ON A NEW TWO-COMPONENT b -FAMILY PEAKON SYSTEM WITH CUBIC NONLINEARITY

. In this paper, we propose a two-component b -family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component b -family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at inﬁnity within its lifespan provided the initial data decay exponentially and algebraically. Classiﬁcation. 35G25, 35L05.

1. Introduction. In this paper, we consider the following b-family of two-component system with cubic nonlinearity: m t + uvm x + bu x vm = 0, n t + uvn x + buv x n = 0, (1.1) which models the unidirectional propagation of shallow water waves over a flat bottom. Here u(t, x) stands for the fluid velocity at time t in the spatial x direction [3,20,31]. The CH equation is also recognized as a model for the propagation of axially symmetric waves in hyperelastic rods [16]. It has a bi-Hamiltonian structure [3,24]and is completely integrable with algebro-geometric solutions on a symplectic submanifold [38]. Its solitary waves vanishing at both infinities are peaked solitons (peakons) [4], and they are orbitally stable [15]. It is also worth pointing out that the peakons replicate a feature that is characteristic for the waves of great height -waves of the largest amplitude that are exact traveling wave solutions of the governing equations for irrotational water waves, cf. [10,43]. The Cauchy problem and initial boundary value problem for the CH equation have been studied extensively [7,8,17,22]. It has been shown that this equation is locally well-posed [7,8,17,42] for initial data u 0 ∈ H s (R), s > 3 2 . Moreover, it has both globally strong solutions [6][7][8] and blow-up solutions at a finite time [6][7][8][9]. On the other hand, it also has globally weak solutions in H 1 (R) [2,14,46]. In comparison with the KdV equation, the advantage of the CH equation lies in the fact that the CH equation has peakons and models wave breaking [4,9] (namely, the wave remains bounded while its slope becomes unbounded in finite time [44]).
The DP equation is regarded as another model for nonlinear shallow water dynamics [11,13]. It was proved in [18] that the DP equation has a bi-Hamiltonian structure and an infinite number of conservation laws, and admits peakon solutions which are analogous to the CH peakons. The DP equation was already extended to a completely integrable hierarchy in a 3×3 matrix Lax pair, which possesses involutive representation of solutions under a Neumann constraint on a symplectic submanifold [39], and furthermore it was proven to have algebro-geometric solutions for such a 3 × 3 integrable system [30]. The Cauchy problem and initial boundary value problem for the DP equation have been studied extensively in [5,22,49,50]. Although the DP equation is very similar to the CH equation in the aspects of integrability, particularly in the form of equation, there are some significant differences between these two equations. One of the remarkable features of the DP equation is that it has not only (periodic) peakon solutions [18,50], but also (periodic) shock peakons [34]. Besides, the CH equation is a re-expression of geodesic flow on the diffeomorphism group [12], while the DP equation is regarded as a non-metric Euler equation [21].
For b = 3 and v ≡ u, the system (1.1) becomes the following Novikov equation which was proposed in [36]: We notice that the nonlinear terms in both CH and DP equations are quadratic with slightly different constant coefficients. However, the Novikov equation is an integrable peakon system with cubic nonlinearity. The first cubic nonlinear peakon system is the following FORQ equation (some times called the modified Camassa-Holm equation) [23,37,40]  which was shown integrable with Lax pair [40] and already generalized to a completely integrable hierarchy including both negative and positive flows with explicit peaked and cusped solutions [41]. The Novikov equation was found to be the second cubic system possessing peakon solutions. Moreover, the Novikov equation's wellposedness, blow-up phenomena and global solutions have been studied extensively in [28,29,45].
In view of the significance of conservation laws in studying the blow-up phenomena and global existence, we shall summarize two useful conservation laws which are for the CH, DP, and Novikov equations in the Table 1. But unfortunately, the above two conservation laws (the H 1 (R) and L 1 (R) norms) are unavailable to the system (1.1).
