Analytical Cartesian solutions of the multi-component Camassa-Holm equations

Hongli An; Liying Hou; Manwai Yuen

doi:10.1080/14029251.2019.1591725

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Volume 26, Issue 2, March 2019, Pages 255 - 272

Analytical Cartesian solutions of the multi-component Camassa-Holm equations

Authors

Hongli An^*

College of Science, Nanjing Agricultural University, Nanjing 210095, People’s Republic of China,kaixinguoan@163.com

Liying Hou

College of Science, Nanjing Agricultural University, Nanjing 210095, People’s Republic of China,lyhou@njau.edu.cn

Manwai Yuen

Department of Mathematics and Information Technology, The Education University of Hong Kong, Tai Po, New Territories, Hong Kong,nevetsyuen@hotmail.com

^*Corresponding author.

Corresponding Author

Hongli An

Received 10 September 2018, Accepted 5 December 2018, Available Online 6 January 2021.

DOI: 10.1080/14029251.2019.1591725 How to use a DOI?
Keywords: Solution; Analytical Cartesian solution; Camassa-Holm equation; Curve integration theory; Multi-component Camassa-Holm equations
Abstract: Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b(t) + A(t)x and no extra constraint on the dimension N is required. The advantage of our method is that we turn the process of analytically solving MCCH equations into algebraically constructing the suitable matrix A(t). As the applications, we obtain some interesting results: 1) If u is a linear transformation on x ∈ ℝ^N, then p takes a quadratic form of x. 2) If A = f(t)I + D with D^T = −D, we obtain the spiral solutions. When N = 2, the solution can be used to describe “breather-type” oscillating motions of upper free surfaces. 3) If A=(α˙iαi)N×N, we obtain the generalized elliptically symmetric solutions. When N = 2, the solution can be used to describe the drifting phenomena of the shallow water flow.
Copyright: © 2019 The Authors. Published by Atlantis 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

Exact solutions of mathematical physics are usually very important. Not only because they can help us to understand physical phenomena they describe in nature, but also because they can serve as benchmarks for checking and improving numerical codes developed for studying more complex problems. Therefore, a lot of powerful methods has been developed, such as inverse scattering method, Hirota direct method, Darboux transformation and Bäcklund transformation et al [1–10]. However, none of these methods is universal due to the diversity and complexity of PDEs. Therefore, it will be of interest to find other effective methods that can lead to exact solutions.

In this paper, we shall adopt a new method to construct solutions of the multi-component Camassa-Holm-type (CH) equations, which takes the following form [11]:

{ρt=−∇ρ⋅u−ρ(∇⋅u),mt=−u⋅∇m−(∇u)T⋅m−m(∇⋅u)−(∇ρ)Tρ,(1.1)

where ρ is the density, and u, m denote the velocity and momentums of fluid on the n-torus 𝕊ⁿ ≅ ℝⁿ/𝕑ⁿ. In general, it is assumed that there exists a linear operator A such that m = Au, and that A is in a form of αμ + β − Δ with {α, β} = {0, 1} and α + β ≠ 2. While μ(u)=∫𝕊nu(x)dx denotes the mean value operator. The above system was introduced as a framework for studying and modeling fluid dynamics, especially for shallow water waves, turbulence modeling and geophysical fluids [12, 13].

To motivate our study, we shall review some related progresses on the multi-component CH system. The original interest in it may go back to the Camassa-Holm equation

mt=−mxu−2uxm, with m=u−uxx(1.2)

which was derived by Camassa, Holm, Johnson and Constantin et al. as a shallow water approximation (see Refs. [14– 17]). As an important integrable model for describing the dynamics of shallow water waves, the CH equation has been studied extensively and intensively in a number of papers [14–26]. For instance, the complete integrability (as an infinite-dimensional Hamiltonian system) was established via inverse scattering method for suitable classes of initial data in [18, 19]. The bi-Hamiltonian structure and an infinite number of conservation laws of the CH equation were constructed in [14]. The links to the first negative flow of the KdV hierarchy were investigated in [20]. Since the traveling waves of greatest height of the governing equations for water waves are usually peaked [21, 22], peakons solutions are important. Such solutions were derived and their dynamics were analyzed in [23, 24]. Other solutions such as multi-soliton solutions, algebro-geometric solutions were studied in [25, 26]. Inspired by the nice properties of CH equation, the two-component CH equation:

{ρt=−ρux−ρxu,mt=−mxu−2uxm−ρρx, with m=u−uxx(1.3)

was introduced by Chen and Falqui in [27, 28] and also derived by Constantin and Ivanov as a model for shallow water waves in [29]. Experts find that it possed similar properties to the classical CH equation (see [30–32]). One of the important models closely related to the CH equation is the μ-Hunter-Saxton (μ-HS) equation

2μ(u)ux=uxxt+2uxuxx+uuxxx,(1.4)

which was proposed by Khesin et al [33] and simultaneously by Lenells et al with the name of μ-CH equation [34]. When μ(u) = 0, it becomes the HS equation

−uxxt=2uxuxx+uuxxx,(1.5)

for modeling the propagation of nonlinear orientation waves in liquid crystals [35]. Both the μ-HS and HS equations have the two-component generalizations, which are considered as the system (1.3) with m = μ(u) − u_xx and m = −u_xx, respectively. Because these equations posses nice mathematical features and physical interpretations, they have gained much attention from integrable systems and PDE areas (see [36–39]).

