ON TRAVELING WAVE SOLUTIONS OF THE θ-equation OF DISPERSIVE TYPE

Transcription

1 ON TRAVELING WAVE SOLUTIONS OF THE -EQUATION OF DISPERSIVE TYPE TAE GAB HA AND HAILIANG LIU Abstract. Traveling wave solutions to a class of dispersive models, u t u txx + uu x = uu xxx +(1 )u x u xx, are investigated in terms of the parameter, including two integrable equations, the Camassa-Holm equation, = 1/3, and the Degasperis-Procesi equation, =1/4, as special models. It was proved in [H. Liu and Z. Yin, Contemporary Mathematics, 011, 56, pp73 94] that when 1/ < 1smooth solutions persist for all time, and when 0 1, strong solutions of the -equation may blow-up in finite time, yielding rich traveling wave patterns. This work therefore restricts to only the range [0, 1/]. It is shown that when = 0, only periodic travel wave is permissible, and when = 1/ traveling waves may be solitary, periodic or kink-like waves. For 0 <<1/, traveling waves such as periodic, solitary, peakon, peaked periodic, cusped periodic, or cusped soliton are all permissible. 1. Introduction In this work, we investigate traveling wave solutions to a class of dispersive models the-equation of the form ([41]) u (1 x)u t +(1 x) x =(1 4) The equation can be formally rewritten as u x, x R, t>0. (1.1) x u t u txx + uu x = uu xxx +(1 )u x u xx, (1.) which when 0 <<1involvesaconvexcombinationofnonlineartermsuu xxx and u x u xx. In (1.1), two equations are worth of special attention: = 1 3 and = 1 4. The -equation when = 1 3 reduces to the Camassa-Holm(CH) equation, modeling the unidirectional propagation of shallow water waves over a flat bottom, 1991 Mathematics Subject Classification. 35Q51, 35Q53. Key words and phrases. Dispersive equations, traveling wave solutions, peakon, soliton. 1

2 TAE GAB HA AND HAILIANG LIU in which u(x, t) denotesthefluidvelocityattimet in the spatial x direction [3, 1, 3]. The CH equation is also a model for the propagation of axially symmetric waves in hyperelastic rods [13, 15]. Taking = 1 4 in (1.1) one finds the Degasperis- Procesi (DP) equation [0]. The DP equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is same as that for the CH equation [, 3]. This class may well have more applications than those mentioned here. In recent years, nonlocal dispersive models as an extension of the classical KdV equation have been investigated intensively at different levels of treatments: modeling, analysis as well as numerical simulation. The model derives in several ways, for instance, (i) the asymptotic modeling of shallow water waves [45,, 3]; (ii) renormalization of dispersive operators [45, 40]; and (iii) model equations of some dispersive schemes [41]. The peculiar feature of nonlocal dispersive models is their capability to capture both smooth long wave propagation and short wave breaking phenomena. The study of traveling wave solutions to dispersive equations proves to be insightful in understanding various wave structures involved in the dispersive wave dynamics, we refer to recent works [16, 9, 33, 34, 4, 46, 51] for such investigations equations. The class of -equations was identified by H. Liu [41] in the study of model equations for some dispersive schemes to approximate the Hopf equation With = 1 b+1 b model, u t + uu x =0. the model (1.1) under a transformation as shown in [39] links to the u t α u txx + c 0 u x +(b +1)uu x +Γu xxx = α (bu x u xx + uu xxx ), which has been extensively studied in recent years [19, 0, 6, 7, 30, 31]. Both classes of equations are contained in the more general class introduced in [40] using

