Variational Theory of Solitons for a Higher Order Generalized Camassa-Holm Equation

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1 International Journal of Mathematical Analysis Vol. 11, 2017, no. 21, HIKAI Ltd, Variational Theory of Solitons for a Higher Order Generalized Camassa-Holm Equation Wilmer L. Molina Department of Mathematics University of Cauca, Kra 3 3N-100, Popayán-Colombia Alex M. Montes Department of Mathematics University of Cauca, Kra 3 3N-100, Popayán-Colombia Jaime Tobar Department of Mathematics University of Cauca, Kra 3 3N-100, Popayán-Colombia Copyright c 2017 Wilmer L. Molina, Alex M. Montes and Jaime Tobar. This is article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper we show the existence of travelling wave solutions for a higher order generalized Cammasa-Holm type equation. We follow a variational approach by characterizing travelling waves as critical points of a suitable functional. The existence result follow as a consequence of the Mountain Pass Theorem. Mathematics Subject Classification: 35Q51, 35C07, 49J35 Keywords: Camassa-Holm Equation, Solitons, Mountain Pass Theorem

2 1008 Wilmer L. Molina, Alex M. Montes and Jaime Tobar 1 Introduction The Camassa-Holm (CH) equation t u 2 x t u + 3u x u = 2 x u 2 xu + u 3 xu, x, t 0, was derived in [1] as a model for dispersive shallow water waves. Here u denotes the fluid velocity in the x direction or, equivalently, the height of the water s free surface above a flat bottom. The nonlinear dispersive equation (CH) has a hamiltonian structure (ver [1], [2]). More precisely, the Camassa-Holm equation can be written as where H is defined as t u = x ( I 2 x ) 1 [H (u)], H(u) = 1 2 ( u 3 + u x u ) dx. A. M. Montes in the work [5] established the existence of solitons (travelling wave solutions of finite energy) in the Sobolev space H 2 () for the following fourth-order p-generalized Camassa-Holm equation ( p + 2 [ t u α 1 x 2 t u+α 2 x 4 t u+ x p + 1 up+1 β u x ( x u) p + p ]) p + 1 ( p+1 xu) = 0, when α 1, α 2 > 0 and β. In the mentioned work the autor showed that u(x, t) = v(x ct) is a travelling wave solution for the previous equation if and only if u(x, y, t) = v(x ct) is a travelling wave solution for the two-dimensional hyperelastic wave equation x ( t u 2 x t u + α 4 x t u + 3u x u β ( 2 x u 2 xu + u 3 xu )) a 2 yu + b 2 x 2 yu = 0, when α 1 = 1, α 2 = α and p = 1. Studies for the hyperelastic equation can be see in [3] and [4]. In this paper we are interested in considering equations type generalized Camassa-Holm of higher order. We study the one-dimensional generalized nonlinear equation M 1 t u + M 2 x u + x [ f(u, x u, 2 xu) ] = 0, (1) where M i (i = 1, 2) is a differential operator of order 2m, with m 2, of the form m M i = ( 1) j α i,j x 2j,

3 Periodic travelling waves 1009 with a i,j constants and the nonlinear terms f is a homogeneous function of degree p + 1 in the variables u, x u, and 2 xu having the form f(q, r, s) = q F (q, r) r rq F (q, r) s rr F (q, r), (2) where F is a homogeneous functions of degree p + 2 of the form F (q, r) = k F j (q, r), j=1 such that for 1 j k, F j (λq, r) = λ p 1,j F j (q, r), F j (q, λr) = λ p 2,j F j (q, r), (3) with p + 2 = p 1,j + p 2,j (if F j depends only in either q or r, we assume that F j is homogeneous of degree p + 2). In particular, we have F is a homogeneous function of degree p + 2, since we have that F (λq, λr) = λ p+2 F (q, r). Notice that if in (1) m = 2, α 1,0 = 1, α 1,1 = α 1, α 1,2 = α 2 and F (q, r) = 1 ( q p+2 + βqr p+1), p + 1 then we obtain the fourth-order p-generalized Camassa-Holm equation. We will establish the existence of solitons for the equation (1); for this, we follow a variational approach by characterizing solitons as critical points of an energy functional (ver section 2) and the existence of critical points follow by using the Mountain Pass Theorem (ver section 3). Throughout this work H k = H k () denotes the usual Sobolev space of order k equipped with the norm [ k v 2 H = (v (j) ) ]dx, 2 k and C denotes a generic constant whose value may change from instance to instance. 2 Variational approach for travelling waves In this section we characterize travelling wave solutions of finite energy or solitons for the equation (1) as critical points of a suitable funtional. First, we establish some basic properties of the homogeneous functions F = m j=1 F j where F j satisfies (3).

