Similarity Reduction and Exact Solutions of a Boussinesq-like Equation

1 Introduction

In 1989, a method called Clarkson and Kruskal (CK) direct method was firstly proposed by Clarkson and Kruskal to seek similarity reduction of the nonlinear partial differential equation, and they obtained many similarity solutions different from ones obtained by the classical Lie group approach [1] and the nonclassical Lie group approach [2]. CK direct method is an ansatz-based method and it does not involve group transformation, so it is widely loved and used by non-mathematicians. By using CK direct method, one can directly obtain the symmetry reduction of high dimensional system and nonlinear system. Up to now, a lot of nonlinear partial differential equations have been reduced to ODEs or low order PDEs, such as the generalised Burgers equations [3], the (2+1)-dimensional KdV equation [4], the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation [5], and the (2+1)-dimensional Lax-Kadomtsev-Petviashvili equation [6].

On the other hand, Lie symmetry analysis is also an important subject and it has been applied to many fields of mathematics and physics, such as Lie algebras [7], classical mechanics [8], and other nonlinear partial differential equations. Lie symmetry is a point transformation. And it was proposed to reduce the order of a given differential equation. With the development of computers, a lot of mathematical and physical equations are solved by the method. And it becomes one of the most common methods with Darboux transformation [9], [10], [11], Bäcklund transformation [12], [13], Painlevé analysis [14], [15], CK direct method and Hirota bilinear approach [16], [17], [18]. Recently, Lie symmetry analysis has been successfully used to find some group invariant solutions for a number of integrable systems such as the Heisenberg equation [19], the Sawada-Kotera equation [20], the (3+1)-dimensional Yu-Toda-Sasa-Fukuyama equation [21], and the modified KdV-Zakharov-Kuznetsov equation [22].

In this paper, we focus on the following Boussinesq-like equation

which was proposed in [23]. As we all know, the Boussinesq equation is a very famous nonlinear evolution equation in describing the propagation of long waves in shallow water. Since then, Boussinesq-like equations were proposed to study variable water depth problems [24], and now the applications of Boussinesq-like equation involve a wide range of fields such as the dynamics of the thin inviscid layer with free surface, the surface waves, the nonlinear string, the shape-memory alloys or other nonlinear phenomena [25], [26], [27], [28]. Different from the regularised Boussinesq equation, (1) retains the fourth spatial derivative u_xxxx but eliminates the dissipative term u_xx. Then it is difficult to analyse because the model is no more fully integrable. In this paper, we study the Boussinesq-like equation with the CK direct method to find the similarity reduction equation and the corresponding exact solutions. We also compare the results obtained by the CK direct method and Lie symmetry method.

In the second section of this paper, we discuss the similarity reduction and the corresponding similarity solutions of the Boussinesq-like equation with the CK direct method. In the third section, the similarity reduction and the corresponding group invariant solutions of the Boussinesq-like equation are discussed by using Lie symmetry method. The group explanation of reduction is introduced to demonstrate the connection of the two methods in the fourth section. And in the last section, we give summary and discussions.

2 Clarkson and Kruskal Direct Method and Similarity Reduction

In order to find similarity solutions of a (1+1)-dimensional partial differential equation, Clarkson and Kruskal have indicated the solution in the form of [29]

On this base, one can prove it is sufficient that the solution u to a nonlinear system may commonly be simplified as the following linear ansatz

Substituting the ansatz (3) into the (1) yields

where primes represent the derivatives with respect to ξ. In order to make (4) be an ODE of P(ξ), the ratio of the coefficients of the derivatives of P(ξ), including the combinations of their products, cannot be anything but the function of ξ. In (4), we use the coefficient of P⁽⁴⁾ (i.e. −βξx4) as the normalizing coefficient and we have

where Γ₁(ξ), Γ₂(ξ),…,Γ₁₃(ξ) are functions of ξ to be determined. Before dealing with the equations above, we propose the following remarks.

Remark 1: If α(x, t) has the form α=α₀(x, t)+ β(x, t)Ω(ξ), then one can take Ω(ξ)=0;

Remark 2: If β(x, t) has the form β=β₀(x, t)Ω(ξ), then one can take Ω(ξ)=1;

Remark 3: If ξ(x, t) can be determined by an equation of the form Γ(ξ)=ξ₀(x, t), then one can take Γ(ξ)=ξ;

Remark 4: In order to facilitate calculation, one can take the original letter for the undermined function obtained by integrating, differentiating, taking the index or logarithm, and scaling transformation.

By using Remarks 1–4, it is easy to obtain the similarity reductions of the Boussinesq-like equation. If ξ_x=0, then ξ and P(ξ) are only the functions of t and the similarity reduction equation is only obviously reduced to the second order ODE. So here, we consider only the ξ_x≠0 case.

For ξ_x≠0, it is easy to obtain

from (12) by using the Remark 2. Here we take

and substitute (19) into (5), by scaling transformation for Γ₁(ξ), we can obtain

Integrating (20) respect to x and using the Remark 4, we have

where θ(t)≡θ are functions of t.

By taking the index on both sides of the (21) and using the Remark 4 again, we can obtain

Integrating (22) respect to x and using the Remark 4, we have

where σ(t)≡σ are functions of t.

Using the Remark 3, we can obtain

Obviously,

From (10), we can get

by using the Remark 1.

Substituting (24)~(26) into (4), we can obtain

Using the coefficient of P⁽⁴⁾ as the normalizing coefficient again, we have

where γ₁(ξ), γ₂(ξ), γ₃(ξ) are functions of ξ to be determined. From (28), we have

where γ(ξ)=(γ1(ξ))12.