For b = 3, the system (1.1) becomes the following Geng-Xue (GX) system (sometimes called the two-component Novikov system) which was proposed in [26]: where m = u−u xx , n = v−v xx . The GX system (1.2) is an integrable generalization of the DP equation with cubic nonlinearity, and is associated with a 3 × 3 Lax pair to guarantee the integrability. This system also admits bi-Hamiltonian structure and regular single peakon solutions. Let us now set up the Cauchy problem for system (1.1) as follows: Using the Green function p(x) 1 2 e −|x| (x ∈ R) and the identity (1 − ∂ 2 x ) −1 f = p * f for all f ∈ L 2 (R), we can rewrite the system (1.3) as follows: where and By using an approach similar to the one in [47], the analytic solutions to the system (1.3) can readily be proved in both variables, globally in space and locally in time. However, the main goal of this paper is to discuss the peakon solutions of the system (1.1), study the blow-up phenomena and global existence for strong solutions to the system (1.3), and investigate the asymptotic behaviors of strong solutions at infinity within its lifespan.
In order to analyze the blow-up phenomena, here we may make good use of the fine structure of the system (1.3). Applying the transport equation theory and one-dimensional Morse type estimates, an important blow-up criterion is obtained. Then we exploit the characteristic ODE related to the system (1.3) to construct some invariant properties of the solutions, and sufficiently utilize the structure of the system itself, which eventually leads to the precise blow-up scenario and global existence for strong solutions to the system (1.3). Overall, we do not use any conservation laws rather than the symmetrical structure of the system (1.3) in the whole paper.
The rest of our paper is organized as follows. In Section 2, we discuss the peakon solutions of the system (1.1). In Section 3, we state the local well-posedness of the system (1.4). In Section 4, we provide the precise blow-up scenario and global existence for strong solutions to the system (1.3). In Section 5, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan.
2. Explicit peakon solutions. Regarding integrability of the b-family cubic system (1.1), so far we know that it is integrable when b = 3 (see the Lax pair in Ref. [26]). For other b's values, within our knowledge, the problem of integrability is still open. But, we may study the single and multi-peakon solutions of the system (1.1). To do so, let us assume that the b-family cubic system (1.1) has N-peakon solutions in the following form: (2.2) For N = 1, one can readily get one single peakons: where c 1 and c 2 are two arbitrary constants. For N = 2, we may rewrite the peakon system (2.2) as (2.4) If q 1 > q 2 , the peakon system (2.4) yields and if q 1 < q 2 , we have a similar system odd symmetric to equation (2.5) through a very simple transform (p 1 , p 2 , r 1 , r 2 , q 1 , q 2 ) → (−p 1 , −p 2 , −r 1 , −r 2 , −q 1 , −q 2 ). So, let us just consider the system (2.5). Apparently, a keen observation on the system (2.5) leads to and there exists a constant c * such that p 1 p 2 = c * r 1 r 2 . (2.7) Let p 1 = c p p 2 , r 1 = c r r 2 . Then, substituting them into equation (2.4) generates the following crucial relationships: Obviously, when b = 2, we have c p = −1, c r = 1 or c p = 1, c r = −1., and when b = 2, c p = −1/c r . Let us discuss the two peakon solutions for b = 2 and b = 2 below, separately.
• For the case b = 2, the system (2.4) can be changed into Through a lengthy procedure to solve the above system, we can arrive at where A i (i = 1, 2, 3) and c are arbitrary constants.
So, we get the following two-peakons for the b-family two-component system (1 where p i (t), r i (t), q i (t) (i = 1, 2) are given through equation (2.9). The graphs of u(t, x) and v(t, x) are shown in Figure (1).
• For the case b = 2, we do not have a general explicit formula for the twopeakon solutions to the system (1.1). But, we do solve for the explicit two-peakon solutions for b = 3 and b = 4. Let us display them below.
When b = 3, solving the system (2.4) generates the following two-peakon solutions to the b-family two component system (1.1): where and A i (i = 1, 2, 3), c are arbitrary constants. The graphs of u(t, x) and v(t, x) are shown in Figure (2). When b = 4, solving the system (2.4) we obtain the following two-peakon solutions to the b-family two component system (1.1): where , c are arbitrary constants. The graphs of u(t, x) and v(t, x) are shown in Figure (3).
Remark 2.1. The two-soliton graphs (2) and (3) for b = 3 and b = 4 do include a singular amplitude p 1 (t) for the function u(t, x) at some time while the amplitude r 1 (t) for the other function v(t, x) immediately vanishes to zero, but fortunately both u(t, x) and v(t, x) are continuous and bounded all the times. The two-soliton graph (1) for b = 2 has no any singularity since both u(t, x) and v(t, x) are continuous at any time.  In order to show the above three two-peakons satisfy the b-family two-component system (1.1), let us take the case b = 4 as the representative to prove. Without loss of generality, let A 1 = c = 0, then the system (2.12) can be rewritten in the following form: and A i (i = 2, 3) are arbitrary constants.
