Existence and regularity of 1D-solitons for a hyperelastic dispersive model

Transcription

1 Boletín de Matemáticas Existence and regularity of 1D-solitons for a hyperelastic dispersive model Existencia y regularidad de 1D-solitones para un modelo hiperelástico dispersivo Alex M. Montes 1,a, icardo Córdoba 1,b Abstract. We show the existence, regularity and analyticity of one-dimensional solitons for a dispersive type equation that models the deformations of a hyperelastic compressible plate. We follow a variational approach by characterizing solitons as critical points of a suitable functional. Our method involves the Mountain Pass Theorem. Keywords: Hypereslactic Equation, Solitons, Variational Approach, Mountain Pass Theorem. esumen. En este trabajo mostramos la existencia, regularidad y analiticidad de solitones uno-dimensionales para una ecuación de tipo dispersivo que modela las deformaciones de una placa hiperelástica. Seguimos una aproximación variacional, en la cual caracterizamos solitones como puntos críticos de una funcional adecuada. Nuestro método involucra el Teorema de Paso de Montaña. Palabras claves: Ecuación Hiperelástica, Solitones, Aproximación Variacional, Teorema de Paso de Montaña. Mathematics Subject Classification: 35Q51, 35K40, 75B20. ecibido: abril de 2015 Aceptado: noviembre de Introduction In this work we consider the following generalized two-dimensional nonlinear dispersive elastic equation x I 2 x + µ 4 x t u + 2 x p + 2 [ p + 1 up+1 γ u x x u p + p p + 1 xu p+1] α 2 yu + β 2 x 2 yu = Departamento de Mátematicas, Universidad del Cauca, Popayán, Colombia a amontes@unicauca.edu.co b ricardocor-10@hotmail.com

2 118 Alex M. Montes & icardo Córdoba Equation 1 was derived by. M. Chen in [6] as a model for the deformations of a hyperelastic compressible plate relative to a uniformly pre-stressed state. In this model u represents vertical displacement of the plate relative to a uniformly pre-stressed state, while x and y are rescaled longitudinal and lateral coordinates in the horizontal plane. To reduce the full three-dimensional field equation to an approximate two-dimensional plate equation, an assumption has been made that the thickness of the plate is small in comparison to the other dimensions. It is also assumed that the small perturbations superimposed on the pre-stressed state only appear in the vertical direction the z-direction and in one horizontal direction the x-direction. Hence the variation of waves in the transverse direction the y-direction is small. Equation 1 is obtained under the additional assumption that the wavelength in the x-direction is short. On the other hand, if the wavelength is large, we obtain the Kadomtsev-Petviashvili KP equation. The parameters in equation 1 are all material constants. The scalar µ describes the stiffness of the plate which is nonnegative. The coefficients α and β are material constants that measure weak transverse effects. The material constant γ occurs as a consequence of the balance between the nonlinear and dispersive effects. Note that there is no dissipation in this model. Equation 1 generalizes several well-known equations including the BBM equation [2] when µ = α = β = γ = 0, the regularized long-wave Kadomtsev- Petviashvili KP equation [3] also referred as KP-BBM equation, see [8] when µ = β = γ = 0, and the Camassa-Holm CH equation [5] when δ = α = β = 0, γ = 1. In contrast to the derivation in [6] of nonlinear dispersive waves in a hyperelastic plate, these particular equations are usually derived as models of water waves. In equation 1, the two spatial dimensions make the analysis very different from the CH equation. The µ terms include a nonlinear term of fourth order, which makes equation 1 very different from the KP-BBM equation. For equations that model the evolution of nonlinear waves, it is very important to determine the existence and uniqueness of solution for the associated initial value problem, and the existence of special solutions as the travelling waves. For instance, travelling wave solutions are important in the study of dynamics of wave propagation in many applied models such as fluid dynamics, acoustic, oceanography, and weather forecasting. An important application is the use of solitons travelling wave of finite energy as an efficient means of long-distance communication. For γ, µ, α, β > 0 and p = 1,. M. Chen see [7] showed, in the nonstandard space Sobolev type W 2 equipped with the norm u 2 [ W = u 2 + u 2 x + u 2 xx + x 1 u y 2 + u 2 y] dxdy, the existence in the weak sense and stability of two-dimensional travelling wave solutions 2D-solitons which propagate with speed wave c > 0, i.d. solutions of the form ux, y, t = vx ct, y. For this,. M. Chen used Boletín de Matemáticas