4 1010 Wilmer L. Molina, Alex M. Montes and Jaime Tobar Lemma 2.1. Let F = m j=1 F j where F j is a homogeneous function satisfying properties (3). Then we have that q q F (q, r) + r r F (q, r) = (p + 2)F (q, r), (4) F (q, r) C( q p+2 + r p+2 ). (5) Proof. 1. We only perform the proof in the case m = 1, p 1 = p 1,1, p 2 = p 2,1 and F = F 1. From the homogeneity with respect to r, we have that F (q, r) = r p 2 F (q, 1), and so r F (q, r) = p 2 r p 2 1 F (q, 1), which implies that The same argument shows that r r F (q, r) = p 2 r p 2 F (q, 1) = p 2 F (q, r). q q F (q, r) = p 1 F (q, r). So, using that p + 2 = p 1 + p 2, we have that q q F (q, r) + r r F (q, r) = (p + 2)F (q, r). 2. First suppose that 0 < q r. Then using that p + 2 = p 1 + p 2, F (q, r) = r p 2 F (q, 1) = r p 2 q p 1 F (1, 1) The result for 0 < r < q is quite similar. ( q + r ) p+2 F (1, 1). Next, a simple calculation shows that the evolution equation (1) has a Hamiltonian structure: Proposition 2.1. The nonlinear evolution equation (1) can be expressed in Hamiltonian form, t u = x M1 1 [G (u)], where G is defined on H m () as G(u) = 1 [ m ] α 2,j ( j 2 xu) 2 + 2F (u, u x ) dx. Now, notice that if u is a solution of the equation (1) of the type u(x, t) = v(x ct), then we see that the soliton-profile v should satisfy the ordinary differential equation [ (cm1 M 2 )v f(v, v, v ) ] = 0,

5 Periodic travelling waves 1011 which, upon integration, yields (cm 1 M 2 )v f(v, v, v ) + A = 0, (6) where A is a constant of integration. Among all the soliton-solutions we shall focus on solutions which have the additional property that the waves are localized and that v and its derivatives decay at infinity, that is, v (n) (z) 0 as z, n Z +. Under this decay assumption the constant of integration in (6) vanishes and then the travelling wave equation takes the form (cm 1 M 2 )v f(v, v, v ) = 0. (7) From the Proposition 2.1 we note that the natural space (energy space) to look for travelling waves is the space H m (). Our strategy to prove the existence of a solution for (7) is consider the functional J c = I c (v) K(v), where the energy I c and the constraint K are functionals defined in H m () given by I c (v) = 1 2 K(v) = [ m (cα 1,j α 2,j )(v (j) ) ]dx, 2 F (v, v ) dx. We will see that solutions of the equation (7) corresponds to critical points of the functional J c. We start by showing some properties of I c and K, assuming that m 2, α 1,j > 0 and α 2,j for j = 1,..., m. Lemma 2.2. Let c > c 0 with c 0 = max{ α 2,j α 1,j : j = 1,..., m}. Then 1. The functionals I c and K are well defined in H m () and smooth. 2. The functional I c is nonnegative. Moreover, there are positive constants C 1 (α i,j, c) < C 2 (α i,j, c) such that C 1 I c (v) v 2 H m () C 2 I c (v). (8) Proof. 1. I c is clearly well defined for v H m (). Moreover, note that if v H m () H 1 () then, using inequality (5) and the fact that the embedding

6 1012 Wilmer L. Molina, Alex M. Montes and Jaime Tobar H 1 () L s () is continuous for s 2, we see that there is a constant C > 0 such that K(v) C ( ) ( ) v p+2 L + v p+2 p+2 L C v p+2 p+2 H + v p+2 1 H C v p+2 1 Hm. (9) So, K is well defined. 2. This property is straightforward. Theorem 2.1. If v 0 is a nontrivial critical point for the functional J c in in the space H m (), then v is a nontrivial solution of the equation (7). Proof. A direct calculation shows for all w H m () that [ m I c(v), w = (cα 1,j α 2,j )v (j) w ]dx (j) = Now, we note that [cm 1 M 2 ] (v) w dx. f(v, v, v ) = q F (v, v ) v rq F (v, v ) v rr F (v, v ) = q F (v, v ) ( r F (v, v )). Then K (v), w = = = [ q F (v, v )w + r F (v, v )w ] dx [ q F (v, v ) ( r F (v, v )) ] w dx f(v, v, v )w dx. Thus, we have for all w H m () that [ ] J c(v), w = (cm 1 M 2 ) v f(v, v, v ) w dx. As a consequence of this we conclude that J c(v) = (cm 1 M 2 ) v f(v, v, v ), meaning that critical points of the functional J c satisfy the travelling wave equation (7).