Hence, we substitute them into the b-family two-component system (1.1) and obtain Therefore, both u(t, x) and v(t, x) given in equation (2.13) are two-peakon solutions to the b-family two-component system (1.1). Similarly, we can prove that both u(t, x) and v(t, x) given in equations (2.10) and (2.11) are also two-peakon solutions to the b-family two-component system (1.1) with b = 2 and b = 3, respectively.
3. Local well-posedness. Applying the Littlewood-Paley decomposition and transport equation theory in Besov spaces [1], which in combination with the classical Kato semigroup theory [32], one may follow the similar arguments as in [48] to obtain the following local well-posedness result for the system (1.4).
Moreover, the solution depends continuously on the initial data, that is, the mapping is continuous.
(2) The maximal existence time T in Theorem 3.1 can be chosen independent of the regularity index s, which will be shown in Remark 4.1 below.
(3) As is well known, the Cauchy problems for the CH, DP and Novikov equations are locally well-posed when the initial data u 0 ∈ H s (R) as long as s > 3 2 [42,45,49]. However, the regularity index in Theorem 3.1 cannot be improved to s > 3 2 , whose reason lies in the following observation: Let (u (i) , v (i) ) (i = 1, 2) be two solutions to the system (1.4) with the initial data (u (1) and v (12) x u (12) x + u (2) x v (2) x v (12) (12) .
Going the similar line as the proof of Lemmas 3.1-3.2 in [48], via the Morse type inequality, one needs to estimate the H s−1 (R) norms of the following cross terms of u x , v (12) x (i = 1, 2) and their first order derivatives: x ) x v (2) , which ultimately leads to s > 5 2 . Nevertheless, this case would not happen to the reduced single equations of system (1.4), such as the CH, DP and Novikov equations.
4. Blow-up and global existence. In this section, we will derive the precise blow-up scenario of strong solutions to the system (1.3), and then state its global existence under some assumptions. Let us first prove a crucial blow-up criterion for the system (1.3). For this, we need some a priori estimates of the following transport equation: In addition, the following one-dimensional Morse-type estimates are also required.
Proof. We will prove the theorem by induction with respect to the regularity index s (s > 1 2 ) as follows.
Step 1. For s ∈ ( 1 2 , 1), by Lemma 4.2 and the system (1.3), we have , u x = (∂ x p) * m, u xx = u − m and ||p|| L 1 = ||∂ x p|| L 1 = 1, together with the Young inequality, for all s ∈ R, we have and Similarly, the identity v = p * n ensures and and Likewise, Thus, we have Taking advantage of Gronwall's inequality, one gets which contradicts the assumption that T < ∞ is the maximal existence time. This completes the proof of the theorem for s ∈ ( 1 2 , 1).
Step 2. For s ∈ [1, 3 2 ), applying Lemma 4.1 to the first equation of the system (1.3), we get with ε 0 ∈ (0, 1 2 ). Using (4.5) and the fact that H Thanks to Gronwall's inequality again, we have which contradicts the assumption that T < ∞ is the maximal existence time. This completes the proof of the theorem for s ∈ [1, 3 2 ).

KAI YAN, ZHIJUN QIAO AND YUFENG ZHANG
which together with (4.6) yields Likewise, Thus, we have This along with (4.7) with s − 1 ∈ (0, 1) instead of s ensures Similar to Step 1, we can easily prove the theorem for s ∈ (1, 2).
Step 4. For s = k ∈ N and k ≥ 2, differentiating the system (1.3) k − 1 times with respect to x, we get which together with Lemma 4.1 imply (4.13) and

Making use of Proposition 4.1 and (4.1)-(4.4) again, one infers
where ε 0 ∈ (0, 1 2 ) and we used the fact that (4.14) Thus, we get Similarly, Then, we have which together with Gronwall's inequality and (4.10) with s = 1 imply which contradicts the assumption that T < ∞ is the maximal existence time. This completes the proof of the theorem for s = k ∈ N and k ≥ 2.
