investigate higher-dimensional non-linear equations and their discrete versions? This will be considered in the future. Moreover, in the extended version of this work, we will discuss more complicated fractional differential equations by using ISM.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding authors.

Author contributions

GQ: Investigation, methodology, software, writing—original draft. MW: Writing—review and editing, software. SS: Formal analysis, writing—review and editing, supervision. All authors contributed to the article and approved the submitted version.

Funding

The work was supported by the National Natural Science Foundation of China (Grant No. 11501419), the Natural Science Foundation of Shaanxi Province, China (Grant No. 2021JM-521) and the Key Research Foundation of Weinan City, China (Grant No. 2019ZDYF-JCYJ-118).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: exact solution, Hamilton–Jacobi equation, complete classification, invariant subspace method, Laplace transform

Citation: Qu G, Wang M and Shen S (2023) Applications of the invariant subspace method on searching explicit solutions to certain special-type non-linear evolution equations. Front. Phys. 11:1160391. doi: 10.3389/fphy.2023.1160391

Received: 07 February 2023; Accepted: 22 February 2023;
Published: 28 March 2023.

Edited by:

Gangwei Wang, Hebei University of Economics and Business, China

Reviewed by:

Guofu Yu, Shanghai Jiao Tong University, China
Junchao Chen, Lishui University, China

Copyright © 2023 Qu, Wang and Shen. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Gaizhu Qu, qugaizhu.hi@163.com; Shoufeng Shen, mathssf@zjut.edu.cn

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.