A New Two-Component Integrable System with Peakon Solutions

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1 Introduction

In recent years, the Camassa-Holm (CH) equation [1]

mt bux + 2mux + mxu = 0, m = u uxx, (1.1) − − where b is an arbitrary constant, derived by Camassa and Holm [1] as a shallow water wave model, has attracted much attention in the theory of soliton and integrable system. As an integrable equation it was implied in the work of Fuchssteiner and Fokas [2] on hereditary symmetries as a very special case. Since the work of Camassa and Holm [1], more diverse studies on this equation have been remarkably developed [3]- [14]. The most interesting feature of the CH equation (1.1) is that it admits peaked soliton (peakon) solutions in the case b = 0 [1, 3]. A peakon is a weak solution in some Sobolev space with corner at its crest. The stability and interaction of peakons were discussed in several references [15]- [19]. Moreover, in [44] the author discussed the potential applications of the CH equation to tsunami dynamics. In addition to the CH equation, other integrable models with peakon solutions have been found [20]- [28]. Among these models, there are two integrable peakon equations with cubic nonlinearity, which are

2 2 mt = bux + m(u u ) , m = u uxx, (1.2) − x x − and

2 mt = u mx + 3uuxm, m = u uxx. (1.3) − Equation (1.2) was proposed independently by Fokas (1995) [5], by Fuchssteiner (1996) [6], by Olver and Rosenau (1996) [4], and by Qiao (2006) [25,26]. Equation (1.2) is the ﬁrst cubic nonlinear integrable system possessing peakon solutions. Recently, the peakon stability of equation (1.2) with b = 0 was worked out by Gui, Liu, Olver and Qu [29]. In 2009, Novikov [28] derived another cubic equation, which is equation (1.3), from the symmetry approach, and Hone and Wang [27] gave its Lax pair, bi-Hamiltonian structure, and peakon solutions. Very recently [30], we derived the Lax pair, bi-Hamiltonian structure, peakons, weak kinks, kink-peakon interac- tional and smooth soliton solutions for the following integrable equation with both quadratic and cubic nonlinearity [5,6]:

1 2 2 1 mt = bux + k m(u u ) + k (2mux + mxu), m = u uxx, (1.4) 2 1 − x x 2 2 − where b, k1, and k2 are three arbitrary constants. By some appropriate rescaling, equation (1.4) was implied in the papers of Fokas and Fuchssteiner [5, 6], where it was derived from the two- dimensional hydrodynamical equations for surface waves. Equation (1.4) can also be derived by applying the tri-Hamiltonian duality to the bi-Hamiltonian Gardner equation [4]. 3

The above shown equations are one-component integrable peakon models. It is very interesting for us to study multi-component integrable generalizations of peakon equations. For example, in [4,31–35], the authors proposed two-component generalizations of the CH equation (1.1) with b = 0, and in [36, 37] the authors presented two-component extensions of the cubic nonlinear equation (1.3) and equation (1.2) with b = 0. In this paper, we propose the following two-component system with cubic nonlinearity and linear dispersion

1 1 mt = bux + 2 [m(uv uxvx)]x 2 m(uvx uxv), 1 − −1 − nt = bvx + [n(uv uxvx)]x + n(uvx uxv), (1.5)  2 − 2 −  m = u uxx, n = v vxx, − −  where b is an arbitrary real constant. This system is reduced to the CH equation (1.1) as v = 2, to the cubic CH equation (1.2) as v = 2u, and to the generalized CH equation (1.4) as − v = k1u + k2. Moreover, by imposing the complex conjugate reduction v = u∗, equation (1.5) is reduced to a new integrable equation with cubic nonlinearity and linear dispersion

1 2 2 1 mt = bux + [m( u ux )]x m(uu∗ uxu∗), m = u uxx, (1.6) 2 | | − | | − 2 x − − where the symbol ∗ denotes the complex conjugate of a potential. The above reductions of the two-component system (1.5) look very like the ones of AKNS system, which can be reduced to the KdV equation, the mKdV equation, the Gardner equation and the nonlinear Schrödinger equation. We prove the integrability of system (1.5) by providing its Lax pair, bi-Hamiltonian structure, and infinitely many conservation laws. Geometrically system (1.5) describes pseudo- spherical surfaces and thus it is also integrable in the sense of geometry. In the case b = 0 (dispersionless case), we show that this system admits the single-peakon of traveling wave solution as well as multi-peakon solutions. In particular, the two-peakon dynamic system is explicitly solved and their interactions are investigated in details. Moreover, we propose the complex valued N-peakon solution and kink wave solution to the cubic nonlinear complex equation (1.6). To the best of our knowledge, equation (1.6) is the first model admitting complex peakon solution and kink solution. The whole paper is organized as follows. In section 2, the Lax pair, bi-Hamiltonian structure as well as infinitely many conservation laws of equation (1.5) are presented. In section 3, the geometric integrability of equation (1.5) are studied. In section 4, the single-peakon, multi- peakon, and two-peakon dynamics are discussed. Section 5 shows that equation (1.6) admits the complex valued peakon solution and kink wave solution. Some conclusions and open problems are described in section 6. 4

2 Lax pair, bi-Hamiltonian structure and conservation laws

Let us consider a pair of linear spectral problems

φ φ 1 α λm 1 = U 1 ,U = − , (2.1) φ φ 2 λn α 2 !x 2 ! − ! φ φ 1 V V 1 = V 1 ,V = 11 12 , (2.2) φ2 ! φ2 ! −2 V21 V11 ! t − 2 where λ is a spectral parameter, m = u uxx, n = v vxx, α = √1 λ b, b is an arbitrary − − − constant, and