3 ON NONLOCAL DISPERSIVE EQUATIONS 3 renormalization of dispersive operators and number of conservation laws, so called the B-equations u t + uu x +[Q B(u, u x )] x =0, (1.3) where Q = 1 e x and B is a quadratic function of u and u x.forthisclassthelocal well-posedness in C([0,T); H 3/+ (R)) C 1 ([0,T); H 1/+ (R)) for (1.3) with initial data u 0 H 3/+ is shown in [40]. In fact, up to a scaling of t t -equation can indeed be rewritten as (1.3) with 1 u B = u x. for = 0, the In the last decades, a lot of analysis has been given to the CH equation and the DP equation, among other dispersive equations. The CH equation has a bi-hamiltonian structure [8, 35] and is completely integrable [3, 7]. Its solitary waves are smooth if c 0 > 0andpeakedinthelimiting case c 0 = 0 [4]. The orbital stability of the peaked solitons is proved in [1], and that of the smooth solitons in [14]. The explicit interaction of the peaked solitons is given in [1]. It has been shown that the Cauchy problem of the CH equation is locally well-posed [8, 43] for initial data u 0 H 3/+ (R). Moreover, it has global strong solutions [6, 8] and also admits finite time blow-up solutions [6, 8, 9]. On the other hand, it has global weak solutions in H 1 (R) [,10,11,47].Theadvantageof the CH equation in comparison with the KdV equation, u t + uu x +Γu xxx =0, lies in the fact that the CH equation has peaked solitons and models the peculiar wave breaking phenomena [4, 9]. For the DP equation an inverse scattering approach for computing n-peakon solutions was presented in [38]. Its traveling wave solutions were investigated in [34, 44]. The formal integrability of the DP equation was obtained in [18] by constructing a Lax pair. It has a bi-hamiltonian structure with an infinite sequence of conserved quantities and admits exact peakon solutions which are analogous to the CH peakons [18]. Global strong solutions to the Cauchy problem of the DP equation are proved

4 4 TAE GAB HA AND HAILIANG LIU in [4, 36, 49] and finite time blow-up solutions in [4, 36, 48, 49]. On the other hand, it has global weak solutions in H 1 (R), see e.g. [4, 49] and global entropy weak solutions belonging to the class L 1 (R) BV (R) andtotheclassl (R) L 4 (R), cf. [5]. Though both the DP and the CH equation share some nice properties, they differ in that the DP equation has not only peakon solutions [18] and periodic peakon solutions [50], for example, u(x, t) = ce x ct, but also shock peakons [37] and periodic shock waves [5], for example u(x, t) = 1 t + c sign(x)e x, c > 0. The issue of how regularity of solutions changes in terms of the parameter was studied in [39], where authors prove that as increases regularity of solutions improves: strong solutions to the -equation subject to smooth initial data for 0 < < 1 may blow up in finite time, while in the case 1 1everystrongsolution to the -equation exists globally in time. This presents a clear picture for global regularity and blow-up phenomena of solutions to the -equation for all 0 <<1. The main quest of this paper is to see how the type of traveling wave solutions varies in terms of the parameter for 0 1/. Traveling waves for 1/ < 1 may be studied in a rather similar fashion, but we choose to skip this range since the relevant traveling waves are less interesting. 1.. Main results. Under the traveling wave transformation u(x, t) = ϕ(ξ), ξ = x ct, c>0, (1.) is reduced to the following ordinary differential equation form After integration we get cϕ + cϕ + ϕϕ = ϕϕ +(1 )ϕ ϕ, = d. (1.4) dξ cϕ + cϕ + 1 ϕ = ϕϕ + 1 (ϕ ) + a, (1.5)

5 ON NONLOCAL DISPERSIVE EQUATIONS 5 where a is an integral constant. We rewrite (1.5) as (4 1) (ϕ ) + ϕ cϕ = 1 (ϕ c) + a. (1.6) Hence Eq.(1.5) holds in weak sense for all ϕ Hloc 1 (R), i.e., it holds (4 1) (ϕ ) v + 1 ϕ v cϕv dy = (ϕ c) v + av dy (1.7) R for all v C 0(R). Our purpose is to find traveling waves of -equation (1.) for 0 1,andthe main results are stated in following theorems. Different types of solutions are to be clarified in subsequent proofs. Theorem 1.1. For 0 << 1 by (1.5) are as follows: R and a R, traveling wave solutions of (1.) given (1) If a c or a c (1 ), then (1.5) has no bounded solutions. () If c solution. <a<0, then (1.5) has smooth periodic solutions and a smooth soliton (3) If a =0, then (1.5) has smooth periodic solutions and a peakon solution. (4) If 0 <a< c (1 ), then (1.5) has smooth periodic solutions, a peaked periodic solution, cusped periodic solutions and a cusped soliton solution. Theorem 1.. For =0and a R, traveling wave solutions of (1.) given by (1.5) are as follows: (1) If a c, then (1.5) has no bounded solutions. () If a> c, then any bounded solution to (1.5) must be periodic. Theorem 1.3. For = 1 (1.5) are as follows: and a R, traveling wave solutions of (1.) given by (1) If a c, then (1.5) has no bounded solutions. () If c <a<0, then (1.5) admits both solitary wave and periodic solutions. (3) If a =0, then the only bounded solution to (1.5) is ϕ =c.