Step 5. For s ∈ (k, k + 1), k ∈ N and k ≥ 2, differentiating the system (1.3) k times with respect to x, we get and which together with Lemma 4.2 with s − k ∈ (0, 1) imply By using Gronwall's inequality, Step 3 with 3 2 + ε 0 ∈ (1, 2) and the similar argument as shown in Step 4, we can arrive at the desired result.
In summary, the above 5 steps complete the proof of the theorem. and some s ∈ ( 1 2 , s). Then Remark 3.1 (1) ensures that there exists a unique H s × H s (resp., H s × H s ) solution (m s , n s ) (resp., (m s , n s )) to the system (1.3) with the maximal existence time T s (resp., T s ). Since H s → H s , it follows from the uniqueness that T s ≤ T s and (m s , n s ) ≡ (m s , n s ) on [0, T s ). On the other hand, if we suppose that T s < T s , then (m s , n s ) ∈ C([0, T s ]; H s × H s ). Hence (m s , n s ) ∈ L 2 (0, T s ; L ∞ × L ∞ ), which is a contradiction to Theorem 4.1. Therefore, T s = T s . Now we turn our attention to the precise blow-up scenario for sufficiently regular solutions to the system (1.3). For this, motivated by [6,35], we first consider the characteristic ordinary differential equation as follows: for the flow generated by uv.
The following lemmas are very crucial to study the blow-up phenomena of strong solutions to the system (1.3).
The following theorem shows the precise blow-up scenario for sufficiently regular solutions to the system (1.3).
Proof. Assume that the solution (m, n) blows up in finite time (T < ∞) and there exists a constant C > 0 such that  With Theorem 4.2 in hand, we conclude this section with the global existence for the strong solutions to system (1.3). On the one hand, Table 1 in the Introduction shows that, unlike the CH and Novikov equations, the H 1 (R) norm is not conserved for the solution to system (1.3), thus it cannot be applied directly to uniformly control the L ∞ (R) norm of the solution by Sobolev's embedding theorem. Besides, for the DP equation, one can obtain that the L ∞ (R) norm of its solution will at most increase linearly in finite time, although the H 1 (R) norm is not a conservation law in this case [33]. On the other hand, Remark 4.2 and Theorem 4.2 tell us that the strong solution will blow up in finite time if the following assumption is not satisfied: Moreover, the L ∞ (R) norms of the solutions to the CH, DP and Novikov equations all do satisfy (4.22). Therefore, the above exponential increase condition (4.22) is a reasonable assumption to study the global strong solution to the system (1.3). Indeed, we here could obtain the global existence under a slight weaker condition.
Moreover, if both m 0 (x) and n 0 (x) do not change signs for all x ∈ R, then T = +∞, i.e. the solution to the system (1.3) exists globally.
Proof. We may assume that m 0 (x), n 0 (x) ≥ 0 for all x ∈ R. Because of (4.19) and (4.20), we have (4.23) Noticing Hence, we have By (4.24) and (4.25), together with the assumption of the theorem, we infer According to Theorem 4.2, T = +∞. Therefore, we have proven the theorem.

Asymptotic behaviors.
In this section, we investigate the asymptotic behaviors of the strong solution to the system (1.4) at infinity within its lifespan as the initial data decay exponentially and algebraically, respectively.
Proof. Let us first introduce the following weight function series {ϕ N (x)} (N = 1, 2, · · ·): with θ ∈ (0, 1). Then a simple calculation yields and By the first equation in the system (1.4), we have Multiplying (5.3) by (ϕ N u) 2n−1 (n ∈ N + ) and integrating the resulted equation on R with respect to x, one gets Note that where we used Hölder's inequality in the last inequality. Thus, we deduce To deal with the second equation in the system (1.4) in a similar way, we obtain Differentiating the first equation in the system (1.4) with respect to x and multiplying the obtained equation by ϕ N (x), one gets Multiplying (5.6) by (ϕ N u x ) 2n−1 (n ∈ N + ) and integrating the resulted equation on R with respect to x, we have As before, by the Hölder inequality, the right hand side of (5.7) can be bounded by Hence, we obtain as well as d dt In addition, observe that if the function f ∈ L p ∩ L ∞ for some 1 ≤ p < ∞ such that f ∈ L q ( ∀ q > p), then lim q→∞ ||f || L q = ||f || L ∞ .
So, making use of ψ N (x) instead of ϕ N (x) and going along the similar line as the proof in Theorem 5.1, one can readily get the following result.