2 α 1 V =λ− α + (uv uxvx)+ (uvx uxv), 11 2 − 2 − 1 1 V = λ− (u αux) λm(uv uxvx), (2.3) 12 − − − 2 − 1 1 V =λ− (v + αvx)+ λn(uv uxvx). 21 2 − The compatibility condition of (2.1) and (2.2) generates

Ut Vx + [U, V ] = 0. (2.4) − Substituting the expressions of U and V given by (2.1) and (2.2) into (2.4), we ﬁnd that (2.4) is nothing but equation (1.5). Hence, (2.1) and (2.2) exactly give the Lax pair of (1.5). Let 0 ∂2 1 K = − , 1 ∂2 0 ! − (2.5) ∂m∂ 1m∂ m∂ 1m ∂m∂ 1n∂ + m∂ 1n + 2b∂ J − − − − . = 1 − 1 1 1 ∂n∂− m∂ + n∂− m + 2b∂ ∂n∂− n∂ n∂− n ! −

Lemma 1 J and K are a pair of Hamiltonian operators.

Proof It is obvious that K is Hamiltonian, since it is a skew-symmetric operator with constant- coeﬃcient. It is easy to check J is skew-symmetric. We need to prove that J satisﬁes the Jacobi identity

ζ, J ′[Jη]θ + η, J ′[Jθ]ζ + θ, J ′[Jζ]η = 0, (2.6) h i h i h i T T T where ζ = (ζ1,ζ2) , η = (η1, η2) , θ = (θ1,θ2) are arbitrary testing functions, and the prime- sign means the Gˆateaux derivative of an operator F on q in the direction σ deﬁned as [6] ∂ F [σ]= F (q)[σ]= F (q + ǫσ). (2.7) ′ ′ ∂ǫ ǫ=0

For brevity, we introduce the notations

1 1 1 A˜ = ∂− (mζ1,x + nζ2,x), B˜ = ∂− (mη1,x + nη2,x), C˜ = ∂− (mθ1,x + nθ2,x), (2.8) 1 1 1 A = ∂− (mζ nζ ), B = ∂− (mη nη ), C = ∂− (mθ nθ ). 1 − 2 1 − 2 1 − 2 By direct calculations, we arrive at

+ ∞ ζ, J ′[Jη]θ = [(θ ,xmx + θ ,xnx)B˜A˜ (ζ ,xmx + ζ ,xnx)B˜C˜ + C˜xB˜xA˜ A˜xB˜xC˜]dx h i 1 2 − 1 2 − Z −∞ + ∞ + [(ζ ,xm ζ ,xn)(BC˜ + CB˜) (θ ,xm θ ,xn)(BA˜ + AB˜)]dx 1 − 2 − 1 − 2 Z −∞+ + ∞[(ζ m + ζ n)BC (θ m + θ n)BA]dx (2.9) 1 2 − 1 2 Z−∞ + ∞ 2b [(ζ ,xη ,x + ζ ,xη ,x)C˜ (η ,xθ ,x + η ,xθ ,x)A˜]dx − 1 2 2 1 − 2 1 1 2 Z−∞ + ∞ + 2b [(ζ η ,x ζ η ,x)C (η ,xθ η ,xθ )A]dx. 2 1 − 1 2 − 1 2 − 2 1 Z−∞ Based on (2.9), we may verify (2.6) directly. This completes the proof of Lemma 1.

Lemma 2 The following relation holds

ζ, J ′[Kη]θ + η, J ′[Kθ]ζ + θ, J ′[Kζ]η + ζ, K′[Jη]θ + η, K′[Jθ]ζ + θ, K′[Jζ]η = 0(2.10). h i h i h i h i h i h i

Proof Direct calculations yield that

+ ∞ ζ, J ′[Kη]θ = [(ζ ,xη ζ ,xη )C˜ (θ ,xη θ ,xη )A˜]dx h i − 2 1 − 1 2 − 2 1 − 1 2 Z −∞+ ∞ [(ζ ,xη ,xx ζ ,xη ,xx)C˜ (η ,xxθ ,x η ,xxθ ,x)A˜]dx − 1 2 − 2 1 − 2 1 − 1 2 Z −∞+ (2.11) + ∞[(ζ η + ζ η )C (η θ + η θ )A]dx 1 2 2 1 − 1 2 2 1 Z −∞+ ∞ [(ζ η ,xx + ζ η ,xx)C (η ,xxθ + η ,xxθ )A]dx. − 1 2 2 1 − 1 2 2 1 Z−∞ Formula (2.10) may be veriﬁed based on (2.11). The proof of Lemma 2 is ﬁnished. From Lemma 1 and Lemma 2, we immediately obtain

Proposition 1 J and K are compatible Hamiltonian operators.