We notice that most existing papers deal with the case n = 1 or 2. Investigations on the multi-component CH system mentioned above are rare, expect for the work done in [11, 34, 40, 41]. In particular, little work has been done on seeking exact solutions. From the view of mathematics and physics as explained in [40, 42, 43], the multi-component CH system is also very interesting. This deeply motivates us to undertake the present investigations.

Here, we would like to seek exact solutions with the velocity field of the form

u(x,t)=A(t)x+b(t), x∈ℝN

for the multi-component CH system. This kind solution is part of a long history of finding exact solutions for fluid flows, especially for the Euler and Navier-Stokes (NS) equations [44–51]. A principle result in this direction is the work of Craik and Criminale [49] which gave a comprehensive analysis of solutions to the incompressible NS equations. We notice the continuity equation of multi-component CH equations (1.1) shares some similarities to the Euler and NS equations. Therefore, a natural question comes to us: Can we devise a plan to derive any solutions for the multi-component CH system? If the answer is positive, can we use such solutions to explain or predict any physical phenomenon? Bearing these questions in mind, we expand the investigation on the multi-component CH equations.

The structure of the paper is as follows: In section 2, we show that the multi-component CH equations admit the analytical Cartesian solutions if A satisfies certain matrix differential equations. In section 3, two solvable reductions are considered. Firstly, if A is an antisymmetric constant matrix, then the multi-component CH equations admit exact Cartesian solutions. Secondly, to construct more general solutions, the technique of matrix decomposition is used to make the matrix ODEs solvable. In particular, when N = 2, we obtain the rotational spiral solution and irrotational elliptical solutions obtained by Zhang, An and Yuen et al. The former can be used to describe the motion of “breather”-type oscillations of free surfaces in the upper ocean and the latter can be used to describe the drifting phenomena. In section 4, we discuss the property of such Cartesian solutions. Finally, a short conclusion is attached.

2. Existence of the exact Cartesian solutions

For convenience, we introduce a transformation via

p=12ρ2(2.1)

then, the multi-component CH equations (1.1) are readily reduced into a form of

{mt+(u⋅∇)m+(∇u)T⋅m+mdivu+∇p=0,pt+2pdiv(u)+(u⋅∇)p=0,(2.2)

wherein m = (αμ + β − Δ)u.

Here, our main goal is to seek suitable function p that enables us to obtain the analytical solutions wherein the velocity function u takes a linear form

u=b(t)+A(x)

for the multi-component CH equations. In the above, b(t) is an N-dimensional vector function and A is an N × N matrix function, which are defined via

b(t)=(b1(t),b2(t),…,bN(t))T, A=(aij(t))N×N.

It is noticed that ρ is redefined by p via (2.1), therefore, we only need to deal with p for solving the multi-component CH equations (2.2).

Theorem 2.1.

Defining B as part of a matrix Riccati equation

B=12[At+(A+AT)A+tr(A)A].(2.3)

If A and B satisfy the following matrix differential equations

BT=B,(2.4)

Bt+2tr(A)B+BA+ATB=0,(2.5)

then the multi-component CH equations (2.2) admit the following explicit analytical solutions

u=b(t)+Ax,(2.6)

p=−(αμ+β)[xTbt+xTbtr(A)+xT(A+AT)b+xTBx−c(t)].(2.7)

In the above, b(t) is a vector function and c(t) is a scalar function c(t), which satisfy the following matrix ODE equations:

dt+2dtr(A)+ATd+2BTb=0,(2.8)

ct+2ctr(A)−bTd=0,(2.9)

with

d=bt+[A+AT+Itr(A)]b.(2.10)

Proof.

We now first prove the proposed analytical solution (2.6) will lead to (2.7) by solving the equation (2.2)₁. Substitution (2.6) into the first equation of (2.2) yields

mt+(u⋅∇)m+(∇u)T⋅m+mdivu+∇p(2.11)

=(λ−Δ)(bt+Atx)+[(b+Ax)⋅∇][(λ−Δ)(b+Ax)](2.12)

+[∇(b+Ax)]T⋅[(λ−Δ)(b+Ax)]+[(λ−Δ)(b+Ax)]div(b+Ax)+∇p(2.13)

=λ[bt+Atx+(b⋅∇)Ax+(Ax⋅∇)Ax+(∇Ax)T⋅(b+Ax)+(b+Ax)tr(A)+1λ∇p](2.14)

=λ{bt+btr(A)+(A+AT)b+[At+(A+AT)A+tr(A)A]x+1λ∇p}=0,(2.15)

with λ = αμ + β.