3 1D-solitons for a hyperelastic model 119 the Concentration-Compactness Theorem. Formally x 1 u y is defined via the Fourier transform as x 1 u y = η ûξ, η. ξ In this paper, using the Mountain Pass Theorem, for p = p 1 /p 2 with p 1, p 2 = 1 p 2 odd, γ and µ, α, β > 0 we establish the existence, regularity and analyticity of one-dimensional travelling wave solutions 1D-solitons for the equation 1. This is, solutions of the form ux, y, t = vx + y ct, 2 which propagate with wavefront normal to z = 1, 1 2, velocity c z, and profile v. First we show the existence of 1D-solitons in the standard Sobolev space H 2 equipped with the norm v 2 H = [ v 2 + v 2 + v 2] dx. 2 We will use a variational approach for which travelling waves corresponding to critical points of a suitable energy functional. Next, we prove that this solutions are regular functions. Moreover, we prove that this solutions admit a Taylor expansion analytic solutions. These characteristics make the 1D-solitons very interesting from the physical and numerical view points Variational approach By a 1D-soliton for the equation 1 we shall mean a solution of the form 2, where c denote the speed of the wave. Then one see that the travelling wave profile v should satisfy the ordinary differential equation [ c + αv c + βv + cµv p + 2 p + 1 vp+1 + γ v [ v p] p + p + 1 v p+1] = 0. 3 Among all the travelling wave solutions of 1 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 k y 0 as y, 0 k 6. We denote by C k 0 the space of real functions with k continuous derivatives vanishing at infinity, with the obvious sense when k =. Under this decay assumption the travelling wave equation takes the form c+αv c+βv +cµv p + 2 p + 1 vp+1 +γ v [v p ] + p p + 1 v p+1 = 0. 4 Boletín de Matemáticas

4 120 Alex M. Montes & icardo Córdoba A classical solution of 4 is a C 4 0 function satisfying 4 in the usual sense. Multiply the travelling wave equation 4 with a test function w C 0, after integration by parts, we obtain [ c + αvw + c + βv w + cµv w ] dx [ p + 2 p + 1 vp+1 w + γ vv p w + 1 ] p + 1 v p+1 w dx = 0. 5 Note that 5 makes sense as soon as v C 2 0, whereas 4 requires four derivatives on v. Let us say provisionally that a C 2 0 function v that satisfies 5 is a weak solution of 4. The following program outlines the main steps of the variational approach in the theory of partial differential equations see Section 8.1 in [4]: Step A. The notion of weak solution is made precise. This involves Sobolev spaces. Step B. Existence of a weak solution is established by a variational method, via the Mountain Pass Theorem in our case. Step C. Weak solution is proved to be of class C 4 0 for example: this is a regularity result. Step D. A classical solution is recovered by showing that any weak solution that is C 4 0 is a classical solution. To carry out Step D is very simple. In fact, suppose that v C 4 0 satisfies 5. Then integrating by parts 5 we obtain for all w C 0 that [ c + αv c + βv + cµv p + 2 p + 1 vp+1] w dx [ + γ v v p p + p + 1 v p+1] w dx = 0. Hence, using that the space C0 is dense in L 2, the equation 4 holds a.e. on and thus everywhere on, since v C0. 4 Finally, we notice that if v C0 6 is a solution for the equation 4 then v is a classical solution for the equation 3. Throughout this work X denotes norm in the Hilbert space X, <, > X is its inner product and X represents the dual space. C denotes a generic constant whose value may change from instance to instance. 2. Existence of weak solutions In this section we will establish the existence of a solution of 4 in the weak sense by using a variational approach in which weak solutions correspond to critical points of a suitable functional. We begin by defining the appropriate functional spaces. The usual Sobolev space H k, k Z + {0}, is the Hilbert Boletín de Matemáticas