6 6 TAE GAB HA AND HAILIANG LIU (4) If a>0, then (1.5) admits both solitary like waves and kink like wave solutions. The traveling wave solutions obtained to solve (1.) contain those for both the CH and the DP equation, and the results generalize some previous traveling wave results [33, 34] for these two special models. Our qualitative analysis through some reduced planar systems is elementary, yet we obtain not only the rich phase portraits but also the existence of seven different types of bounded traveling wave solutions: periodic, solitary, peakon, peaked periodic, cusped periodic, cusped soliton and kink-like wave solutions. The rest of this paper is organized as follows. In, we prove Theorem 1.1 by analyzing traveling wave solutions for 0 <<1/. In 3, we show how traveling wave solutions are established to complete Theorem 1.. A detailed account of traveling wave solutions for the case of =1/ ispresentedin 4. Some concluding remarks are given in 5.. Traveling waves for 0 << 1 :ProofofTheorem Traveling wave system and phase portraits. Let w = ϕ,then(1.5)can be rewritten as dϕ dξ = w, dw dξ = ϕ cϕ (1 )w a (ϕ c). (.1) This is valid only when ϕ(ξ) = c. For ϕ = c,wemakethetransformationdξ = (ϕ c)dτ, thensystem(.1)becomes Note that dϕ dτ =(ϕ c)w, dw = dτ ϕ cϕ (1 )w a. dw dϕ = ww ϕ = ϕ cϕ (1 )w a, ϕ c (.)

7 ON NONLOCAL DISPERSIVE EQUATIONS 7 then systems (.1) and (.) have the same first integral H(ϕ, w) = c ϕ 1 w + ϕ a 1 = h, (.3) where h is an arbitrary constant. Hence both system (.1) and (.) have the same topological phase portraits except near the line ϕ = c. In order to find out bounded orbits for (.), we consider c in such a case (.) has four equilibriums, which are denoted by and ( c,w 1)= We see that (ϕ 1, 0) = c c +a, 0, (ϕ, 0) = c + c +a, 0 c, Proposition.1. Assume that c c c a, ( c c (1 ),w )=, <a< c (1 ) c c a. (1 ) since ϕ 1 <c<ϕ < c. (.4) <a< c (1 ). (i) (ϕ 1, 0), ( c,w 1) and ( c,w ) are saddle points and (ϕ, 0) is a center point. (ii) If c <a<0, then for ϕ [ϕ 1, c ), the homoclinic orbit connecting with (ϕ 1, 0) lies in between two separatrices passing ( c,w 1) and ( c,w ), respectively. (iii) If a =0, then separatrices of (ϕ 1, 0) is same as separatrices of ( c,w 1) and ( c,w ), respectively for ϕ< c. (iv) If 0 <a< c (1 ), then the connecting orbit between ( c,w 1) and ( c,w ) lies in between two separatrices passing (ϕ 1, 0) for ϕ [ϕ 1, c ). Phase diagrams for these cases are shown in Fig. 1. Proof. (i) Thecharacteristicequationfor(.)whenlinearizedatanequilibrium (ϕ,w )is D(λ) = w λ (ϕ c) (ϕ c) (1 )w λ = λ + pλ + q =0,