For the convenience of subsequent computations, an auxiliary matrix is introduced via

B=(bij)N×N=12[At+(A+AT)A+tr(A)A],(2.16)

with

bij=12(aij,t+∑k=1N(aikakj+akiakj)+(a11+⋯+ann)aij),

therefore, the equation (2.11) can be readily rewritten into the component form

Qi(x1,…,xN,t)≡−bit−tr(A)bi−∑k=1N[(aik+aki)bk−2bikxk]=1λ∂p∂xi, i=1,2,…,N.(2.17)

It is noticed that for solving p(x, t) from the above equation, all the N equations must be compatible with each other. In other words, the vector functions (Q₁, Q₂,...,Q_N) should be a potential field of p wherein the sufficient and necessary conditions are

∂Qi(x1,…,xN,t)∂xj=∂Qj(x1,…,xN,t)∂xi, i, j=1,2,…,N,(2.18)

which holds if and only if

bij=bji, i, j=1,2,…,N.

The above condition implies that B=(bij)N×N=12[At+(A+AT)A+tr(A)A] is a symmetric matrix, which is just the condition (2.4).

The condition (2.18) shows that the function p(x, t) can be written into a complete differential form

dp(x,t)=∑i=1N∂p(x,t)∂xidxi=∑i=1NλQi(x1,…,xN,t)dxi.

Therefore, we obtain that the second kind of curvilinear integral of p(x, t) is independent of path. So that it allows us to take a special integration route from (0, 0,...,0) to (x₁, x₂,...,x_N)

p(x,t)=∑i=1N∫(0,0,…,0)(x1,x2,…,xN)λQi(x1,x2,…,xN,t)dxi=λ[∫0x1Q1(x1,0,…,0,t)dx1+∫0x2Q2(x1,x2,0,…,0,t)dx2+⋯+∫0xNQN(x1,x2,…,xN,t)dxN]

Directly complicated calculation shows that

p(x,t)=∑i=1N∫(0,0,…,0)(x1,x2,…,xN)λQi(x1,x2,…,xN,t)dxi=λ[∫0x1Q1(x1,0,…,0,t)dx1+⋯+∫0xNQN(x1,x2,…,xN,t)dxN]=−λ[∑i=1N[bit+∑k=1Nakkbi+∑k=1N(aik+aki)bk]xi+∑i=1Nbiixi2+2∑i,k=1,i<kNbikxixk−c(t)]]=−λ[xTbt+xTbtr(A)+xT(A+AT)b+xTBx−c(t)].

At this point, we have finished proving that the functions given by (2.6) and (2.7) satisfy the first equation of (2.2). In the sequel, we shall prove that such functions also satisfy the second equation of (2.2). On use of the relations (2.5), (2.8) and (2.9), we have

pt+2pdiv(u)+(u⋅∇)p=−λ{xT[bt+tr(A)b(A+AT)b]t+xTBtx−ct+2tr(A)[xT(bt+tr(A)b) +xT(A+AT)b+xTBx−c]+(Ax+b)T[bt+tr(A)b(A+AT)b+2Bx]}=−λ{xT[Bt+2tr(A)B+2ATB]x+xT[dt+2dtr(A)+ATd+2BTb]−[ct+2tr(A)c−bTd]}=0,(2.19)

with d = b_t + [A + A^T + I tr(A)]b. While the condition

xT[Bt+2tr(A)B+2ATB]x=0,

implies that B_t + 2tr(A)B + 2A^TB is antisymmetry, namely

[Bt+2tr(A)B+2ATB]T=−[Bt+2tr(A)B+2ATB].

Hence we obtain the following relation

Bt+2tr(A)B+BA+ATB=0

as described in (2.5) in Theorem 2.1. The proof is completed.

To conclude, here we have theoretically obtained the existence of explicit Cartesian vector solutions of (2.2) in Theorem 2.1. The Cartesian solutions are mainly governed by A via the relation (2.5), which is a complex matrix ODE system involving N² scalar equations. As we know, compared with a scalar equation, it is usually very difficult to construct general solutions of a given matrix ODE system. Therefore, to make some solutions available, special techniques are devised in the subsequent section.

3. Special reductions and corresponding solutions

Now, we return back to the condition (2.5) given in Theorem 2.1. Due to the inherent complexity to solve matrix ODE, our attentions are restricted to the cases that lead to some special solutions.

I. The first reduction: A is an antisymmetric constant matrix

It is noted that when A is an antisymmetric constant matrix, the solution can be readily constructed, which is established in the following theorem:

Theorem 3.1.

If A is an anti-symmetric constant matrix, then the multi-component CH equation (2.2) admits a general solution via

u=b(t)+A(x)(3.1)

p=−(αμ+β)[xTbt−c(t)],(3.2)

where b(t) and c(t) satisfies

b(t)=exp(At)b1+b2,(3.3)

c(t)=b1TAb1t+b2Texp(At)b1+b3,(3.4)

where b₁ and b₂ are constant vectors of integration, and b₃ is a constant of integration.

Proof.

Since the solution derived here is just a special case of Theorem 2.1. we just need to validate that the conditions (2.4), (2.5), (2.8) and (2.9) can be satisfied in Theorem 3.1.