Furthermore, we have 6

Proposition 2 Equation (1.5) can be rewritten in the following bi-Hamiltonian form

δH δH T δH δH T (m , n )T = J 1 , 1 = K 2 , 2 , (2.12) t t δm δn δm δn where J and K are given by (2.5), and

1 + H = ∞(uv + u v )dx, 1 2 x x Z −∞+ (2.13) 1 ∞ 2 2 H = [(u vx + u vx 2uuxv)n + 2b(uvx uxv)]dx. 2 4 x − − Z−∞ Next we construct conservation laws of equation (1.5). Let ϕ = φ2 , where φ and φ are φ1 1 2 determined through equations (2.1) and (2.2). From (2.1), one can easily verify that ϕ satisﬁes the Riccati equation

1 2 1 ϕx = λmϕ + αϕ λn. (2.14) −2 − 2 Equations (2.1) and (2.2) give rise to α 1 1 (ln φ )x = + λmϕ, (ln φ )t = (V + V ϕ) , (2.15) 1 − 2 2 1 −2 11 12 which yields conservation law of equation (1.5):

ρt = Fx, (2.16) where ρ = mϕ, (2.17) 2 1 1 1 F = λ− (u αux)ϕ λ− (αuv αuxvx + uvx uxv)+ m(uv uxvx)ϕ. − − 2 − − 2 − Usually ρ and F are called a conserved density and an associated ﬂux, respectively. In the case b = 0, we are able to derive the explicit forms of conservation densities by expanding ϕ in powers of λ in two ways. The ﬁrst one is to expand ϕ in terms of negative powers of λ as

∞ j ϕ = ϕjλ− . (2.18) j X=0 Substituting (2.18) into (2.14) and equating the coeﬃcients of powers of λ, we arrive at

n mnx mxn 2mn ϕ = , ϕ = − − , (2.19) 0 − m 1 2m2n p and the recursion relation for ϕj:

1 1 ϕj+1 = mϕ ϕj ϕj,x 2 m i+k=j+1, i,k 1 ϕiϕk , j 1. (2.20) 0 − − ≥ ≥ h P i 7

Inserting (2.18), (2.19) and (2.20) into (2.17), we obtain the following inﬁnitely many conserved densities and the associated ﬂuxes of equation (1.5): 1 ρ = √ mn, F = √ mn(uv uxvx), 0 − 0 2 − − mnx mxn 2mn 1 1 ρ = − − , F = (uv uxvx + uvx uxv)+ ρ (uv uxvx), (2.21) 1 2mn 1 −2 − − 2 1 − 1 ρj = mϕj, Fj = (u ux)ϕj 2 + ρj(uv uxvx), j 2, − − 2 − ≥ where ϕj is given by (2.19) and (2.20). The second expansion of ϕ is in the positive powers of λ,

∞ j ϕ = ϕjλ . (2.22) j X=0 Substituting (2.22) into (2.14) and comparing powers of λ produce

ϕ j = 0, j 0, 2 ≥ 1 1 (2.23) ϕ = (v + vx), ϕ j ϕ j ,x = m ϕiϕk, j 1. 1 2 2 +1 − 2 +1 2 ≥ i+k=2j, 0 i,k 2j X≤ ≤ By inserting (2.22) and (2.23) into (2.17), we arrive at

ρ j = 0, A j = 0, j 0, (2.24) 2 2 ≥ and 1 1 ρ1 = m(v + vx), A1 = (u ux)ϕ3 + m(uv uxvx)(v + vx), 2 − 4 − (2.25) 1 ρ j = mϕ j , A j = (u ux)ϕ j + m(uv uxvx)ϕ j , j 1, 2 +1 2 +1 2 +1 − 2 +3 2 − 2 +1 ≥ where the odd-index ϕ2j+1 is deﬁned by the recursion relation

1 1 ϕ j = (1 ∂x)− m ϕiϕk , j 1. (2.26) 2 +1 2 −   ≥ i+k=2j, 0 i,k 2j X≤ ≤   Formula (2.24) means that the even-index conserved densities and associated ﬂuxes are trivial. Formulas (2.25) and (2.26) show that the nontrivial high-order odd-index conserved densities may involve in nonlocal expressions in u and v.

Remark 1. Here, we have derived two sequences of inﬁnitely many conserved densities and the associated ﬂuxes for equation (1.5). The conserved densities in the sequence (2.21) become singular when the denominators have zero points. The conserved densities in the sequence (2.25) have no singularity, but they might involve in nonlocal expressions. 8

3 Geometric integrability

Based on the work of Chern and Tenenblat [38] and the subsequent works [39,40], a diﬀerential equation for a real valued function u(x,t) is said to describe pseudo-spherical surfaces if it is the necessary and suﬃcient condition for the existence of smooth functions fij, i = 1, 2, 3, j = 1, 2, depending on x, t, u and its derivatives, such that the one-forms ωi = fi1dx + fi2dt satisfy the structure equations of a surface of constant Gaussian curvature equal to 1 with metric ω2 + ω2 − 1 2 and connection one-form ω3, namely

dω = ω ω , dω = ω ω , dω = ω ω . (3.1) 1 3 ∧ 2 2 1 ∧ 3 3 1 ∧ 2 Let us consider

1 (α λ)x (λ α)x f = λ[e − m e − n], 11 −2 − 1 1 (λ α)x (α λ)x 1 (λ α)x (α λ)x f = λ− [e − (v + αvx) e − (u αux)] + λ[e − n e − m](uv uxvx), 12 2 − − 4 − − f21 = λ, (3.2) 2 α 1 f = λ− α + (uv uxvx)+ (uvx uxv), 22 2 − 2 − 1 (α λ)x (λ α)x f = λ[e − m + e − n], 31 −2 1 1 (λ α)x (α λ)x 1 (λ α)x (α λ)x f = λ− [e − (v + αvx)+ e − (u αux)] λ[e − n + e − m](uv uxvx), 32 −2 − − 4 − and introduce the following three one-forms

ω1 = f11dx + f12dt, ω2 = f21dx + f22dt, ω3 = f31dx + f32dt. (3.3)

Through a direct computation, we ﬁnd that the structure equations (3.1) hold whenever u(x,t) and v(x,t) are solutions of system (1.5). Thus we have

Theorem 1 System (1.5) describes pseudo-spherical surfaces.