8 8 TAE GAB HA AND HAILIANG LIU where p =(1 3)w and q = 4(1 )(w ) 4(ϕ c)(ϕ c). The theory of a planar dynamical system (see e.g., [17]) yields that (ϕ,w )isasaddlepointifq<0, (ϕ,w )isacenterorspiralpointifq>0andp =0. Since H(ϕ, w w )=H(ϕ, w w), in such a case (ϕ, 0) can only be a center. When (ϕ,w )=(ϕ 1, 0), from (.4) we see that q = 4(ϕ 1 c)(ϕ 1 c) < 0. When (ϕ,w )=(ϕ, 0), p =0andq = 4(ϕ c)(ϕ c) > 0. Hence (ϕ 1, 0) is a saddle point and (ϕ, 0) is a center point. When ϕ = c, q = 4(1 )(w i ) < 0, i =1,. Hence, both ( c,w 1)and( c,w ) are saddle points. (ii) If c <a<0. The separatrices of ( c,w )aregivenby w + ϕ a 1 =0 and ϕ = c. These three separatrices together with ϕ = ϕ 1 forms a closed domain denoted by Ω. Note that for any interior point of Ω, w + ϕ a > 0, hence, H(ϕ, w) > 0. 1 Moreover, H(ϕ 1, 0) = c ϕ 1 1 (ϕ 1 ) 1 a > 0. Therefore (ϕ, w) :H(ϕ, w) =H(ϕ 1, 0),ϕ ϕ 1, c Ω. This together with the fact that (φ 1, 0) is a saddle point ensures that the orbit connecting with (φ 1, 0) is homoclinic. (iii) Ifa =0. Theseparatricesof( c,w )aregivenby w ϕ =0 and ϕ = c. Since H(ϕ 1, 0) = 0, the orbit passing through (ϕ 1, 0) coincides with the orbit passing through ( c,w ). By a similar argument as in (ii), we obtain (iv).

9 ON NONLOCAL DISPERSIVE EQUATIONS 9 According to the above analysis, we draw the phase portraits of (.1) and (.) which are shown in Fig.1. w c/ w * w c/ w * w c/ w * * * c/ * * c/ * * c/ c/ w * c/ w * i c / a ii a iii a c Figure 1. Phase portraits of system (.1) and (.).. Bounded traveling wave solutions. There is a possibility that some interval I satisfying ϕ(ξ) = c on ξ I may exist for 0 a< c (1 ) (see (ii) and(iii) of Fig.1). This traveling wave is called stumped wave. We now show this is impossible except a = c (1 ). Lemma.1. If there exists some interval [ξ 1,ξ ] such that ϕ(ξ) = c on ξ [ξ 1,ξ ], then a must be c (1 ). Proof. In the weak formulation (1.7), we take v C 0(R) butwithv =0outside [ξ 1,ξ ]toobtain ξ (4 1) (ϕ ) v + ξ 1 ϕ v cϕvdy = From ϕ(ξ) = c, ϕ (ξ) =0onξ (ξ 1,ξ ), it follows that ξ c (1 ) ξ 1 vdy = ξ 1 (ϕ c) v + avdy. ξ 1 ξ ξ 1 avdy.

10 10 TAE GAB HA AND HAILIANG LIU Hence a = c (1 ). Remark.1. By Lemma.1, there are no stumped traveling waves for c c (1 ). <a< From (.3) it follows that (ϕ ) = F (ϕ), (.5) where F (ϕ) = F 0(ϕ) h (c ϕ) 1, where F 0 (ϕ) =(c ϕ) 1 ϕ a. (.6) 1 In order to analyze all bounded solutions to (.5), we investigate the qualitative behavior of solutions to (.5) near points where F has a zero or a pole, which corresponds to the set {ϕ = 0} and {ϕ = c } in the phase plane (ϕ, ϕ ). The following lemma describes the behavior of smooth solutions to (.5) at the point where F (ϕ) =0. Lemma.. [33] (i) Assume F (ϕ) has a simple zero at ϕ =. Then for each ξ 0, there exists at least one nontrivial solution such that ϕ(ξ 0 )= satisfying ϕ(ξ) = (ξ ξ 0) F ()+O (ξ ξ 0 ) 4 as ξ ξ 0. (.7) (ii) If F (ϕ) has a double zero at ϕ =, then there exists only one bounded solution ϕ = ϕ(ξ) such that for some constant α. ϕ(ξ) α exp ξ F () as ξ, (.8) We would like to point out that the function F in [33] corresponds to the case =1/3. For a range of parameter 0 <<1/, F given in (.6) is well defined, and averysimilarprooftothatin[33]leadstotheaboveresult,asclaimedin[33]for the case =1/3.