7 Periodic travelling waves Existence of critical points Our approach to show the existence of a nontrivial critical point for functional J c is to use the Mountain Pass Theorem. Theorem 3.1. (Mountain Pass Theorem) Let X be a Hilbert space, ϕ C 1 (X, ), e X and r > 0 such that e X > r and ϑ = inf ϕ(v) > ϕ(0) ϕ(e). v X =r Then, given n N, there is v n X such that where ϕ(v n ) δ, ϕ (v n ) 0 in X and δ ϑ, δ = inf max ϕ(π(t)), and Π = {π C([0, 1], X) : π(0) = 0, π(1) = e}. π Π t [0,1] Before we go further, we establish an important result for our analysis, which is related with the characterization of vanishing sequences in H m (), m 2. Hereafter, B r (ζ) denotes the ball in of center ζ and radius r > 0. Lemma 3.1. If (v n ) n is a bounded sequence in H m () with m 2 and there is a positive constant r > 0 such that lim sup (v n ) 2 dx = lim sup (v ζ n) 2 dx = 0, ζ then we have that B r(ζ) lim K(v n) = 0. B r(ζ) Proof. Let ζ and r > 0. Using the Hölder inequality and the fact that the embedding H 1 (B r (ζ)) L q (B r (ζ)) is continuous for q 2, we see for m 2 that v n p+2 dx v n L 2 (B r(ζ)) v n p+1 L 2(p+1) (B r(ζ)) B r(ζ) C v n L 2 (B r(ζ)) v n p+1 C v n p+1 H m (B r(ζ)) ( H 1 (B r(ζ)) sup ζ B r(ζ) v n 2 dx) 1/2. Now, covering by balls of radius r in such a way that each point of is contained in at most two balls, we find ( 1/2 v n p+2 dx 2C v n p+1 H m () v n dx) 2. sup ζ B r(ζ)

8 1014 Wilmer L. Molina, Alex M. Montes and Jaime Tobar Thus, under the assumptions of the lemma, v n p+2 dx = 0. lim In a similar fashion we obtain that lim v n p+2 dx = 0. So that lim K(v [ n) C lim v p+2 + v p+2] dx = 0. Now, we want to verify for J c the Mountain Pass Theorem hypotheses given in Theorem 3.1. Lemma 3.2. Let m 2, α 1,j > 0 and α 2,j for j = 1,..., m. If c > c 0 with c 0 = max{ α 2,j α 1,j : j = 1,..., m}, then 1. There exists ρ > 0 small enough such that ϑ(c) := inf J c(z) > 0. z H m () =ρ 2. There is e H m () such that e H m ρ and J c (e) If δ(c) is defined as δ(c) = inf π Π max t [0,1] J c(π(t)), Π = {π C ( [0, 1], H m) : π(0) = 0, π(1) = e}, then δ(c) ϑ(c) and there is a sequence (v n ) n in H m () such that J c (v n ) δ, J c(v n ) 0 in (H m ()). Proof. From inequalities (8)-(9), we have for any v H m () that Then for ρ > 0 small enough such that we conclude for v H m () = ρ that J c (v) C 1 v 2 H m C v p+2 H m (C 1 C v p H m) v 2 H m. C 1 ρ p C > 0, J c (v) (C 1 ρ p C) ρ 2 := ɛ > 0.

9 Periodic travelling waves 1015 In particular, we also have that For any t we see that ϑ(c) = inf J c (z) ɛ > 0. z H m=ρ J c (tv 0 ) = t 2 [ I c (v 0 ) t p ] K(v) dx. Using the hypothesis, it is not hard to prove that there exist v 0 C 0 () H m () such that K(v 0 ) > 0. So that, lim J c(tv 0 ) =, t because 0 I c (v 0 ) C 2 v 0 2 H m. Then, there is t 0 > 0 such that e = t 0 v 0 H m () satisfies that t 0 v 0 H m = e H m > ρ and that J c (e) J c (0) = The following theorem is our main result. The third part follows by applying Theorem Theorem 3.2. Let m 2, α 1,j > 0 and α 2,j for j = 1,..., m. If c > c 0 with c 0 = max{ α 2,j α 1,j : j = 1,..., m}, then the equation (7) admits nontrivial solutions in the space H m (). Proof. We will see that δ(c) is in fact a critical value of J c. Let (v n ) n be the sequence in H m () given by previous lemma. First note from (3) that δ(c) ϑ(c) ɛ > 0. (10) Using (4) we see that [ K (v), v = v q F (v, v ) + v r F (v, v ) ] dx = (p + 2)K(v) Thus, Then we obtain that J c(v), v = 2I c (v) (p + 2)K(v) (11) = 2J c (v) pk(v). I c (v n ) = p + 2 p J c(v n ) 1 p J c(v n ), v n.