When A is an antisymmetric constant matrix, the conditions (2.4) and (2.5) can be easily checked. Insertion of A into (2.8) and (2.9) delivers

btt−Abt=0, ct−bTbt=0.(3.5)

Computation shows the solutions of them are just (3.3) and (3.4).

As the application of Theorem 3.1. here we give an illustrative example:

Example 3.1.

Taking N = 2, for the 2-dimensional multi-component CH equations, setting the anti-symmetric matrix A as

A=(0a−a0),

where a is an arbitrary constant. According to (3.3) and (3.4), we have

b(t)=(c1c2)cosat+(c2−c1)sinat+(c3c4), c(t)=c5

where c_i (i = 1,...,5) are constants of integration. Therefore, the solution of the 2D multi-component CH equations is

u=(u1u2)=(0a−a0)(x1x2)+(c1c2)cosat+(c2−c1)sinat+(c3c4),p=−(αμ+β)[(c2cosat−c1sinat)ax1−(c2sinat+c1cosat)ax2−c5].(3.6)

To shed lights on the behaviors that the solution derived may exhibit, we perform the numerical simulations in Fig. 1. There we choose c₁ = c₂ = 1, c₃ = c₄ = 0. From the figure, we can see that the steady flow is rotational and the velocities of all fluid particles point directly inwards to the origin, which are quite different from those in the radially symmetric solution case.

Example 3.2.

Taking N = 3, for the 3-dimensional multi-component CH equations, setting the anti-symmetric matrix A as

A=a3(01−1−1011−10), b(t)=(b1(t)b2(t)b3(t))=((3c1−c2)cosat−(c1+3c2)sinat−(3c1+c2)cosat−(c1−3c2)sinat2c1sinat+2c2cosat),

So according to (3.1) and (3.2) we can get the following rotational solution:

u1=b1(t)+a3(x2−x1), u2=b2(t)+a3(x3−x1), u3=b3(t)+a3(x1−x2),p=−(αμ+β)(b1tx1+b2tx2+b3tx3−c(t)).(3.7)

II. The second type reduction: A is a decomposable t-dependent matrix

The special case considered here is A being a certain decomposable time-dependent matrix. In particular, we discuss the two cases: one is A = f(t)I + D with D antisymmetric and the other is A=(α˙iαi)N×N. Under these two cases, the corresponding analytical solutions can be obtained.

Theorem 3.2.

Suppose that the matrix A can be decomposed into

A=D+E, DT=−D,(3.8)

where D = A^off and E = A^diag change the off-diagonal part and diagonal part of A, respectively. If D and E satisfy the following matrix differential equations:

Dt=−2ED−Dtr(E),(3.9)

C1≡Ett+6EEt+3tr(E)(Et+2E2)+4E3+Etr(Et)+2E[tr(E)]2=0,(3.10)

C2≡EtD−DEt+tr(E)(ED−DE)+2(E2D−DE2)=0,(3.11)

then

B=12[Et+2E2+tr(E)], (3.12)

is a symmetric matrix satisfying the conditions (2.4) and (2.5) given in Theorem 2.1.

Proof.

With the aid of (3.8) and (3.9), one can have

B=12[At+(A+AT)A+tr(A)A]=12[Dt+Et+(D+E+DT+ET)(D+E)+tr(E)(D+E)]12[Dt+Et+2ED+2E2+Dtr(E)+Etr(E)]12[Et+2E2+tr(E)],

which shows the condition (2.4) is satisfied and B is a symmetric matrix.

In the sequel, we shall show the condition (2.5) can be also satisfied under conditions given in this theorem. On using (3.12) and (3.8), we can obtain

2[Bt+2tr(A)B+BA+ATB]=Ett+4EEt+tr(E)Et+Etr(Et)+2tr(E)[Et+2E2+Etr(E)] +[Et+2E2+Etr(E)](D+E)+(DT+E)[Et+2E2+Etr(E)]=Ett+6EEt+3tr(E)(Et+2E2)+4E3+Etr(Et)+2E[tr(E)]2 +EtD−DEt+tr(E)(ED−DE)+2(E2D−DE2)≡C1+C2=0.

The proof is completed.

In the following, we shall mainly focus on two special cases. One case is when E = f(t)I, D ≠ 0 to be discussed in Corollary 3.1. The other case is when E=(α˙iαi)N×N and D = 0 to be discussed in Corollary 3.2. There exact solutions can be obtained.

Corollary 3.1.

In Theorem 3.2, setting

E=f(t)I,(3.13)

where I is a unitary matrix, then the matrix differential equation (3.9) admits the solution of

D=e−(N+2)∫f(t)dtC,(3.14)

where C is an antisymmetric constant matrix, that is

C=(cij)N×N, cii=0, cij=−cji, i≠j.(3.15)

While Eq. (3.10) is reducible to

ftt+2(2N+3)fft+2(N+1)(N+2)f3=0.(3.16)

Proof.

On inserting (3.13) into (3.9), one can have

Dt=−(N+2)fD.(3.17)

Hence, the solution of D is derived in (3.14). While substituting (3.13) into (3.7), one can directly obtain (3.16) and the condition (3.11) is satisfied automatically.