11 ON NONLOCAL DISPERSIVE EQUATIONS 11 Next, we consider the behavior of solutions to (.5) near points where F has a pole. A pole of a function is defined as: Definition.1. Let r 0 and f be a given function. We say that f has a r-th pole at p if lim f(x)(x p) r =0. x p Remark.. If F (ϕ) has an r-th pole at ϕ =, then no classical solution exists such that ϕ(ξ 0 )=. If there is any weak solution such that ϕ(ξ 0 )=, it must satisfy lim ξ ξ0 ϕ (ξ) ±. h h h * * * c/ * * c/ * c/ h h i c / a ii a iii a c Figure. Graph of F 0 Using the above as a guide we now discuss different cases for F (ϕ) definedin (.6). Note that F0(ϕ) =(c ϕ) 1 3 ( ϕ +cϕ +a) = (c ϕ) 1 3 (ϕ ϕ 1)(ϕ ϕ ). (.9) Hence, ϕ 1 and ϕ are critical points of F 0 (ϕ) andf 0 (ϕ 1)=h 1 and F 0 (ϕ )=h where h 1 = H(ϕ 1, 0), h = H(ϕ, 0). Three cases of F 0 (ϕ) are shown in Fig.. Once a is fixed, a change in h will shift the graph vertically up and down. Hence we can determine which h s that yield bounded traveling waves. Case (1) : c <a<0. There are three cases to consider.

12 1 TAE GAB HA AND HAILIANG LIU c/ c/ c/ c/ c/ c/ c/ Figure 3. Types of graph of F (a) h 1 <h<h. For ϕ< c, F has three simple zero at z 1,z,z 3 such that z 1 <ϕ 1 <z <ϕ < z 3 < c.thegraphoff is like (1) of Fig.3. Since F (ϕ) > 0forz <ϕ<z 3 < c and F (z )=F (z 3 )=0,byLemma.,forsomesmall>0, lim ξ ξ 0 ϕ (ξ) < 0, lim ξ ξ 0 + ϕ (ξ) > 0 and lim ξ ξ 1 ϕ (ξ) > 0, lim ξ ξ 1 + ϕ (ξ) < 0, where ϕ(ξ 0 )=z and ϕ(ξ 1 )=z 3. Moreover since ϕ =0forz <ϕ<z 3, ϕ is strictly decreasing or increasing until one reaches the points ϕ = z or z 3.Thiswhen combined with (ii) of Proposition.1 shows that ϕ is a smooth periodic traveling wave. (b) h = h. F has a simple zero at z 4,doublezeroatϕ satisfying z 4 <ϕ. The graph of F is like () of Fig.3. Since for ϕ<z 4, F (ϕ) > 0andF (z 4 )=0, lim ξ ξ ϕ (ξ) > 0 and lim ξ ξ + ϕ (ξ) < 0,