10 1016 Wilmer L. Molina, Alex M. Montes and Jaime Tobar But from (8) we conclude for n large enough that C 1 v n 2 H m I c(v n ) p + 2 p (δ(c) + 1) + v n H m. Then we have shown that (v n ) n is a bounded sequence in H m (). We claim that ɛ 1 = lim sup (v n ) 2 dx > 0 or ɛ 2 = lim sup (v ζ B 1 (ζ) n) 2 dx > 0. ζ B 1 (ζ) Suppose that lim sup ζ B 1 (ζ) Hence, from Lemma 3.1 we conclude that Then, we have from (10)-(11) that (v n ) 2 dx = lim sup (v n) 2 dx = 0. ζ B 1 (ζ) lim K(v n) = 0. 0 < ɛ δ(c) = J c (v n ) 1 2 J c(v n ), v n + o(1) p 2 K(v n) + o(1) o(1), but this is a contradiction. Thus, there is a subsequence of (v n ) n, denoted the same, and a sequence ζ n such that (v n ) 2 dx ɛ 1 or (v 2 n) 2 dx ɛ 2 2. (12) B 1 (ζ n) B 1 (ζ n) Now, we define the sequence ṽ n (x) = v n (x + ζ n ). For this sequence we also have that ṽ n H m = v n H m, J c (ṽ n ) δ and J c(ṽ n ) 0 in (H m ()). Then (ṽ n ) n is a bounded sequence in H m (). Thus, for some subsequence of (ṽ n ) n, denoted the same, and for some v H m () we have that ṽ n v, as n (weakly in H m ()). Since the embedding H m (Ω) L 2 (Ω) is compact for all bounded open set Ω, we see that ṽ n v in L 2 loc().

11 Periodic travelling waves 1017 Note that v 0 because using (12) we have that v 2 dx = lim (ṽ n ) B 1 (0) B 2 dx = lim n ) 1 (0) B 1 (ζ n)(v 2 dx ɛ 2 or (v ) 2 dx ɛ 2 2. B 1 (0) Moreover, if w C0 (), then for K = supp w we have that [ m I c(v), w = (cα 1,j α 2,j )v (j) w ]dx (j) K [ m = lim (cα 1,j α 2,j )(ṽ n ) K (j) w ]dx (j) = lim I (ṽ n ), w. Now (taking a subsequence, if necessary) noting that q F (ṽ n, ṽ n) q F (v, v ), r F (ṽ n, ṽ n) q F (v, v ) in L 2 loc(), we have that and K K Then we conclude that q F (ṽ n, ṽ n)w dx r F (ṽ n, ṽ n)w dx K K q F (v, v )w dx q F (v, v )w dx. K (v), w = lim K (ṽ n )w, and also that J c(v), w = lim J c(ṽ n ), w = 0. If w H m (), by using density, there is w k C0 () such that w k w in H m (). Hence, J c(v), w J c(v), w w k + J c(v), w k Thus, we have already established that J c(v) (H m ) w w k H m + J c(v), w k 0. J c(v) = 0. In other words, v H m () is a nontrivial weak solution for equation (7). Acknowledgments. A. M. Montes was supported by University of Cauca under the projet I.D W. L. Molina and J. Tobar are supported by University of Cauca.

12 1018 Wilmer L. Molina, Alex M. Montes and Jaime Tobar eferences [1]. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. ev. Lett., 71 (1993), [2] A. Constantin, The Hamiltonian structure of the Camassa-Holm equation, Exposition. Math., 15 (1997), no. 1, [3]. M. Chen, Some nonlinear dispersive waves arising in compressible hyperelastic plates, International Journal of Engineering Science, 44 (2006), [4]. M. Chen, The Cauchy problem and the stability of solitary waves of a hyperelactic dispersive equation, Indiana University Mathematics Journal, 57 (2008), [5] A. M. Montes, 1D-Solitons for a generalized dispersive equation, Communications in Mathematical Analysis, 18 (2015), no. 1, [6] M. Willem, Minimax Methods, Progress in Nonlinear Differential Equations and Their Applications, Vol. 24, eceived: November 7, 2017; Published: November 23, 2017