5 1D-solitons for a hyperelastic model 121 space defined as the closure of C 0 with respect to the inner product and the norm v, w H k = k n=0 n=0 v n w n dx, k v 2 H = [ v n ] 2 dx. k Now, if we assume that v, w H 2, from the Young inequality, we see that [ c + αvw + c + βv w + cµv w ] dx Cc, α, β, µ v 2 H + 2 w 2 H. 2 So that, the first integral in 5 is well defined. In a similar way, using the Hölder inequality and the fact that for q 2 the embedding H 1 L q is continuous, we see that v p+1 w dx, v p+1 w dx v p+1 + v p+1 w L 2p+1 L 2p+1 L 2 C v p+1 H + v p+1 1 H w 1 L 2 In addition, 2C v p+1 H 2 w H 2. vv p w dx v L 4 v p L 4p w L 2 C v L 4 v p H 1 w L 2 C v p+1 H 2 w H 2. Therefore, the second integral in 5 is well defined. Then we have the following definition. Definition 2.1. We say that v H 2 is a weak solution of 4 if for all w H 2 the integral equation 5 holds. Next, we will see that weak solutions of the equation 4 corresponds to critical points of the functional J c defined as where I c v = 1 2 Gv = 1 p + 1 J c v = I c v + Gv, [ c + αv 2 + c + βv 2 + cµv 2] dx, [ v p+2 + γvv p+1] dx. First we will show in the following lemma some properties for I c and G, assuming that p = p 1 /p 2 with p 1, p 2 = 1 p 2 odd, γ and µ, α, β > 0. Boletín de Matemáticas

6 122 Alex M. Montes & icardo Córdoba Lemma 2.2. Let c > 0. Then 1. The functionals I c and G are well defined in H The functional I c is nonnegative. Moreover, there are C 1 α, β, µ, c < C 2 α, β, µ, c such that C 1 v 2 H I cv C 2 2 v 2 H2. 6 Proof. 1. I c is clearly well defined for v H 2. In addition, note that if v H 2 then, using the fact that the embedding H 1 L q is continuous for q 2, we see that there is a constant C = Cp, γ > 0 such that Gv C v p+2 L + v p+2 L 2 v p+1 C v p+2 L 2p+1 H. 7 2 So, G is well defined. 2. This property is straightforward. We define C 1, C 2 by C 1 = min{c + α, c + β, cµ}, C 2 = max{c + α, c + β, cµ}. Proposition 2.3. If v is a nontrivial critical point for the functional J c in the space H 2, then v is a nontrivial weak solution for the equation 4. Proof. If v, w H 2, a direct calculation shows that [ I cv, w = c + αvw + c + βv w + cµv w ] dx, [ p + 2 G v, w = p + 1 vp+1 w + γ vv p w + 1 ] p + 1 v p+1 w dx. In particular, if v H 2 is a critical point for the functional J c we see that for all w H 2, I cv, w G v, w = J cv, w = 0. 8 Therefore, v is a solution of the integral equation 5. Our approach to show the existence of a nontrivial critical point for the functional J c is to use the Mountain Pass Theorem without the Palais-Smale condition see M. Willem [10], A. Ambrosetti et al. [1] to build a Palais-Smale sequence for J c for a minimax value and use a local embedding result to obtain a critical point for J c as a weak limit of such Palais-Smale sequence. Theorem 2.4. 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 ϕv > ϕ0 ϕe. v X =r Boletín de Matemáticas