Recall that a diﬀerential equation is geometrically integrable if it describes a nontrivial one- parameter family of pseudo-spherical surfaces. It follows that

Corollary 1 System (1.5) is geometrically integrable.

According to [38]- [41], we have the following fact

Proposition 3 A geometrically integrable equation with associated one-forms ωi, i = 1, 2, 3, is the integrability condition of a one-parameter family of sl(2,R)-valued linear problem

dΦ=ΩΦ, (3.4) 9 where Ω is the matrix-valued one-form

1 ω ω ω Ω= Xdx + T dt = 2 1 − 3 . (3.5) 2 ω1 + ω3 ω2 ! −

Therefore, the one-forms (3.3) and (3.4) yield an sl(2,R)-valued linear problem Φx = XΦ and Φt = T Φ, whose integrability condition is the two-component system (1.5). The expression (3.5) implies that the matrices X and T are

1 λ λe(λ α)xn X − , = (α λ)x 2 λe − m λ ! − − (3.6) 1 λ 2α + α (uv u v )+ 1 (uv u v) [λ 1(v + αv )+ λ n(uv u v )]e(λ α)x T − 2 x x 2 x x − x 2 x x − . = 1 − λ − (α λ)x 2 α − 1 2 [λ− (u αux)+ m(uv uxvx)]e − λ− α (uv uxvx) (uvx uxv) ! − − 2 − − − 2 − − 2 − 4 Peakon solutions to system (1.5) in the case b =0

In this section, we shall derive the peakon solutions to the two-component system (1.5) with b = 0 in two situations. The ﬁrst situation is the peakon solutions with the same peakon position. The second situation is the peakon solutions with diﬀerent peakon positions, which is studied by C.J. Cotter et al. [35] for a cross-coupled CH equation.

4.1 Peakon solutions to the two-component system (1.5) with the same peakon position

Let us suppose that a single peakon solution of (1.5) with b = 0 is of the following form

x ct x ct u = c1e−| − |, v = c2e−| − |, (4.1) where the two constants c1 and c2 are to be determined. With the help of distribution theory, we are able to write out ux, m and vx, n as follows

x ct ux = c1sgn(x ct)e−| − |, m = 2c1δ(x ct), − − − (4.2) x ct vx = c sgn(x ct)e−| − |, n = 2c δ(x ct). − 2 − 2 − Substituting (4.1) and (4.2) into (1.5) with b = 0 and integrating in the distribution sense, one can readily see that c1 and c2 should satisfy

c c = 3c. (4.3) 1 2 − In particular, for c = c , we recover the single peakon solution u = √ 3ce x ct of the cubic 1 2 ± − −| − | CH equation (1.2) with b = 0 [29, 30]. 10

Let us now assume a two-peakon solution as follows:

x q1(t) x q2(t) x q1(t) x q2(t) u = p1(t)e−| − | + p2(t)e−| − |, v = r1(t)e−| − | + r2(t)e−| − |. (4.4)

In the sense of distribution, we have

x q1 x q2 ux = p1sgn(x q1)e−| − | p2sgn(x q2)e−| − |, m = 2p1δ(x q1) + 2p2δ(x q2), − − − − − − (4.5) x q1 x q2 vx = r sgn(x q )e−| − | r sgn(x q )e−| − |, n = 2r δ(x q ) + 2r δ(x q ). − 1 − 1 − 2 − 2 1 − 1 2 − 2 Substituting (4.4) and (4.5) into (1.5) with b = 0 and integrating through test functions yield the following dynamic system:

1 q q p ,t = p (p r p r )sgn(q q )e 1 2 , 1 2 1 1 2 − 2 1 1 − 2 −| − | 1 q2 q1  p2,t = p2(p2r1 p1r2)sgn(q2 q1)e−| − |, 2 − −  1 1 q1 q2  q1,t = p1r1 (p1r2 + p2r1) e−| − |,  − 3 − 2 (4.6)  q = 1 p r 1 (p r + p r ) e q2 q1 ,  2,t 3 2 2 2 1 2 2 1 −| − |  − 1 − q q r ,t = r (p r p r )sgn(q q )e 1 2 , 1 − 2 1 1 2 − 2 1 1 − 2 −| − |  r = 1 r (p r p r )sgn(q q )e q2 q1 .  2,t 2 2 2 1 1 2 2 1 −| − |  − − −  Guided by the above equations, we may conclude the following relations:

p1 = Dp2, p1r1 = A1, p2r2 = A2, (4.7) where D, A1 and A2 are three arbitrary integration constants.