13 ON NONLOCAL DISPERSIVE EQUATIONS 13 where ϕ(ξ )=z 4. If there is a bounded solution, then solution must converge to equilibrium point as ξ ±. But, equilibrium points ϕ 1 and ϕ are above z 4. Hence, there are no bounded solutions. (c) h = h 1. F has a double zero at ϕ 1, a simple zero at z 5 satisfying ϕ 1 <z 5 and F<0for z 5 <ϕ< c. The graph of F is like (3) of Fig.3. Since F (ϕ) > 0forϕ 1 <ϕ<z 5 < c and F (z 5 )=0,byLemma., lim ξ ξ 3 ϕ (ξ) > 0 and lim ξ ξ 3 + ϕ (ξ) < 0, where ϕ(ξ 3 )=z 5,andϕ ϕ 1 as ξ ±.Thereforeϕ is a smooth solitary wave case () : If a =0,wehavethreecasestoconsider. (a) 0<h<h. For ϕ< c, F has three simple zeros at z 6,z 7,z 8 such that z 6 <ϕ 1 <z 7 <ϕ < z 8 < c. Since F (ϕ) > 0forz 7 <ϕ<z 8 and F (z 7 )=F (z 8 )=0,ϕ is strictly monotone until one reaches the points ϕ = z 7 or z 8. Hence ϕ is a smooth periodic traveling wave by (iii) of Proposition.1. (b) h = h. F has a simple zero at z 9,adoublezeroatϕ such that z 9 <ϕ 1 <ϕ and F>0 for ϕ<z 9. So, F is well defined on ϕ z 9. Hence we can consider the area for ϕ z 9.But,sinceequilibriumpointsϕ 1 and ϕ are above z 9,therearenobounded solutions. (c) h =0. F has a double zero at origin. By Lemma., ϕ 0asξ ±.ThegraphofF is like (4) of Fig.3. Since F (ϕ) > 0for0<ϕ< c, ϕ is strictly monotone. Moreover ϕ increases and reaches the point ϕ = c.thereforeϕis a peaked solitary wave. case (3) : If 0 <a< c (1 ),wehavefivecasestoconsider. (a) h = h.

14 14 TAE GAB HA AND HAILIANG LIU F has a simple zero at z 10,adoublezeroatϕ satisfying z 10 <ϕ 1 <ϕ and F (ϕ) > 0forϕ<z 10. Since equilibrium points ϕ 1 and ϕ are above z 10,thereare no bounded solutions. (b) 0<h<h. F has three simple zeros at z 11,z 1,z 13 such that z 11 <ϕ 1 <z 1 <ϕ <z 13 < c. Since F (ϕ) > 0forz 1 <ϕ<z 13 and F (z 1 )=F (z 13 )=0,ϕ is strictly monotone until one reaches the points ϕ = z 1 or z 13. Hence ϕ is a smooth periodic traveling wave by (iv) ofproposition.1. (c) h =0. a F has two simple zeros at ± such that a 1 1 <ϕ a 1 < 1 <ϕ < c. a The graph of F is like (5) of Fig.. Since F (ϕ) > 0for < ϕ < c and 1 a F =0,ϕis strictly monotone until one reaches the point ϕ = a 1 Moreover, ϕ increases and reaches the point ϕ = c. periodic traveling wave by (iv) ofproposition.1. (d) h 1 <h<0. 1. Therefore, ϕ is a peaked F has two simple zeros at z 14,z 15 satisfying z 14 <ϕ 1 <z 15 <ϕ and F has the following form F = (ϕ z 14)(ϕ z 15 )f 1 (ϕ), where f (c ϕ) 1 1 (ϕ) > 0 for ϕ< c. The graph of F is like (6) of Fig.3. So for z 15 <ϕ< c, F (ϕ) > 0andF has a 1 -th pole at c. Hence, F (ϕ) as ϕ c. Thus, by Lemma., ϕ is a cusped periodic traveling wave. form (e) h = h 1. F has a double zero at ϕ 1 and F (ϕ) > 0forϕ 1 <ϕ< c.sof has the following F = (ϕ ϕ 1) f (ϕ), where f (c ϕ) 1 (ϕ) > 0 for ϕ< c.