7 1D-solitons for a hyperelastic model 123 Then, given n N, there is v n X such that where δ = inf π Π ϕv n δ, ϕ v n 0 in X and δ ϑ, PS max ϕπt, and Π = {π C[0, 1], X : π0 = 0, π1 = e}. t [0,1] Before we go further, we establish an important result for our analysis. B r ζ denotes the ball in of center ζ and radius r > 0. Lemma 2.5. If v n n is a bounded sequence in H 2 and there is a positive constant r > 0 such that lim sup v n 2 dx = 0. n ζ B rζ Then we have that lim Gv n = 0. 9 n 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 that v n v n p+1 dx v n L 2 B rζ v n p+1 L 2p+1 B rζ B rζ C v n L2 B rζ v n p+1 C v n p+1 H 2 B rζ sup ζ H 1 B rζ B rζ 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 v n v n p+1 dx 2C v n p+1 H 2 sup ζ B rζ Thus, under the assumptions of the lemma, v n v n p+1 dx = 0. lim n v n 2 dx 1/2. In a similar fashion we obtain that lim n vp+2 n dx = 0. So that lim Gv n = 1 n p + 1 lim n [ v p+2 n + γv n v n p+1] dx = 0. Boletín de Matemáticas

8 124 Alex M. Montes & icardo Córdoba Now, we want to verify the Mountain Pass Theorem hypotheses given in Theorem 2.4 and to build a Palais-Smale sequence for J c. Lemma 2.6. Let c > 0. Then 1. There exists ρ > 0 small enough such that ϑc := inf J cz > 0. z H 2 =ρ 2. There is e H 2 such that e H 2 ρ and J c e If δc is defined as δc = inf π Π max J cπt, Π = {π C [0, 1], H 2 : π0 = 0, π1 = e}, t [0,1] then δc ϑc and there is a sequence v n n in H 2 such that J c v n δ, J cv n 0 in H Proof. From inequalities 6-7, we have for any v H 2 that Then for ρ > 0 small enough such that we conclude for v H 2 = ρ that In particular, we also have that J c v C 1 v 2 H2 Cλ, p v p+2 H 2 C 1 C v p H v 2 2 H 2. C 1 ρ p C > 0, 11 J c v C 1 ρ p C ρ 2 := ɛ > 0. ϑc = inf J cz ɛ > 0. z H 2 =ρ For any t we see that [ J c tv 0 = t 2 I c v 0 tp v p γv 0 v p p+1 ] dx. Using the hypothesis, it is not hard to prove that there exist v 0 C 0 H 2 such that Gv 0 > 0. So that, lim J ctv 0 =, t because 0 I c v 0 C 2 c v 0 2 H 2. Then, there is t 0 > 0 such that e = t 0 v 0 H 2 satisfies that t 0 v 0 H 2 = e H 2 > ρ and that J c e J c 0 = 0. The third part follows by direct applying Theorem 2.4. Boletín de Matemáticas

9 1D-solitons for a hyperelastic model 125 The following theorem is our main result in this section. Theorem 2.7. Let c > 0. The equation 4 admits nontrivial weak solutions in the space H 2. Proof. We will see that δc is in fact a critical value of J c. Let v n n be the sequence in H 2 given by previous lemma. First note from 12 that Using the definition of J c we have that Then we obtain that δc ϑc ɛ > J cv, v = 2I c v p + 2Gv 13 = 2J c v pgv. I c v n = p + 2 p J cv n 1 p J cv n, v n. But from 6 we conclude for n large enough that C 1 v n 2 H 2 I cv n p + 2 p δc v n H 2. Then we have shown that v n n is a bounded sequence in H 2. We claim that ɛ = lim sup v n 2 dx > 0. n ζ Suppose that B 1ζ lim sup v n 2 dx = 0. n ζ B 1ζ Hence, from Lemma 2.5 we conclude that Then, we have from that lim Gv n = 0. n 0 < ɛ δc = J c v n 1 2 J cv n, v n + o1 p 2 Gv n + o1 o1, 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 B 1ζ n Boletín de Matemáticas

