Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation

In this paper, we mainly study the Cauchy problem of an integrable dispersive Hunter-Saxton equation in periodic domain. Firstly, we establish local well-posedness of the Cauchy problem of the equation in \begin{document}$H^s (\mathbb{S}), s > \frac{3}{2},$\end{document} by applying the Kato method. Secondly, by using some conservative quantities, we give a precise blow-up criterion and a blow-up result of strong solutions to the equation. Finally, based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation. By using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equations, we get the travelling wave solutions of the original equation.

1. Introduction. In the paper we consider the Cauchy problem for the following periodic integrable dispersive Hunter-Saxton equation [28]    u xt = u + 2uu xx + u 2 x , u(t, x)| t=0 = u 0 (x), u(t, x + 1) = u(t, x). (1.1) Recently, Hone, Novikov and Wang in [28] present a classification of nonlinear partial differential equations of second order of the general form It is known that the above form contains many interesting equations, especially some valuable integrable ones.
The first important integrable member of the form (1.2) is the short-pulse equation [47] in case that c j = 0 for j = 0, 1, 2, 3 and d 0 = d 1 = 0, d 3 = 2d 2 , or more simplified The short-pulse equation is derived by Schäfer and Wayne in [47] as an approximate model of Maxwell's equations describing the propagation of ultra-short optical pulses in nonlinear media. The local well-posedness of the short-pulse equation in H 2 on the line was showed by Schäfer and Wayne in their original paper [47]. They also proved the non-existence of smooth traveling wave solutions [47]. Later, by using some conserved quantities, Pelinovsky and Sakovich [46] extended the local solution into a global solution. The blow-up phenomena both on the line and in the periodic domain were investigated by Liu, Pelinovsky and Sakovich in [40].
The second important integrable member of (1.2) is the Ostrovsky-Hunter equation [3] u xt = u + (u 2 ) xx . This equation is known under different names, such as the Vakhnenko equation [42], the short-wave equation [29], and the reduced Ostrovsky equation [44], which models small-amplitude long waves in rotating fluids of finite depth, under the assumption of no-high frequency dispersion.
Local existence of solutions of the Ostrovsky-Hunter equation in H s (R) for s > 3 2 was obtained in [48]. But for sufficiently steep initial data u 0 ∈ C 1 (R), local solutions break [39,4] in a finite time in the standard sense of finite-time wave breaking that occurs in the inviscid Burgers equation u t + uu x = 0. Then, by using a new transformation of the reduced Ostrovsky equation to the integrable Tzitzéica equation, Grimshaw and Pelinovsky [26] proved global existence of smallnorm solutions in H 3 (R). Another series equation of the type (1.2) is the nonlinear Klein-Gordon equation. Considering a linear coordinate transform x = ξ+τ 2 , t = ξ−τ 2 , which leads to ∂ xt = ∂ 2 ξ − ∂ 2 τ := . After this transform, and letting c j = d j = 0 for j = 1, 2, 3, we get the well-known nonlinear Klein-Gordon equation Some sufficient conditions on quadratic (d 0 = 0) or cubic (c 0 = 0) nonlinearities were given in [24], to achieve the global existence and find sharp asymptotic of small solutions to the Cauchy problem with small and regular initial data having a compact support. Moreover it was proved that the asymptotic profile of solutions differs from that of the linear Klein-Gordon equation. Compactness condition on the data was removed in [27] in the case of a real-valued solution. When the initial data are complex-valued, the global existence and L ∞ time decay estimates of small solutions to the Klein-Gordon equation with cubic nonlinearity were obtained in [50].