If A1 = A2, we arrive at the following solution of (4.6):

1 (D2A A )sgn(C )e−|C1|t p1 A1 A1 p (t)= Be 2D 1 1 1 , p (t)= , r (t)= , r (t)= , 1 − 2 D 1 p 2 p 1 2 (4.8) 1 1 2 C1 1 q (t)= A + (D A + A )e−| | t + C , q (t)= q (t) C , 1 − 3 1 2D 1 1 2 1 2 1 − 1 where B and C1 are two arbitrary non-zero constants. In this case, the collision between two peakons will never happen since q (t)= q (t) C . For example, as A = B = D = 1, C = 2, 2 1 − 1 1 1 (4.8) is reduced to

p1(t)= p2(t)= r1(t)= r2(t) = 1,

1 2 1 2 q (t)= + e− t + 1, q (t)= + e− t 1. 1 − 3 2 − 3 − Thus, the associated solution of (1.5) with b = 0 becomes

x+( 1 +e−2)t 1 x+( 1 +e−2)t+1 u(x,t)= v(x,t)= e−| 3 − | + e−| 3 |. (4.9) 11

u(x,t) u(x,t)

0.8 1

0.4 0.8

0.6 0 -4 -2 0 2 4 x 0.4

-0.4

0.2

-0.8 0 -4 -2 0 2 4 x

Figure 1: The M-shape two-peakon solution u(x, t) Figure 2: The N-shape peak-trough solution u(x, t) in (4.9) at the moment of t = 0. in (4.10) at the moment of t = 0.

This wave has two peaks, and looks like a M-shape soliton. See Figure 1 for this M-shape two- peakon solution. As A = B = D = 1, C = 2, the associated solution of (1.5) with b = 0 1 − − 1 becomes

x+( 1 e−2)t 1 x+( 1 e−2)t+1 u(x,t)= v(x,t)= e− 3 − − + e− 3 − , (4.10) − | | | | which has one peak and one trough and looks like N-shape soliton solution. See Figure 2 for this N-shape two-peakon solution. As B = 2D = 1, A1 = C1 = 2, (4.8) becomes

1 3 e−2t 3 e−2t p (t)= p (t)= e− 2 , r (t) = 2r (t) = 2e 2 , 1 2 2 1 2 (4.11) 2 5 2 2 5 2 q (t)= + e− t + 1, q (t)= + e− t 1. 1 − 3 2 2 − 3 2 − and the associated solution of (1.5) with b = 0 becomes

3 e−2t x+( 2 + 5 e−2)t 1 x+( 2 + 5 e−2)t+1 u(x,t)= e− 2 e−| 3 2 − | + 2e−| 3 2 | , (4.12) 3 e−2t x+( 2 + 5 e−2)t 1 x+( 2 + 5 e−2)t+1  v(x,t)= e 2 2e− 3 2 − + e− 3 2 .  | | | |  From (4.11), one can easily see that the amplitudes p1(t) and p2(t) of potential u(x,t) are two monotonically decreasing functions of t, while the amplitudes r1(t) and r2(t) of potential v(x,t) are two monotonically increasing functions of t. Figures 3 and 4 show the proﬁles of the potentials u(x,t) and v(x,t). 12

u(x,t) v(x,t)

5 5

4 4

3 3

2 2

1 1

0 0 -10 -50 5 10 -10 -5 0 5 10 x x

Figure 3: The two-peakon solution u(x, t) in (4.12). Figure 4: The two-peakon solution v(x, t) in (4.12). Red line: t = 5; Blue line: t = 2; Brown line: Red line: t = 5; Blue line: t = 2; Brown line: − − − − t = 0; Green line: t = 2; Black line: t = 5. t = 0; Green line: t = 2; Black line: t = 5.

If A = A , we may obtain the following solution of (4.6): 1 6 2

3(A D2−A ) 1 A A t 2 1 e− 3 |( 1− 2) | p A A 2D(A −A ) 1 1 2 p1(t)= Be 1 2 , p2(t)= , r1(t)= , r2(t)= , D p1 p2 2 A D A 1 1 3( 2 + 1) (A1 A2)t q1(t)= A1t + sgn[(A1 A2)t] e− 3 | − | 1 , (4.13) −3 2D(A1 A2) − − 2− A D A 1 1 3( 2 + 1) (A1 A2)t q2(t)= A2t + sgn[(A1 A2)t] e− 3 | − | 1 , −3 2D(A1 A2) − − − where B is an arbitrary integration constant. Let us study the following special cases of this solution. 1 Example 1. Let A1 = 1, A2 = 4, B = 1, D = 2 , then

p1(t)= r1(t) = 1, p2 = r2(t) = 2, q (t)= 1 t + 2sgn(t) e t 1 , (4.14)  1 − 3 −| | −  q (t)= 4 t + 2sgn(t) e t 1 . 2 − 3 −| | −  The associated two-peakon solution of (1.5) becomes

x+ 1 t 2sgn(t)(e−|t| 1) x+ 4 t 2sgn(t)(e−|t| 1) u(x,t)= v(x,t)= e−| 3 − − | + 2e−| 3 − − |. (4.15)

As t< 0 and t is going to 0, the tall peakon with the amplitude 2 chases after the short peakon with the amplitude 1. The two-peakon collides at time t = 0. After the collision (t > 0), the peaks separate (the tall peakon surpasses the short one) and develop on their own way. See Figure 5 for the detailed development of this kind of two-peakon. 13

u(x,t) u(x,t)

3 1

2.5 0.8

2 0.6

1.5

0.4 1

0.2 0.5

0 0 -15-10 -5 0 5 10 15 -15-10 -5 0 5 10 15 . x x

Figure 5: The two-peakon solution u(x, t) in (4.15). Figure 6: The two-peakon solution u(x, t) in (4.17). Red line: t = 5; Blue line: t = 2; Brown line: Red line: t = 5; Blue line: t = 1; Brown line: − − − − t = 0 (collision); Green line: t = 2; Black line: t = 5. t = 0 (collision); Green line: t = 1; Black line: t = 5.