10 126 Alex M. Montes & icardo Córdoba Now, we define the sequence ṽ n x = v n x + ζ n. For this sequence we also have that ṽ n H 2 = v n H 2, J c ṽ n δ and J cṽ n 0 in H 2. Then ṽ n n is a bounded sequence in H 2. Thus, for some subsequence of ṽ n n, denoted the same, and for some v H 2 we have that ṽ n v, as n weakly in H 2. Since the embedding H 2 Ω L 2 Ω is compact for all bounded open set Ω, we see that ṽ n v in L 2 loc. Note that v 0 because using 14 we have that v 2 dx = lim ṽ n 2 dx = lim n n B 10 B 10 B 1ζ n v n 2 dx ɛ 2. Moreover, if w C0, then for K = supp w we have that I cv, w = [c + αvw + c + βv w + cµv w ] dx K = lim [c + αṽ n w + c + βṽ nw + cµṽ nw ] dx n K = lim n I cṽ n, w. Now taking a subsequence, if necessary noting that ṽ n p+1 v p+1, ṽ n p+1 v p+1, ṽ n ṽ n p vv p in L 2 loc, we have that ṽ n p+1 w dx and K Then we conclude that and also that K K v p+1 w dx, K ṽ n ṽ n p w dx ṽ n p+1 w dx K vv p w dx. G v, w = lim n G ṽ n, w, J cv, w = lim n J cṽ n, w = 0. K v p+1 wdx If w H 2, by using density, there is w k C 0 such that w k w in H 2. Hence, J cv, w J cv, w w k + J cv, w k J cv H 2 w w k H 2 + J cv, w k 0. Boletín de Matemáticas

11 Thus, we have already established that 1D-solitons for a hyperelastic model 127 J cv = 0. In other words, v H 2 is a nontrivial weak solution for the equation egularity and analyticity of solutions In this section, we will establish that any weak solution of the equation 4 is a regular and analytic function. For this, we will use that for s 0 there are K 1, K 2 > 0 such that for all v H s, K 1 v 2 H 1 + ξ 2 s v 2 dξ K s 2 v 2 H s, where the Fourier transform of a function v defined on is given by vξ = 1 e ixξ vxdx. 2π 1 2 In addition, we will use the following result see Proposition 16 c in F. H. Soriano [9]. Proposition 3.1. For s > 1 there exists C 3 > 0 such that for all p and k Z +, k 1+ +k p=k 1 k s k p + 1 s Cp 1 3 k + 1 s. Theorem 3.2. Let c > 0. If v H 2 is a weak solution of the equation 4 then v is a classic solution. Moreover, v is a analytic function, this is, for each ζ 0 there is > 0 such that k D k v ζ 0 x ζ 0 k k! converges absolutely in to vx for all x B ζ 0. Proof. First we will establish that v H l for any l 0, if v H 2 is a weak solution of 4. Thus, using that if f H l, l 1 + k, then f C k 0, we conclude that v is a classical solution. From the facts H 1 L and H 1 L 2p+1 we have that the functions f = p + 2 p + 1 vp+1, g = γv [v p ], h = γp p + 1 v p+1 15 belong to L 2. Since [ vv p 1 v ] 2 dx v 2 L v 2p 1 L v 2 L 2 C v 2p+1 H 2 Boletín de Matemáticas

12 128 Alex M. Montes & icardo Córdoba and v 2p+1 dx, v 2p+1 dx C v 2p+1 H. 2 Taking Fourier transform on the equation 4 we obtain that v satisfies vξ = fξ + ĝξ + ĥξ c + α + c + βξ 2 + cµξ 4. Hence, there exists M = Mα, β, µ, c such that 1 + ξ 2 4 v 2 dξ M f 2 + ĝ 2 + ĥ 2 dξ = M f 2 L + 2 g 2 L + 2 h 2 L. 2 This implies that v H 4. Now, since H l is a algebra for l 1, by using 15 we see that f, g, h H 2. Then, repeating the previous argument we have that v H 8. A simple bootstrapping argument then yields that v H l for all l 0. Next, we will prove the analyticity of v. First we establish the result under the assumption of the existence of > 0 such that for all k N 0, D k v H 2 k! C k k, 16 where D k v = v k. If ζ 0, we will show that there exists r > 0 such that we have the following Taylor expansion for v in B r ζ 0, vx = k D k vζ 0 x ζ 0 k. k! If we set ζ = x ζ 0, then by the Taylor Theorem with remainder we have that vx = N 1 k=0 D k vζ 0 ζ k + E N x, k! E N x = DN vζ 0 + tζ ζ N. N! On the other hand, using 16 and the embedding H l L for l 1, if k N 0 we have that D k vx D k v L D k v H 2 C k k! k If we take r > 0 in such a way that 2r < 1, we conclude for ζ < r that E N x C rn N CrN C2 N. In other words, the Taylor series for v converges in B ζ 0. Boletín de Matemáticas