In this paper, we study the equation (1.1), which is one of the integrable generalized short-pulse equation of the form (1.2), that means it possesses an infinite hierarchy of local higher symmetries [28]. We can consider (1.1) as the Hunter-Saxton (HS) equation with a dispersion term u. In fact, dropping the dispersion term u from right-hand side of (1.1) and replacing u with 1 2 u, we can get the following HS equation The equation (1.4) was derived by Hunter and Saxton as an asymptotic model of liquid crystals [30]. The x derivative of the HS equation corresponds to geodesic flow on an infinite-dimensional homogeneous space with constant positive curvature [36]. The HS equation also has a bi-Hamiltonian structure [30,43] and is completely integrable [1,31]. The initial value problems for the HS equation on the line and the circle were studied in [30,54]. Global solutions of the HS equation was investigated in [5] The HS equation (1.4) also arises in a different physical context as the highfrequency limit [22,31] of the Camassa-Holm (CH) equation. The CH equation can be regarded as a shallow water wave equation [8,18]. It is completely integrable [2,9,17,19]. It has a bi-Hamiltonian structure [25], and has peakon solutions of the form ce −|x−ct| with c > 0, which are orbitally stable [21]. It is worth mentioning that the peakons are suggested by the form of the Stokes water wave of greatest height, cf. [11,12,15,16,41,51]. The local well-posedness for the Cauchy problem of the CH equation was investigated in [13,23,37]. Blow-up phenomena and global existence of strong solutions were discussed in [10,13,14]. The global weak solutions [20,53], conservative and dissipative solutions were studied in [6,7] respectively.
However the Cauchy problem of the equation (1.1) has not been studied yet. In the paper we first establish the local well-posedness of the Cauchy problem (1.1) in H s (S), s > 3 2 , by using the Kato method. Then, we deduce some useful conservation laws of the equation, and use these obtained conservative quantities to control the H 1 norm of the solution. Next, we give a precise blow-up criterion, and apply this precise criterion to obtain a blow-up result and the precise blow-up rate for strong solutions to the equation (1.1). Moreover, we study the travelling wave solutions. Based on a sign-preserve property, we transform the original equation into the sinh-Gordon equation, using the travelling wave solutions of the sinh-Gordon equation and a period stretch between these two equation, we then get the travelling wave solutions of the original equation (1.1).
2. Local well-posedness. In order to establish local well-posedness of the Cauchy problem (1.1), we first consider the following Cauchy problem x ∈ R. (2.1) Comparing (2.1) with the original equation (1.1), here we minus the average of (u − u 2 x ) to insure that the integral x 0 of the right hand side of (2.1) is a continuous periodic function when u belongs to some Sobolev spaces. Integrating both sides of (2.1) with respect to x, and by choosing a specific boundary term, we can transform (2.1) into a transport-like equation x ∈ R. (2.2) is chosen to insure that the quantity 1 0 (u − u 2 x )(t, y)dy is conserved, which will be shown a posteriori.
Let S denote a circle of unit length. The usual Sobolev norm of a function f (x) on S is defined based on it's Fourier seriesf (n), n ∈ Z, more precisely, we have For simplicity, sometimes we will denote the H s (S) norm by · s . Local wellposedness of the Cauchy problem (2.2) with initial data u 0 ∈ H s (S), s > 3 2 can be obtained by using the framework presented in [54] for the following Hunter-Saxton equation, Similar to the proof of the well-posedness of the above HS equation, we have the following local well-posedness result for (2.2) by the Kato method.
Then there exists a maximal T = T (u 0 ) > 0, and a unique solution u to (2.2), such that And, the solution depends continuously on the initial data, i.e., the mapping is continuous. Moreover, for t ∈ [0, T ) we have the following conserved quantity 2) with respect to x, we deduce that u satisfies the equation (1.1). On the other hand, any solution u(t, x) of (1.1) entails that Thus, by integrating (1.1) with respect to x, one can easily see that u(t, x) is also the solution of (2.2).
Applying Theorem 2.1 and Remark 2.2, we have the following local well-posedness result for (1.1).
Suppose that u 0 satisfies the following condition: Then there exists a maximal T = T (u 0 ) > 0, and a unique solution u to (1.1), such that And, the solution depends continuously on the initial data, i.e., the mapping is continuous. Moreover, for t ∈ [0, T ) we have the following conserved quantity To prove Theorem 2.1, we first recall the Kato method for the Cauchy problem for abstract quasi-linear equations of evolution. For convenience we state the relevant theorem in the simplest form sufficient for the present purpose. Consider the Cauchy problem for the quasi-linear equation of evolution Let X, Y be reflexive Banach spaces with Y continuously and densely imbedded in X. Let Q be an isomorphism (bi-continuous linear map) of Y onto X. Assume that the function A, defined on Y , and f (t, ·) satisfies the following conditions: and extends also to a map from X into X. For all t ∈ [0, ∞), f is uniformly bounded on bounded sets in Y , and Here µ 1 , µ 2 , µ 3 , and µ 4 depend only on max y Y , z Y .