15 ON NONLOCAL DISPERSIVE EQUATIONS 15 The graph of F is like (7) of Fig.3. Hence F (ϕ) > 0forϕ 1 <ϕ< c and F has a 1 -th pole at c.thereforebylemma.,ϕ is a cusped solitary wave with ϕ ϕ 1 as ξ ±. 3. Traveling waves for =0:ProofofTheorem1. In this section, we study traveling wave solutions to the -equation with =0, i.e., u t u txx + uu x = u x u xx. (3.1) Substituting =0into(1.5),wehave cϕ + cϕ + 1 ϕ = 1 (ϕ ) + a, (3.) where a is an integral constant. Let w = ϕ then we have the traveling wave system from (3.) ϕ = w, w = ϕ 1 c ϕ + 1 c w + a. (3.3) To find out bounded orbits for (3.3), we consider a> c since in such a case (3.3) has two critical points, (ϕ 1,w )=(c c +a, 0) and (ϕ,w )=c + c +a, 0), respectively. Furthermore, from the theory of planar dynamical system (see e.g., [17]), we know that (c c +a, 0) is a saddle point and (c + c +a, 0) is a center or spiral point. From (3.3) it follows that upon integration we find dw dϕ = ϕ 1 c ϕ + 1 c w + a, w 1 w = ϕ a + b exp c ϕ, (3.4) where b is an integral constant. This suggests that any orbit must be symmetric about ϕ-axis.

16 16 TAE GAB HA AND HAILIANG LIU We first look at the particular orbit passing through the saddle point (c c +a, 0), which corresponds to b = b =c( 1 c +a c)exp c +a 1. c Hence two branches of the orbit passing through (c c +a, 0) is determined by 1 w = ϕ a + b exp c ϕ. To proceed we need the following lemma. Lemma 3.1. For any fixed c, if c <a<0 or a>0, then b > a. Proof. Define k(a) =c( c +a c)exp( 1 c c +a 1) a. We first consider for a>0. We know that d 1 k(a) = exp c +a 1 da c 1 > 0 and k(0) = 0. Hence k(a) is increasing and positive function for a>0. Therefore, b > a for a>0. By a similar argument, we also have b > a for c <a<0. We now examine to see how many times the above mentioned orbit crosses ϕ-axis. At crossing points (ϕ, 0), we have 1 ϕ a + b exp c ϕ =0. (3.5) Note that ϕ 1 = c c +a is a root of (3.5). Define g(ϕ) =ϕ a and h(ϕ) = b exp( 1 ϕ), then intersections of g(ϕ) andh(ϕ) aresolutionsof(3.5). c We distinguish a>0fromthecase c <a 0. For a>0, we have ϕ 1 < 0andb > a by Lemma 3.1. Also, g(ϕ) andh(ϕ) have following properties: (i) g(ϕ) and h(ϕ) are monotonically decreasing for ϕ 0. (ii) g(ϕ 1)=h(ϕ 1)andg (ϕ 1)=h (ϕ 1).

17 ON NONLOCAL DISPERSIVE EQUATIONS 17 Hence Eq.(3.5) has only one solution, ϕ 1,whichimpliesthattheorbitpassing through the point (ϕ 1, 0) touches ϕ-axis only once. Therefore (c+ c +a, 0) must be a center, with a phase portrait shape like Fig.4. Thus any trajectory issuing from (ϕ, 0) with ϕ >c c +a must be a closed trajectory, i.e., any bounded traveling wave solution is periodic. The case c <a 0canbetreatedinsimilar fashion. * * Figure 4. Phase portrait of (3.3) 4. Traveling waves for = 1:ProofofTheorem1.3 In this section, we study traveling wave solutions to the -equation with = 1, i.e., u t u txx + uu x = 1 uu xxx + 1 u xu xx. (4.1) Substituting = 1 into (1.5), we have the following ordinary differential equation cϕ + cϕ + 1 ϕ = 1 ϕϕ + a. (4.) We first consider the case of a =0. (4.)witha =0becomes (ϕ c)(ϕ ϕ) =0. Hence, the only bounded solution for ξ R is ϕ =c. Conversely, taking ϕ =c into (4.), we obtain a =0. Therefore,ϕ =c if and only if a =0.