It is noticed that the equation (3.16) can be rewritten into a form of

[ft+(N+1)f2]t+2(N+2)f[ft+(N+1)f2]=0(3.18)

Thus, in what follows, we shall show its solvability in two cases:

Case 1. If f_t + (N + 1) f² = 0, then the system (3.18) has a solution
f=1(N+1)t+k1,(3.19)
where k₁ is a constant of integration. Then insertion it into (3.14) gives
D=C[(N+1)t+k1]N+2N+1=1[(N+1)t+k1]N+2N+1(cij)N×N, 1≤i<j≤N.(3.20)
with C is an antisymmetric matrix of integration whose elements satisfy (3.15).
Case 2. If f_t + (N + 1) f² ≠ 0, on introduction a function g via
f=∂tlng(3.21)
where ∂t=∂∂t, therefore, the equation (3.18) is now reformulated into
∂tlnggtt+Ngt2g2=∂tlng−2(N+2).(3.22)

Integration (3.22) with respect to t shows that
ggtt+Ngt2=k2g−2(N+1),
which can be readily written into
(gtgN)=k2g−(N+3).

Therefore, we can obtain the following relation
(gN+1)tt=k2(N+1)g−(N+3).(3.23)

In the above, k₂ is an integration constant. Here we go with k₂ ≠ 0. By multiplying gtN+1 on both sides of (3.23), integration shows

gtN+1=±N+1gk3g2−k2,(3.24)

whence

∫gN+1dgk3g2−k2=±t+k4.(3.25)

Its solutions can be readily obtained by trigonometric substitutions, whose forms are dependent on the parameters k₂ and k₃. Here we just take k₂ > 0 and k₃ > 0 as the example to illustrate. Under this condition, we introduce the trigonometric substitution of this form

secφ=k2/k3g:=1kg,(3.26)

then substitution it into (3.25), we obtain

kN+2k2∫secN+2φdφ=kN+2k2(N+1)[−secNφtanφ+N∫secN−1φdφ]=±t+k4.(3.27)

Once N is given, the value of (3.27) can be known. So that g and f can be generated accordingly via (3.21). Once f is generated, the function of D will be obtained via (3.14). Therefore the analytical solutions under Case 2 is derived.

Remark 3.1.

For other choices of k₂ and k₃, one needs to introduce the corresponding trigonometric substitution to find the iterative relation similar to (3.27). Here the calculations procedures are omitted.

Remark 3.2.

We needs to point out that when k₂ = 0, from (3.23), we can obtain that g is governed by

gN+1=h1t+h2.(3.28)

Insertion it into (3.21) shows that

f=gtg=gN+1t(N+1)gN+1=h1(N+1)(h1t+h2)(3.29)

If we choose h₁ = 1 and h₂ = k₁, it is nothing but the solution given in (3.19) in Case 1. Thus, we conclude that the solutions (3.27) obtained in Case 2 are more general.

As the application of the above corollary, we give some examples:

Example 3.3.

Taking N = 2, according to Case 1, we choose k₁ = 0, b(t) = 0, c(t) = 0 and A as

A=D+f(t)I=13t(113t3−13t31),

so that

B=12[Et+2E2+Etr(E)]=19t2(1001).

Therefore, we can obtain an exact solution

u1=13t(x1+x23t3), u2=13t(x2+x13t3)(3.30)

p=−αμ+β9t2(x12+x22).

It is a kind of spiral solution that presented by Zhang and Zheng for Euler equations [52]. Interestingly, here we obtain for the multi-component CH equation. The behaviors of the spiral solution is exhibited in Fig. 2 and Fig. 3. We find from Fig. 2 that the velocities of all fluid particles point directly from the origin to the outside, which are quite different from the rotational solution obtained in (3.6). From Fig. 3, we find the surfaces of p jump up and down with time changing. It is remarkable that such jumping up and down phenomena just coincide with the motion of the “breather-type” oscillations of upper free surfaces in the ocean.

Corollary 3.2.

In Theorem 3.2, if setting

E=(α˙iαi)N×N, D=0(3.31)

Then it is obvious that the relations (2.9)–(2.11) in Theorem 3.2are satisfied. Meanwhile, B=12[At+(A+AT)A+tr(A)A] satisfies the conditions (2.4) and (2.5).

Example 3.4.

Taking N = 1, for 1-dimensional CH equation, we choose c(t) = 0 and A=α˙1α1, so that

B=12[At+(A+AT)A+tr(A)A]=12(α¨1α1+2α˙12α12)(3.32)

Therefore, we obtain an analytical solution

u=α˙1α1x+b(t), p=−αμ+β2[(bt+3bα1α1)x+2(α¨1α1+2α˙12α12)x2−c(t)](3.33)

where α_i is governed by an Emden equation

β¨1=C1β113, β1=α13.(3.34)

In particular, if we choose b(t) = 0 and c(t) = 0, this solution is just what has obtained by Yuen [53].

Example 3.5.