13 1D-solitons for a hyperelastic model 129 To complete the proof, we only need to prove that there exists > 0 such that 16 holds for all k 0. We will argue by induction on k. Since v H l, l 0, we have the result for k = 0, 1. Now, suppose that 16 holds for fixed k Z + and which will be chosen later. If we apply the operator D k to equation 4, then multiply with D k v, after integration by parts, we obtain that 2I c D k v = p + 2 p + 1 D k v p+1, D k v L 2 17 D + γ k [vv p ], D k v + 1 L D k [ v p+1], D k v. 2 p + 1 L 2 But by using Hölder inequality, D k [vv p ], D k v L 2 D k [vv p ] L 2 D k v L 2 C D k [vv p ] L 2 D k v H 2. In addition, we see that D k v p+1, D k v L 2 D k v p+1 L 2 D k v L 2. Also we have that D k[ v p+1], D k v L 2 D k [ v p+1] L 2 D k v L 2. Then applying in 17 the previous estimates and inequality 6 we obtain that D k v H 2 C 1 D k v p+1 L 2 + D k [vv p ] L 2 + D k[ v p+1] L 2. We want to estimate the terms of right hand side. For simplicity first we consider p = 1. Thus, note that if u, w H l for any l 1, we have for k Z + that D k uw = D k u k 1 k w + D m k m ud m w + ud k w. Then we see, for example, that m=1 D k [ v 2] = 2D k v v + k 1 m=1 Using the induction hypothesis we have that D k v v L 2 D k v L 2 v L C 2 D k+1 v L 2 v H 1 C 2 D k 1 v H 2 v H 2 C 2 C 2 k 1 k 1! k 2 k + 1! C k k+1 k m D k m v D m v k + 22 C 2 C k 2 k + 1. Boletín de Matemáticas

14 130 Alex M. Montes & icardo Córdoba Also we obtain that D k m v D m v L 2 D k m+1 v L 2 D m v L C 2 D k m 1 v H 2 D m v H 1 C 2 D k m 1 v H 2 D m v H 2. Hence, using induction hypothesis on the right hand side and Proposition 3.1, we obtain that k 1 m=1 k 1 k D k m v D m v m L 2 C 2 m=1 k! k m!m! Dk m 1 v H 2 D m v H 2 k 1 C 2 C 2 k! k 1 C 2 C 2 k! k 1 m=1 k + 1! C k k+1 1 k m 3 m k k k + 22 C 2 C 3 C k 2 k + 1 k 1+k 2=k 1 Note that there exists M > 0 such that k+22 k 2 k+1 < M. So, taking > 0 large enough such that C 2 C 3 CM 2 < 1,. we conclude that Next, we see that D k [v 2 k + 1! ] L 2 C k k+1. D k v v L 2 D k v L 2 v L C 2 D k 1 v H 1 v H 1 C 2 D k 1 v H 2 v H 2, and also that D k v v L 2 D k+1 v L 2 v L C 2 D k 1 v H 2 v H 2. Moreover, D k m vd m v L 2 D k m 1 v H 1 D m v H 1 C 2 D k m 1 v H 2 D m v H 2. Then we obtain that D k vv k + 1! L 2 C k k+1. Boletín de Matemáticas