18 18 TAE GAB HA AND HAILIANG LIU For a = 0,letw = ϕ,(4.)canberewrittenas ϕ = w, w = ϕ cϕ a ϕ c, when ϕ = c. To find out bounded orbits for (4.3), we consider a> c (4.3) since in such acase(4.3)hastwocriticalpoints,whicharedenotedby(ϕ 1,w )=(c c +a, 0) and (ϕ,w ) = (c + c +a, 0), respectively. We see that for c < a < 0, ϕ 1 <ϕ < c and for a>0, ϕ 1 < c <ϕ.system(4.3)hasthefirstintegral H(ϕ, w) =w ϕ +4aln 1 ϕ c = h, (4.4) where h is an arbitrary constant. Hence by the qualitative theory of planar dynamical system, we have the following. (i) If c <a<0, then (ϕ 1, 0) is a saddle point, and (ϕ, 0) is a center point. (ii) If a>0, then both (ϕ 1, 0) and (ϕ, 0) are saddle points. w w * * * * i c / a ii a Figure 5. Phase portraits of system (4.3) According to the above properties, we obtain the phase portraits of system (4.3) (see Fig.5). Let G(ϕ) =G 0 (ϕ)+h, where G 0 (ϕ) =ϕ 4a ln 1. ϕ c

19 ON NONLOCAL DISPERSIVE EQUATIONS 19 From (4.4), we have w = G(ϕ). (4.5) Note that G 0(ϕ) =(ϕ c) 1 (ϕ ϕ 1)(ϕ ϕ ). Hence, ϕ 1 and ϕ are critical points of G 0 (ϕ) andg 0 (ϕ 1)=h 3 and G 0 (ϕ )=h 4 where h 3 = H(ϕ 1, 0), h 4 = H(ϕ, 0). Two cases of G 0 (ϕ) are shown in Fig.6. Once a is fixed, a change in h will shift the graph vertically up and down. Hence we can determine which h s that yield bounded traveling waves. h * * h c h h * c * i c / a ii a Figure 6. Graph of G 0 For c <a<0, by same argument as case (1) of subsection., we obtain the result such as () of Theorem.1. For a>0, we consider following cases. (a) h = h 3 or h 4. For h = h 3, G has a double zero at ϕ 1 and G(ϕ) > 0forϕ 1 <ϕ<c. Also G(ϕ) + as ϕ c but, ϕ never touches the line c. ThereforebyLemma., ϕ is a cusped like solitary wave with ϕ ϕ 1 as ξ ±. This wave is similar to shapes connected two kink waves. Similarly, for h = h 4, ϕ is cusped like solitary wave with ϕ ϕ as ξ ±. (b) h 4 <h< h 3 or h< h 4.

20 0 TAE GAB HA AND HAILIANG LIU For h 4 <h< h 3, G has two simple zeros at z 16 and z 17 such that z 16 <ϕ 1 < z 17 < c and G(ϕ) > 0forz 17 <ϕ<c. Also, G(ϕ) + as ϕ c but, ϕ never touches the line c. By Lemma., lim ξ ξ 5 ϕ (ξ) < 0 and lim ξ ξ 5 + ϕ (ξ) > 0, where ϕ(ξ 5 )=z 17. Moreover, since ϕ =0forz 17 <ϕ<c, ϕ is strictly decreasing or increasing until it reaches the point ϕ = z 17 or c. Since ϕ is not touched at c and there is no center point for z 17 <ϕ<c, implying that this process occurs only once. Therefore, ϕ is a kink like traveling wave. Similarly, for h< h 4, ϕ is also a kink like traveling wave. 5. Concluding remarks In this paper, we have investigated traveling wave solutions to the -equation for 0 1, including two well-studied integrable equations, the Camassa-Holm equation, = 1,andtheDegasperis-Procesiequation, = 1. Hence this paper 3 4 generalizes some known traveling wave results [33, 34] for these two special models. As shown in [39], when 0 1,strongsolutionsofthe-equation may blow-up in finite time, correspondingly, the traveling waves for [0, 0.5] are very rich. Our study shows that when =0,onlyperiodictravelwaveispermissible,and when =0.5 travelingwavesmaybesolitary,periodicorkink-likewaves. For 0 <<1/, traveling waves such as periodic, solitary, peakon, peaked periodic, cusped periodic, or cusped soliton are all permissible. Acknowledgments Liu s research was partially supported by the National Science Foundation under Grant DMS and DMS

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