Example 2. Let A1 = 1, A2 = 4, B = 1, D = 1, then we have

3 e−|t| p1(t)= p2(t)= e− 2 , 3 e−|t| 3 e−|t|  r1(t)= e 2 , r2(t) = 4e 2 ,  1 5 t (4.16)  q1(t)= t + sgn(t) e−| | 1 ,  − 3 2 − q t 4 t 5 sgn t e t . 2( )= 3 + 2 ( ) −| | 1  − −  The associated two-peakon solution of (1.5) becomes 3 e−|t| x+ 1 t 5 sgn(t)(e−|t| 1) x+ 4 t 5 sgn(t)(e−|t| 1) u(x,t)= e− 2 e−| 3 − 2 − | + e−| 3 − 2 − | , (4.17) 3 e−|t| x+ 1 t 5 sgn(t)(e−|t| 1) x+ 4 t 5 sgn(t)(e−|t| 1)  v(x,t)= e 2 e− 3 − 2 − + 4e− 3 − 2 − .  | | | | 3 e−|t| For the potential u(x,t), the two-peakon solution possesses the same amplitude e− 2 , which reaches the minimum value at t = 0. Figure 6 shows the proﬁle of the two-peakon dynamics for the potential u(x,t). For the potential v(x,t), the two-peakon solution with the amplitudes 3 e−|t| 3 e−|t| e 2 and 4e 2 collides at t = 0. At this moment, the amplitudes attain the maximum 3 x value and the two-peakon overlaps into one peakon 5e 2 e−| |, which is much higher than other moments. See Figures 7 and 8 for the 2-dimensional and 3-dimensional graphs of the two-peakon dynamics for the potential v(x,t). Example 3. Let A = 1, A = 4, B = 1, D = 1, then we have 1 2 − 3 e−|t| p1(t)= p2(t)= e 2 , − 3 e−|t| 3 e−|t|  r1(t)= e− 2 , r2(t)= 4e− 2 ,  1 5 −t (4.18)  q1(t)= t sgn(t) e−| | 1 ,  − 3 − 2 − q t 4 t 5 sgn t e t . 2( )= 3 2 ( ) −| | 1  − − −   14

v(x,t) v(x,t)

15 20

10 10

-20 5 0 -20 -10 -10 0 0 0 x t 10 10 -10 -5 0 5 10 20 x 20

Figure 7: The two-peakon solution v(x, t) in (4.17). Figure 8: 3-dimensional graph for the two-peakon Red line: t = 4; Blue line: t = 1; Brown line: solution v(x, t) in (4.17). − − t = 0 (collision); Green line: t = 1; Black line: t = 4.

The associated two-peakon solution of (1.5) becomes

3 e−|t| x+ 1 t+ 5 sgn(t)(e−|t| 1) x+ 4 t+ 5 sgn(t)(e−|t| 1) u(x,t)= e 2 e−| 3 2 − | e−| 3 2 − | , − (4.19) 3 e−|t| x+ 1 t+ 5 sgn(t)(e−|t| 1) x+ 4 t+ 5 sgn(t)(e−|t| 1)  v(x,t)= e− 2 e− 3 2 − 4e− 3 2 − .  | | − | | For the potential u(x,t), the peakon-antipeakon collides and vanishes at t = 0. After the collision, the peakon and antipeakon reemerge and separate. For the potential v(x,t), the peakon and trough overlap at t = 0, and then they separate. Figures 9 and 10 show the peakon-antipeakon dynamics for the potentials u(x,t) and v(x,t). In general, we suppose an N-peakon solution has the following form

N N x qj (t) x qj (t) u(x,t)= pj(t)e−| − |, v(x,t)= rj(t)e−| − |. (4.20) j j X=1 X=1 Substituting (4.20) into (1.5) with b = 0 and integrating through test functions, we obtain the N-peakon dynamic system as follows

1 N qj q qj qi pj,t = pj pirk (sgn(qj qk) sgn(qj qi)) e k , 2 i,k=1 − − − −| − |−| − | 1 1 N qj qi qj q qj,t = pjrj pirk (1 sgn(qj qi)sgn(qj qk)) e k , (4.21)  6 P− 2 i,k=1 − − − −| − |−| − | 1 N qj q qj qi  rj,t = rj pirk (sgn(qj qk) sgn(qj qi)) e k . − 2 i,kP=1 − − − −| − |−| − |  P 4.2 Peakon solutions to the two-component system (1.5) with diﬀerent peakon positions

In this section, we discuss the N-peakon solutions with diﬀerent peakon positions based on the work in [35]. Let us assume that the N-peakon solutions of the two potentials u and v with 15

u(x,t) v(x,t)

1 1 x -10 -5 0 5 10 0 0.5

-1

0 -10 -5 0 5 10 x -2

-0.5 -3

-1

Figure 9: Peakon-antipeakon solution u(x, t) in Figure 10: Peakon-antipeakon solution v(x, t) in (4.19). Red line: t = 6; Blue line: t = 2; At (4.19). Red line: t = 5; Blue line: t = 2; Brown − − − − t = 0 they collide and vanish; Green line: t = 2; line: t = 0 (collision); Green line: t = 2; Black line: Black line: t = 6. t = 5. diﬀerent peakon positions are given in the form