If N > 1, we choose A=(α˙iαi)N×N, so that

B=(bij)N×N=12[At+(A+AT)A+tr(A)A]=12(α¨iαi+2α˙i2αi2+α˙iαi∑k≠inα˙kαk)(3.35)

So that we obtain the general analytical solution

ui=α˙iαixi+bi(t),p=−(αμ+β)[xTbt+xTb(2α˙i2α˙i2+∑k=1nα˙kαk)+12xT(α¨iαi+2α˙i2αi2+α˙iαi∑k≠inα˙kαk)N×Nx−c(t)](3.36)

where α_i is governed by the generalized Emden equation

β¨i+13β˙i∑k≠inβ˙kβk=Ciβi−13∏k=1nβk23, βi=αi3,(3.37)

with C_i is the integration constant. We find that it is just a kind of elliptically symmetric solutions generalized what are obtained by An and Yuen [41]. In order to show the behaviors the solution may posses, we present the numerical simulations when N = 2. It is seen from Fig. 4 that the velocity field u is irrotational. From Fig. 5, we find the center of the flow moves forward with time changing. Such moving behaviors that the solution exhibits is just the drifting phenomena of the shallow water flow.

4. Quasi-linear superposition principle of solutions

In order to shed some light on the properties and structures of the solutions, we now return back to the solutions (2.6) and (2.7). We take b(t) = 0 and c(t) = 0 for simplicity so that the solutions is reducible to

u(x)=Ax, p=−(αμ+β)xTBx.(4.1)

This renders one to consider that the solution u(x) is generated by a linear transformation A on x ∈ ℝ^N:

A:x→Ax=u,

while p = p(x, t) is a quadratic form associated with the matrix B = (A + A^T)A + tr(A)A

B:x→−xTBx=p(x,t).

It is noticed that there exists a quasi-linear superposition principle for Cartesian solution (4.1) that is analogous to the linear equations:

Theorem 4.1.

Suppose that A and B satisfy the conditions in Theorem 2.1, and u(x), u(y) are two solutions whose take the form of (4.1). Then

u(x+y)=u(x)+u(y),p(x+y)=p(x)+p(y)−2xTBy

is a solution of the multi-component CH equations (2.2).

5. Conclusions and discussions

The multi-component CH system is an important mathematical physics model, which has been wide used in fluid mechanics, geophysics, oceanic dynamics and nonlinear optics et al. In the paper, based on the matrix theory and curve integration theory, we have shown that the multi-component CH equations admit analytical Cartesian solutions. Such solutions globally exist and admit a quasi-linear superposition principle under the condition b(t) = 0 and c(t) = 0. As applications, we give some interesting solutions and their numerical simulations. Among these solutions, some are more general than other researchers obtained before. While, to our knowledge, some are completely new. However, there are still some interesting problems that need further consideration:

1)
It is seen that seeking the suitable matrix solution A in (2.5) proves key to derivation of analytical solutions of the multi-component CH equations. However, due to the inherent complexity of solving matrix different equations, it constitutes an obstacle to give the general solutions of A. So that, in this paper, we only consider three special cases: A being an antisymmetric constant matrix, A = f(t)I +D with D^T = −D, and A=(α˙iαi)N×N. Therefore, it would be of interest to consider how to find the more general reductions for (2.5) leads to the more general solutions of multi-component CH equation available.
2)
It is noted that the solutions u we obtained take the linear form with respect to the spatial variables x. Thus, it is natural to inquire whether the nonlinear form type solutions u exist, for example, can we find u be of quadratic form? Since we know that when u is the linear form, p takes the relevant quadratic form. Hence, it is worthy of investigating when u is a quadratic form, whether the suitable function p exists. If the answer is positive, how can we use the solutions to explain any related physical phenomena?
3)
We find that the continuity function of the multi-component CH equations takes the same form as that of Euler equations, Navier-Stokes equations and Euler-Poisson equations, which are important fluid models. Therefore, we believe that the method proposed in this paper provides a possible way to solve these fluid equations.
4)
It is known the multi-component CH system is the higher dimensional variation of classical CH equation. The important features of the latter equation is its complete integrablity and admittance the peakon solutions. Therefore, it would be worthy of considering whether the form equations have the same features.

Based on the importance and applications of the multi-component CH system, all these interesting problems mentioned above will be deeply investigated in our future work.

Acknowledgments

The authors would like to express their sincere gratitude to the referees for their kind comments and valuable suggestions. This work is supported by the Fundamental Research Funds for the National Natural Science Foundation of China under Grant No. 11775116, 11601232 and 11505094, the Central Universities under Grant No. KYZ201649 and the research grant RG 94/2016-2017R from the Education University of Hong Kong.

References

[1]M.J. Ablowitz and P.A. Clarkson, Soliton, Nonlinear Evolution Equations and Inverse Scatting, Cambridge University Press, New York, 1991.

[2]C. Rogers and W. Schief, Bäcklund and Darboux Transformations Geometry and Modern Applications in Soliton Theory, Cambridge University Press, New York, 2002.

[3]R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitions, Phys. Rev. Lett, Vol. 27, 1971, pp. 1192.