Theorem 2.4. (Kato [32]) Assume that (i), (ii), and (iii) hold. Given φ ∈ Y, there is a maximal T > 0 depending only on φ Y and a unique solution v to (2.4) Obviously, Q is an isomorphism of H r (S) into H r−1 (S). In order to prove Theorem 2.1, by applying above theorem, we only need to verify that A(u) and f (t, u) satisfy the conditions (i)-(iii).
To purse our goal, we need the following lemmas.
[32] Let s, t be real numbers such that −s < t ≤ s, and f ∈ H s (S), g ∈ H t (S), Then where C is a positive constant depending on s, t.
The above two lemmas can be proved in the same way as in Lemmas (2.6)-(2.9) in [54] by replacing u∂ x with −2u∂ x . At last, we have the following estimate for the right-hand side term f (t, u) of (2.2).
Then f (t, u) is uniformly bounded on bounded sets in H r (S) and satisfies is a constant function of x, by the embedding H r−1 → L ∞ and Lemma 2.5, for ∀ p ∈ R we have

(2.7)
Note that H r−1 (S) is a Banach algebra. Using (2.7) we can get This proves (1). Let w = 0 in the above inequality, we obtain that f is uniformly bounded on bounded set in H r (S). To prove (2), we have where we applied Lemma 2.5 with s = r − 1, t = r − 2.
Proof of Theorem 2.1. Combining Theorem 2.4 and Lemmas 2.6-2.8, we can get the local well-posedness result of Theorem 2.1. For the conservative property, multiplying (2.1) by u x , and integrating by parts in the unit circle S, we have This completes the proof of Theorem 2.1.

Blow-up.
In this section, we study the wave breaking phenomena of Eq.(1.1). It turns out that there are some smooth initial data for which the slope of corresponding solutions will blow up in finite time.
The following conserved quantities are key to our main blow-up result. Then the corresponding solution u to (1.1) with the initial data u 0 satisfies the following conservation laws: Or, more precisely, we have here K ≥ 0 is a constant only depending on u 0 , and as a consequence Proof. Multiplying (1.1) by u x , and integrating by parts in the unit circle S, we get which implies the desired conserved quantity for u x L 2 . On the other hand, integrating (1.1) directly over S yields By the conserved quantity u x L 2 , we obtain Obviously, we have K ≥ 0 only depending on u 0 . Then we have According to the equality 3.1, we finally get for any (x, t) ∈ S × [0, T ). This completes the proof.
In fact, by the above conservation law of u x L 2 and the boundedness of u L ∞ , the H 1 norm of u can be controlled in the periodic case. We now present a precise blow-up criterion for (1.1) as follows. Proof. Applying Theorem 2.3 and a simple density argument, it's sufficient to consider the case where u ∈ C ∞ 0 . To begin with, for the H 1 norm of u, we have Differentiating both sides of (1.1) with respect to x, taking L 2 inner product with u xx , and then integrating by parts, we obtain Suppose that u x is bounded from above on [0, T ) and T < ∞. we get An application of Gronwall's inequality yields By (3.2) and the above inequality, we obtain . This contradicts the assumption that T < ∞ is the maximal existence time.
In our next study, we will use the following result.
The function m(t) is absolutely continuous on [0, T ) with dm dt = v x,t (ξ(t), t) a.e. on [0, T ). Now we can state the following main theorem, which shows that there is some initial data such that the corresponding solution to (1.1) will blow up in finite time.
Theorem 3.4. Given u 0 ∈ H s , s > 3, satisfies the following condition Then the corresponding solution u(t, x) of (1.1) blows up in finite time.
Proof. Let T > 0 be the maximal existence time of the solution u(·, t) of (1.1) with initial data u 0 ∈ H 3 . By the local well-posedness of Theorem 2.