First, because ξ=θ(t)x+σ(t) and the right-hand side of the (31) is a linear function related to x, we can assume

with A and B being arbitrary constants.

It is easy to obtain

Substituting (32) into (31), we can obtain

Comparing the coefficients of x on both sides of the (34), we have

Substituting (35) and (36) into (29) and (30), we can obtain

Now the similarity reduction of the Boussinesq-like equation reads

with (24), (35), and (36).

Finally, by substituting (28)~(30), (33) and (37) into (27), we can obtain the corresponding similarity reduction equation for P,

If A=0, from (35) and (36), we can obtain the general solution for θ and σ:

with a₀ and b₀ being arbitrary constants, and the similarity reduction of the Boussinesq-like equation has the form

then the similarity reduction (41) is a travelling reduction and (39) is reduced to

By integrating (42) twice, we can obtain the general ODE

with c₀ and c₁ being arbitrary constants. Without loss of generality, we take c₀=c₁=0, then the (43) becomes

and (44) is the Duffing equation which has been studied extensively. Here we give only some solutions of the Duffing equation:

1. Bell-shaped soliton solution

   (45)P1=±B[sech(±Bξ)].

2. Jacobi elliptic sine function solution

   (46)P2=a[sn(±a2−B2ξ, m)],

   where a is a constant and m=±aB2−a2.

3. Jacobi elliptic cosine function solution

   (47)P3=a[cn(±amξ, m)],

   where a is a constant and m=±a2a2−B2.

4. The third kind of Jacobi elliptic function solution

   (48)P4=a[dn(±aξ, m)],

where a is a constant and m=±2a2−B2a.

There are different types of the exact solutions of the Duffing equation, thus we can obtain the corresponding solutions of the Boussinesq-like equation from (41).

If A≠0, from (35) and (36), we can obtain the general solution for θ and σ:

with a₀ and b₀ being arbitrary constants, and the similarity reduction of the Boussinesq-like equation has the form

with P(ξ) being determined by (39).

3 Lie Symmetry Analysis and Group Invariant Solutions

In this section, we perform Lie symmetry analysis on the Boussinesq-like equation in order to obtain similarity reduction. In the classical Lie group theory, we know that if the Boussinesq-like equation is invariant under one-parameter Lie group of transformation,

then, the group invariants ξ, P(ξ) can be obtained by dealing with characteristic equation,

Now we take u→u+εσ into (1) and extract only coefficients of ε, then we can obtain the following symmetry equation:

and the corresponding symmetry is

Substituting (56) into (55) and using (1) again, we can obtain

with C₁, C₂ and C₃ being arbitrary constants. we will study three cases by the different selections of the arbitrary constants.

Case 1. C₁=1, C₂=C₃=0.

Considering the characteristic equation,

From equation dxx=dt2t, we can obtain

and from equation dt2t=du−u, we can obtain

with c₁, c₂ being arbitrary constants. In the classical Lie group theory, we can define

then we can obtain a similarity reduction of the Boussinesq-like equation

The final corresponding reduction equation for V can be obtained by substituting (65) into (1),

It is easy to see that (66) is just a special case of (39).

Case 2. C₁=0, C₂=k, C₃=1.

Considering the characteristic equation,

Similarly, we can obtain a travelling wave reduction of the Boussinesq-like equation

The final corresponding reduction equation for V can be obtained by substituting (68) into (1),

It is obvious that (69) is equivalent to (42).

Case 3. C₂=1, C₁=C₃=0.

Considering the characteristic equation,

Similarly, we can obtain a similarity reduction of the Boussinesq-like equation

The corresponding reduction equation for V can be obtained by substituting (71) into (1),

i.e.

with a, b being arbitrary constants. Then a similarity solution of the Boussinesq-like equation can be obtained by substituting (73) into (71),

4 The Connection of the CK Direct Method and Lie Approach

In the second section and third section, we can see that CK direct method is efficient and convenient because of not using group theory. And the similarity reductions by using it contain the similarity reductions obtained by the classical Lie symmetry analysis.

In the following paper, we will demonstrate that the similarity reduction obtained by CK direct method is the same as that one by the Lie symmetry analysis [30].

Now we give a group explanation of the reduction (38) with (24). In reality, from (24)

we have

i.e.

In the same way, we have

i.e.

Therefore, the corresponding symmetry of the reduction in the sense of CK direct method has the form

Substituting (35), (36) and (80) into (55), we can obtain

i.e. T=θ⁻² for A≠0 and T=1 for A=0. If T is determined by (81), then σ given by (80) is really a solution of (55). That means the similarity reduction obtained by CK direct method also can be obtained by the classical Lie approach.

5 Summary and Discussion

Clarkson and Kruskal direct method is performed on the Boussinesq-like equation, the similarity reduction and similarity solutions of the Boussinesq-like equation have been studied. On the other hand, the similarity reduction and group invariant solutions of the Boussinesq-like equation also have been obtained with Lie symmetry analysis. Further, the group explanation of the reduction has been discussed to compare the results obtained by the two methods.

From the analysis of this paper, we establish the relation between the Boussinesq-like equation and the Duffing equation which has been studied extensively, and get some new forms of solution of the Boussinesq-like equation. We also demonstrate the similarity reduction obtained by CK direct method is the same as that one by the Lie symmetry analysis. In addition, these methods are also convenient to solve high dimensional nonlinear systems and many coupled integrable systems.