[4]S.Y. Lou and J.Z Lu, Special solutions from the variable separation approach: the Davey-Stewartsone quation, J. Phys. A: Math. Gen, Vol. 29, 1996, pp. 4209-4215.

[5]W.X. Ma, A transformed rational function method and exact solutions to the 3+1-D Jimbo-Miwa equation, Chaos, Solitons & Fractals, Vol. 42, 2009, pp. 1356-1363.

[6]E.G. Fan, Extended tanh-function method and its applications to nonlinear equations, Phys. Lett. A, Vol. 277, 2000, pp. 212-218.

[7]Y. Liu and B. Li, Dynamics of rogue waves on multisolition background in the Benjamin Ono equation, Pramana. J. Phys, Vol. 88, 2017, pp. 57.

[8]Z.Y. Yan, New explicit travelling wave solutions for two new integrable coupled nonlinear evolution equations, Phys. Lett. A, Vol. 292, 2001, pp. 100-106.

[9]J. Chen and Z. Ma, Consistent Riccati expansion solvability and soliton -cnoidal wave interaction solition of a (2+1)-d KdV equation, Appl. Math. Lett, Vol. 64, 2017, pp. 87-93.

[10]M. Chen and B. Li, Classification and recursion operators of dark Burgers equations, Z. Naturforsch A, Vol. 357, 2018, pp. 175-180.

[11]M. Kohlmann, A note on multi-dimensional Camassa-Holm-type systems on the torus, J. Phys. A: Math. Theor, Vol. 45, 2012, pp. 125205.

[12]R. Danchin, A note on well-posedness for Camassa-Holm equation, J. Diff. Equa, Vol. 192, 2003, pp. 429-444.

[13]D. Holm, L.N. Lennon, and C. Tronci, Singular solutions of a modified two-component Camassa-Holm equation, Phys, Rev. E, Vol. 79, 2009, pp. 016601.

[14]R. Camassa and D.D. Holm, An integrable shallow water wave equation with peaked solitons, Phys. Rev. Lett, Vol. 71, 1993, pp. 1661-1664.

[15]R.S. Johnson, Camassa-Holm, Korteweg-de Vries and related models for water waves, J. Fluid Mech, Vol. 455, 2002, pp. 63-82.

[16]A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations, Arch. Ration. Mech. Anal, Vol. 192, 2009, pp. 165-186.

[17]D. Ionescu-Kruse, Variational derivation of the Camassa-Holm shallow water equation, J. Nonlinear Math. Phys, Vol. 14, 2007, pp. 303-312.

[18]A. Constantin and H.P. McKean, A shallow water equation on the circle, Comm. Pure Appl. Math, Vol. 52, 1999, pp. 949-982.

[19]A. Constantin, V.S. Gerdjikov, and R.I. Ivanov, Inverse scattering transform for the Camassa-Holm equation, Inverse Problems, Vol. 22, 2006, pp. 2197-2207.

[20]B. Fuchessteiner, Some tricks from the symmetry-toolbox for nonlinear equations: Generalizations of the Camassa-Holm shallow water equation, Physica D, Vol. 95, 1996, pp. 229.

[21]A. Constantin, The trajectories of particles in Stokes waves, Invent. Math, Vol. 166, 2006, pp. 523-535.

[22]A. Constantin and J. Escher, Particle trajectories in solitary water waves, Bull. Amer. Math. Soc, Vol. 44, 2007, pp. 423-431.

[23]A. Constantin and W. Strauss, Stablity of peakons, Comm. Pure Appl. Math., Vol. 53, 2000, pp. 603-610.

[24]L. Molinet, On well-posedness results for Camassa-Holm equation on line: a survey, J. Nonlinear Math. Phys, Vol. 11, 2004, pp. 521-533.

[25]Z. Qiao, The Camassa-Holm Hierarchy, N-dimensional integrable systems and algebro-geometric solution on a symplectic submanifold, Comm. Math. Phys, Vol. 239, 2003, pp. 309-321.

[26]F. Gesztesy and H. Holden, Algebro-geometric solutions of the Camassa-Holm Hierarchy, Rev. Mat. Iberoamericana, Vol. 19, 2003, pp. 73-142.

[27]M. Chen, S.Q. Liu, and Y. Zhang, A two-component generalization of the Camassa-Holm equation and its solutions, Lett. Math. Phys, Vol. 75, 2006, pp. 1-15.

[28]G. Falqui, On a Camassa-Holm type equation with two dependent variables, J. Phys. A: Math. Gen, Vol. 39, 2006, pp. 327-342.

[29]A. Constantin and R.I. Ivanov, On an integrable two-component Camassa-Holm shallow water system, Phys. Lett. A, Vol. 372, 2008, pp. 7129-7132.

[30]J. Escher, M. Kohlmann, and J. Lenells, The geometry of the two-component Camassa-Holm and Degasperis-Procesi equations, J. Geom. Phys, Vol. 61, 2011, pp. 436-452.

[31]Z. Guo and Y. Zhou, On solutions to a two-component generalized Camassa-Holm equation, Stud. Appl. Math, Vol. 124, 2010, pp. 307-322.