N N x qj (t) x sj (t) u(x,t)= pj(t)e−| − |, v(x,t)= rj(t)e−| − |, (4.22) j j X=1 X=1 where qi(t) = sj(t), 1 i, j N. With the help of delta functions, we have 6 ≤ ≤ N N m(x,t) = 2 pj(t)δ(x qj(t)), n(x,t) = 2 rj(t)δ(x sj(t)). (4.23) − − j j X=1 X=1 Substituting (4.22) and (4.23) into (1.5) with b = 0 and integrating through test functions, we arrive at the following system regarding pj, qj, rj, sj:

1 N qj s qj qi pj,t = pj pirk (sgn(qj sk) sgn(qj qi)) e k , 2 i,k=1 − − − −| − |−| − | 1 N sj s sj qi  rj,t = rj pirk (sgn(sj sk) sgn(sj qi)) e−| − k|−| − |, − 2 P i,k=1 − − − (4.24)  1 N qj qi qj s  qj,t = pirk (1 sgn(qj qi)sgn(qj sk)) e−| − |−| − k|,  − 2 Pi,k=1 − − − 1 N sj qi sj s sj,t = pirk (1 sgn(sj qi)sgn(sj sk)) e k . − 2 Pi,k=1 − − − −| − |−| − |   P Diﬀerent from the N-peakon dynamic system of the coupled CH equation proposed in [35], our system (4.24) can not directly be rewritten in the standard form of a canonical Hamiltonian system. It is interesting to study whether (4.24) is able to be rewritten as an integrable Hamil- tonian system by introducing a Poisson bracket. We will investigate the related topics in the near future. 16

u(x,t),v(x,t)

0.8

0.6

0.4

0.2

0 -4 -2 0 2 4 x

Figure 11: The single-peakon solution (4.28) at t = 0. Solid line: u(x, 0); Dashed line: v(x, 0).

For N = 1, (4.24) is reduced to

1 2 q s p ,t = p r sgn(q s )e 1 1 , 1 2 1 1 1 − 1 −| − | 1 2 s1 q1  r1,t = p1r sgn(s1 q1)e−| − |, 2 1 − (4.25)  1 q1 s1  q1,t = p1r1e−| − |,  − 2 1 s1 q1 s1,t = 2 p1r1e−| − |.  −  From the last two equations of (4.25), we obtain

s1 = q1 + A1, (4.26) where A = 0 is an integration constant. Without loss of generality, we suppose A > 0. 1 6 1 Substituting (4.26) into (4.25) leads to

1 −A e 1 A2t p1 = A3e− 2 , 1 −A A2 e 1 A2t  r1 = e 2 , A3 (4.27)  1 A1  q1 = e− A2t,  − 2 s = 1 e A1 A t + A , 1 − 2 − 2 1   where A2 and A3 are integration constants. In particular, we take A1 = ln2 and A2 = A3 = 1, then the single-peakon solutions with diﬀerent peakon positions become

1 t x+ 1 t 1 t x+ 1 t ln 2 u = e− 4 e−| 4 |, v = e 4 e−| 4 − |. (4.28)

See Figure 11 for the proﬁle of this single-peakon solution at t = 0. We have not yet explicitly solved (4.24) with N = 2. This is due to the complexity of (4.24) with N = 2, which is a coupled ordinary diﬀerential equation with eight components. 17

5 Solutions to the integrable system (1.6)

As mentioned above, system (1.5) is cast into the integrable cubic nonlinear equation (1.6) under the complex conjugate reduction v = u∗. Thus equation (1.6) possesses the following Lax pair

φ φ 1 α λm 1 = U 1 ,U = − , (5.1) φ φ 2 λm α 2 !x 2 ! − ∗ ! φ φ 1 V V 1 = V 1 ,V = 11 12 , (5.2) φ φ −2 V V 2 !t 2 ! 21 − 11 ! with α = √1 λ2b, and − 2 α 2 2 1 V =λ− α + ( u ux )+ (uu∗ u∗ux), 11 2 | | − | | 2 x − 1 1 2 2 V = λ− (u αux) λm( u ux ), (5.3) 12 − − − 2 | | − | | 1 1 2 2 V =λ− (v + αvx)+ λn( u ux ). 21 2 | | − | | Next, we show that the dispersionless version of equation (1.6) with b = 0 admits the complex valued N-peakon solution, while the dispersion version of equation (1.6) with b = 0 allows the 6 complex valued kink wave solution.

5.1 Complex valued peakon solution of (1.6) with b =0

Let us assume that a complex valued N-peakon solution of (1.6) with b = 0 has the following form

N x qj (t) u = pj(t)+ √ 1rj(t) e−| − |, (5.4) − j=1 X where pj(t), rj(t) and qj(t) are real valued functions. Substituting (5.4) into (1.6) with b = 0 and integrating through real valued test functions, and separating the real part and imaginary part, we ﬁnally obtain that pj(t), rj(t) and qj(t) evolve according to the dynamical system

N qj qk qj ql pj,t =rj plrk (sgn(qj qk) sgn(qj ql)) e−| − |−| − |,  − − −  l,kX=1   N  qj qk qj ql  rj,t =pj plrk (sgn(qj ql) sgn(qj qk)) e−| − |−| − |, (5.5)  − − −  l,kX=1 N 1 1  2 2 qj ql qj qk  qj,t = (pj + rj ) (plpk + rlrk) (1 sgn(qj ql)sgn(qj qk)) e−| − |−| − |.  6 − 2 − − −  l,k=1  X   18