15 In a similar fashion we obtain that 1D-solitons for a hyperelastic model 131 D k v 2 k + 1! L 2 C k k+1. In other words, we have shown for large enough that D k v 2 L 2 + D k vv L 2 + D k[ v 2] k + 1! L 2 C k k+1. Now we consider the case p 1. To illustrate the type of computation we only consider the typical term D k [v p+1 ], for this we will use the general rule D k [v p+1 ] = k 0+k 1+ +k p=k k D k0 v D k1 v D kp v, k 0, k 1..., k p where the sum extends over all p + 1-tuples k 0, k 1,..., k p of non-negative integers with p k i = k and i=0 k k! = k 0, k 1,..., k p k 0!k 1! k p!. Define the set Ak, p as Ak, p = Then we see that { k 0, k 1,..., k p : 0 k i k 1 and D k [v p+1 ] = p+1d k v v p + Ak,p p i=0 } k i = k. k D k0 v D k1 v D kp v. k 0, k 1..., k p But we have that D k v v p L 2 D k v L 2 v p L C 2 p D k 1 v H 2 v p H 2 k + 1! C k k+1 C 2 C p 2 k + 22 k 2 k + 1, Boletín de Matemáticas

16 132 Alex M. Montes & icardo Córdoba and also that k D k 0 v k 0,..., k p D kp v L 2 Ak,p C 2 p Ak,p C 2 p C p+1 k! k 1 C 2 p C p+1 k! k 1 k + 1! C k k+1 k! k 0! k p! Dk0 1 v H 2 D k1 v H 2 D kp v H 2 Ak,p 1 k 3 0 k k p k 0+k 1+ +k p=k 1 C 3 C 2 C p 2 k + 22 k 2 k + 1 Then we can see for large enough that 1 k k k p D k v p+1 L 2 + D k [v v p ] L 2 + D k[ v p+1] k + 1! L 2 C k k By using 19 we will establish that D k+1 v k + 1! H 2 C k + 2 k+1. To do this, we apply operator D k+1 to equation 4 and compute the L 2 - inner product with D k+1 v. Thus, we have that 2I c D k+1 v = p + 2 D k+1 v p+1, D k+1 v 20 p + 1 L 2 D + γ k+1 [vv p ], D k+1 v + 1 D k+1 [ v p+1], D k+1 v. L2 p + 1 L 2 Next, we see that D k+1[ v v p], D k+1 v L 2 = D k [ v v p], D k+3 v L 2 D k[ v v p] L 2 D k+3 v L 2 D k[ v v p] L 2 D k+1 v H 2. In a similar way we have that D k+1 v p+1, D k+1 v L 2 D k v p+1 L 2 D k+1 v H 2, and D k+1[ v p+1], D k+1 v L 2 D k [ v p+1] L 2 D k+1 v H 2. Boletín de Matemáticas

17 1D-solitons for a hyperelastic model 133 Therefore, I c D k+1 v C 1 D k v p+1 L 2 + D k[ v v p] L 2 + D k[ v p+1] L 2 Then, from 6 and 19 we conclude that D k+1 v H 2. D k+1 v H 2 C 1 D k v p+1 L 2 + D k[ v v p] L 2 + D k[ v p+1] L 2 as desired. k + 1! C k + 2 k+1, Acknowledgement A. Montes was supported by Universidad del Cauca under the project Id Córdoba is a student at the Maestría en Ciencias Matemáticas Program at Universidad del Cauca. AM and C were suported by Colciencias under the research project No eferences [1] A. Ambrosetti and P. abinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal , [2] T. Benjamin, J. Bona, and J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. oyal Soc. London, A , [3] J. Bona, Y. Liu, and M. Tom, The Cauchy problem and stability of solitary.wave solutions for the LW-KP equations, J. Diff. Eq , [4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitex, [5]. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. ev. Lett , [6]. Chen, Some nonlinear dispersive waves arising in compressible hyperelastic plates, International Journal of Engineering Science , [7], The cauchy problem and the stability of solitary waves of a hyperelastic dispersive equation, Indiana U. J. Math , Boletín de Matemáticas

18 134 Alex M. Montes & icardo Córdoba [8] J. Saut and N. Tzvetkov, Global well-posedness for the KP-BBM equations, AMX , [9] F. Soriano, On the Cauchy Problem for a K.P.-Boussinesq-Type System, Journal of Differential Equations , [10] M. Willem, Minimax Methods, Progress in Nonlinear Differential Equations and Their Applications Boletín de Matemáticas