[32]D.D. Holm and R.I. Ivanov, Two-component CH system: inverse scattering, peakons and geometry, Inverse Prob, Vol. 27, 2011, pp. 045013.

[33]B. Khesin, J. Lenells, and G. Misiolek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Math. Ann, Vol. 342, 2008, pp. 617-656.

[34]J. Lenells, G. Misiolek, and F. Tiǧlay, Integrable evolution equations on spaces of tensor densities and their peakon solutions, Commun. Math. Phys, Vol. 299, 2010, pp. 129-161.

[35]J.K. Hunter and R. Saxton, Dynamics of director fields, SIAM J. Appl. Math, Vol. 51, 1991, pp. 1498-1521.

[36]O. Lechtenfeld and J. Lenells, On the N = 2 supersymmetric Camassa-Holm and Hunter-Saxton equations, J. Math. Phys, Vol. 50, 2009, pp. 012704.

[37]H. Wu and M. Wunsch, Global existence for the generalized two-component Hunter-Saxton system, J. Math. Fluid Mech, Vol. 14, 2012, pp. 455-469.

[38]D. Zou, A two-component µ-Hunter-Saxton equation, Inverse Prob., Vol. 26, 2010, pp. 085003.

[39]J.K. Hunter and Y. Zheng, On a completely integrable nonlinear hyperbolic variational equation, Physica D, Vol. 79, 1994, pp. 361-386.

[40]F. Gay-Balmaz, Well-posedness of higher dimensional Camassa-Holm equations, Bull. Transilvania Univ. Brasov, Vol. 2, 2009, pp. 51.

[41]H.L. An, M.K. Kwong, and M.Y. Yuen, Perturbational self-similar solutions for the 2-component Degasperis-Procesi system via a characteristic method, Electronic. J. Diff. Equ, Vol. 2017, 2017, pp. 1-12.

[42]D.D. Holm and J.E. Marsden, Momentum maps and measure-valued solutions (peakons, filaments and sheets) for the EPDiff equation The Breadth of Symplectic and Poisson Geometry, 2005, pp. 203-235. (Progress in Mathematics vol. 232) (Boston, MA: Birkhǎuser)

[43]H.P. Kruse, J. Scheurle, and W. Du, D. Bambusi, G. Gaeta, and M. Cadoni (editors), A two-dimensional version of the Camassa-Holm equation, World Scientific, New York, 2001, pp. 120-127. SPT 2001: Symmetry and Perturbation Theory (Sardinia)

[44]H. Kwak and H. Yang, An aspect of sonoluminescence from hydrodynamic theory, J. Phys. Soc. Japan, Vol. 64, 1995, pp. 1980-1992.

[45]V. Simonsen and J. Meyer-ter-Vehn, Self-similar solutions in gas dynamics with exponential time dependence, Phys. Fluids, Vol. 9, 1997, pp. 1462-1469.

[46]P. Sachdev, K. Joseph, and M. Haque, Exact solutions of compressible flow equations with spherical symmetry, Stud. Appl. Math, Vol. 114, 2005, pp. 325-342.

[47]H.L. An, E.G. Fan, and H.X Zhu, Elliptical vortex solutions, integrable Ermakov structure and Lax pair formulation of the compressible Euler equations, Phys. Review E, Vol. 91, 2015, pp. 013204.

[48]P.G. Drazin and N. Riley, The Navier-Stokes Equations: A Classification of Flows and Exact Solutions, London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, 2006. Book 334

[49]A.D. Craik and W.O. Criminale, Evolution of wavelike disturbances in shear flows: A class of exact solutions of the Navier-Stokes equations, Proc. R. Soc. Lond. A, Vol. 406, 1986, pp. 13-26.

[50]K.K. Chow, E.G. Fan, and M.W. Yuen, The analytical solutions for the n-dimensional damped compressible Euler equations, Stud. Appl. Math, Vol. 138, 2017, pp. 294-316.

[51]H.L. An, E.G. Fan, and M.W. Yuen, The Cartesian vector solutions for the N-dimensional compressible Euler equations, Stud. Appl. Math, Vol. 134, 2015, pp. 101-119.

[52]T. Zhang and Y. Zheng, Exact sprial solutions of the two-dimensional Euler equations, Disc. Con. Dyn. Sys, Vol. 3, 1997, pp. 117-133.

[53]M.W. Yuen, Self-similar Blowup Solutions to the 2-component Camassa-Holm Equations, J. Math. Phys, Vol. 51, 2010, pp. 093524.

<Previous Article In Issue

Download article (PDF)

Next Article In Issue>

Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 2
Pages: 255 - 272
Publication Date: 2021/01/06
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
DOI: 10.1080/14029251.2019.1591725 How to use a DOI?
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/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Hongli An
AU  - Liying Hou
AU  - Manwai Yuen
PY  - 2021
DA  - 2021/01/06
TI  - Analytical Cartesian solutions of the multi-component Camassa-Holm equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 255
EP  - 272
VL  - 26
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1591725
DO  - 10.1080/14029251.2019.1591725
ID  - An2021
ER  -

download .riscopy to clipboard