As we have assumed that S u 0 dx = K ≥ 1, then, from Lemma 3.1 we know Considering that in the periodic case, u 0 (·) can't be monotonic function or a constant over S (this contradicts the assumption (3.3)). Therefore, m 0 := sup x∈S ∂ x u 0 (x) > 0. By the above inequality, we can get which leads to As we have mentioned that m 0 > 0, there exist such that m(t) → +∞, as t → T 1 for K > 1 ( or t → T 2 for K = 1 ). According to the previous blow-up criterion in Lemma 3.2, this proves the wave breaking theorem.
Finally, we give the exact blow-up rate for blowing-up solutions. The proof is trivial and similar to Theorem 4.7 in [38], here we omit it. In order to simplify the problem and make all the transforms meaningful and invertible, we'll omit some regularity discussion at first. For the original ideal of this transformation, the readers can refer to Hone, Novikov and Wang [28].
Reciprocal transformation: Taking the x derivative of (1.1), we can get the following equations readily,  Assume that m 0 (x) = 1 + 4∂ xx u 0 (x) > 0 for any x ∈ R, and let T be the maximal existence time of the solution u(x, t) to (1.1) with the initial data u 0 (x). Then Proof. We prove the lemma in Lagrangian coordinates. Consider the following initial value problem: According to the standard ODE theory, we infer that (4.2) has a unique solution q ∈ C 1 (R × [0, T ); R). Moreover, the map q(·, t) is an increasing diffeomorphism of R with By (4.2) and (4.1), we deduce that for any fixed x ∈ R, This implies Since m 0 > 0, and the map q(·, t) is an increasing diffeomorphism, we obtain that m(x, t) > 0 for any (x, t) ∈ R × [0, T ).
Hereafter we shall assume that m 0 (x) > 0, ∀x ∈ S. By Lemma 4.1 we have m(x, t) > 0, ∀(x, t) ∈ S × [0, T ). Under this assumption, the original equation (1.1) can be written in the conservative form We now introduce a coordinate transformation (x, t) → (ξ, t) according to Thanks to the equality (4.3), and p(x, t) > 0 for ∀(x, t) ∈ R × [0, T ), it is easy to see that the above transformation is one-one and on-to from R to R with a period stretch from 1 to : This new period is independent of t, that means the map (4.4) is also a bijective from S = R/Z to S = R/kZ for each t ∈ [0, T ), and has an inverse map satisfying By this relation, we can easily get that Now, define that v(ξ, t) = ln p(x(ξ, t), t). By the above relation we can deduce that v satisfies the following Gorden-type equation : This is the Sinh-Gordon equation, which has been studied by many authors, its traveling wave solution over R was constructed in [52] with a complicated expression. As we mainly pay attention to its periodic solution, we'll get a more natural and simpler form. We look for its travelling wave solutions of the following form: v(ξ, t) = v(z), z = ξ − ct. (4.8) Applying (4.8) into (4.7) leads to the following ordinary differential equation for v(z) : The main results are included in the following theorem.
Theorem 4.2. For every c > 0, there exists a smooth periodic solution with the following form : here, for some n ∈ N + .
Proof. Multiplying both sides of (4.9) with v (z), we obtain d dz Since we are looking for a periodic smooth solution, there exists at least one minimum v(z 0 ) in a period such that v (z 0 ) = 0 and v (z 0 ) ≥ 0. By (4.9) we know v(z 0 ) ≤ 0. Without losing generality, set z 0 = 0 and denote v(0) = −v 0 ≤ 0. Then, we get the following equality 1 2 c(v (z)) 2 + cosh v(z) = cosh v 0 . By the definition of T c , we can continuously extend v(z) periodically over R. It's easy to check that v(z) is a periodic continuous solution of (4.9). In other words, v(z) is a continuous solution of (4.9) over S = R/T c Z. By virtue of 1D Laplace equation over circle, v(z) is automatically smooth. This proves the first part of the theorem. As for the period scaling, by the equality (4.6), we can calculate the minimum positive period of p(x, t) by As the solution of original equation (1.1) has a period equal to 1, so does p(x, t).
In order to suit this periodic condition, there must exist an positive integer n, such that nT * = 1. This completes the proof.
Remark 4.3. If c < 0, we can infer from (4.12) that v (z) can't change sign, so v(z) is a monotonic function then it must be 0, i.e., we can't get non-trivial periodic solutions if c < 0. This means the waves only travel from left to right (or anti-clock direction in the circle case) and not the inverse.