For N = 1, (5.5) becomes

1 2 2 p ,t = 0, r ,t = 0, q ,t = (p + r ), (5.6) 1 1 1 −3 1 1 which gives

1 p = c , r = c , q = (c2 + c2)t, (5.7) 1 1 1 2 1 −3 1 2 where c1 and c2 are real valued integration constants. Thus we arrive at the single-peakon solution

c2+c2 x+ 1 2 t x+ 1 c 2t u = (c + √ 1c )e−| 3 | = ce−| 3 | | |, (5.8) 1 − 2 where c = c + √ 1c , and c is the modulus of c. 1 − 2 | | For N = 2, we may solve (5.5) as

1 2 q1(t)= 3 A1t +Γ1(t), − 1 2  q2(t)= A t +Γ1(t), − 3 2   p1(t)= A1 sin(Γ2(t)+ A3),  (5.9)  p2(t)= A2 sin(Γ2(t)+ A4),  r1(t)= A1 cos(Γ2(t)+ A3),  r t A t A ,  2( )= 2 cos(Γ2( )+ 4)   where 

3A1A2 cos(A3 A4) 1 2 2 3 (A1 A2)t Γ1(t)= 2 2− sgn(t) e− | − | 1 , A1 A2 − | − | (5.10) 3A1A2 sin(A3 A4) 1 (A2 A2)t Γ (t)= − e− 3 | 1− 2 |, 2 A2 A2 1 − 2 and A , , A are real valued integration constants. Hence the two-peakon solution reads 1 · · · 4

1 2 1 2 √ 1(Γ2(t)+A3) x+ A t Γ1(t) √ 1(Γ2(t)+A4) x+ A t Γ1(t) u = A √ 1e− − e−| 3 1 − | + A √ 1e− − e−| 3 2 − |, (5.11) 1 − 2 − where the Euler formula e√ 1x = cosx + √ 1sinx is employed. − −

5.2 Complex valued kink solution of (1.6) with b =0 6 We suppose that a complex valued kink wave solution of equation (1.6) with b = 0 has the form 6 x ct u = C + √ 1C sgn(x ct) e−| − | 1 , (5.12) 1 − 2 − − 19

where the real constant c is the wave speed, and C1 and C2 are two real constants to be determined. Substituting (5.12) into equation (1.6) with b = 0 and integrating through real 6 valued test functions, and separating its real part and imaginary part, we ﬁnally arrive at 1 c = b, C2 + C2 = b. (5.13) −2 1 2 − Formula (5.13) implies that the wave speed is exactly 1 b, where b< 0 is the coeﬃcient of − 2 the linear dispersive term. Thus the complex valued weak kink solution becomes

1 x+ 1 bt u = Csgn(x + bt) e−| 2 | 1 , (5.14) 2 − where C = C + √ 1C , and C 2 = b, b< 0. We remark that in (5.14) only the constant C 1 − 2 | | − is complex, the variables x and t are real valued variables.

6 Conclusions and discussions

In our paper, we propose a new integrable two-component system with cubic nonlinearity and linear dispersion. The system is shown to possess Lax pair, bi-Hamiltonian structure, and inﬁnitely many conservation laws. Geometrically, this system describes a nontrivial one-parameter family of pseudo-spherical surfaces. In the dispersionless case, we show the system admits N-peakon solution and explicitly solve the system for the single-peakon and the two-peakon dynamical system. Moreover, we propose a scalar integrable complex cubic nonlinear equation and ﬁnd the complex valued N-peakon solution and kink wave solution to the integrable complex equation. In [37], the authors introduced an integrable two-component extension of the dispersionless version of cubic nonlinear equation (1.2) (or the FORQ equation called in some literature)

mt = [m(uv uxvx + uxv uvx)]x, − − − nt = [n(uv uxvx + uxv uvx)]x, (6.1)  − − −  m = u uxx, n = v vxx. − − We remark that the dispersionless version of our two-component system (1.5) with b = 0 is not equivalent to system (6.1). System (1.5) in our paper is able to be reduced to the CH equation, but system (6.1) is not, which apparently implies that these two equations are not equivalent. In fact, both system (1.5) with b = 0 and system (6.1) belong to a more general negative ﬂow in a hierarchy. For the details of this topic, one may see our very recent paper [42]. It is an interesting task to study whether there are other new features in the structure of solutions for our two-component system, and particularly for our complex equation with a linear dispersive term. Also other topics, such as smooth soliton solutions [43], cuspons, peakon stability, and algebra-geometric solutions, remain open for our system (1.5) and (1.6).

Data accessibility. There are no primary data in this article. 20

Competing interests. There are no competing interests. Authors’ contributions. This is a theoretical paper. The author Xia is the ﬁrst author. The author Qiao is the corresponding author. Acknowledgements. The authors would like to express their sincerest thanks to the reviewers for their valuable comments and suggestions. The author Xia also thanks Ruguang Zhou for helpful discussions. Funding statement. The author Xia was supported by the National Natural Science Foun- dation of China (Grant Nos. 11301229 and 11271168), the Natural Science Foundation of the Jiangsu Province (Grant No. BK20130224) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 13KJB110009). The author Qiao was par- tially supported by the National Natural Science Foundation of China (Grant No. 11171295 and 61328103) and also thanks the U.S. Department of Education GAANN project (P200A120256) to support UTPA mathematics graduate program.

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