Presenter: Noriyoshi Fukaya

Transcription

1 Y. Martel, F. Merle, and T.-P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), Presenter: Noriyoshi Fukaya 1 Introduction In this talk, we study the paper Y. Martel, F. Merle, and T.-P. Tsai, Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gkdv Equations, Comm. Math. Phys. 31 (00), In this talk, we consider the generalized Korteweg de Vries equations { u t + (u xx + u p ) x = 0, (t, x) R R, (gkdv) u(0, x) = u 0 (x), x R, for p =, 3, or 4 and u 0 := (R, R). It is known that (gkdv) is global wellposed in the energy space, that is, for u 0 (R), there exists a unique solution u C(R, ) of (gkdv). Moreover, the solution u satisfies the following two conservation laws: u(t) = u 0, E(u(t)) := 1 u x (t) 1 p + 1 u(t) p+1 = E(u 0 ). Eq. (gkdv) has explicit traveling wave solutions, called solitons, of the form u c (t, x) = Q c (x ct), 1

2 Presenter: Noriyoshi Fukaya where c > 0 is the speed of the soliton, and Q c (x) = c 1/(p 1) Q( ( ) 1/(p 1) p + 1 c x), Q(x) = cosh ( p 1. x) Note that Q c is the only positive solution in of Q xx + Q p = cq. It is known that for c > 0, the soliton Q c (x ct) is stable in the following sense: δ 0 > 0, α 0 > 0 s.t. u 0 Q c < α 0 x: [0, + ) R s.t. u(t) Q c ( x(t)) < δ 0 (see [1, 6]). Moreover, Martel and Merle proved the asymptotic stability of the family of solitons { Q c ( x 0 ct) c > 0, x 0 R } in the following sense: c > 0, α 0 > 0 s.t. u 0 Q c < α 0 c + > 0, x: [0, + ) R s.t. u(t, + x(t)) Q + in. In [5], Martel, Merle, and Tsai proved the stability and asymptotic stability of the sum of N solitons in the following sense. Theorem 1 ([5]). Let p =, 3, or 4. Let 0 < c 0 1 < < c 0 N. Then γ 0, A 0, L 0, α 0 > 0 s.t. the following is true: Let u 0, L > L 0, α < α 0, and x 0 1,..., x 0 N R satisfy u 0 Q c 0 ( x 0 ) < α, and =,..., N, x 0 > x L. Let u(t) be the solution of (gkdv). Then x 1,..., x N : [0, + ) R s.t. the following is true. (i) (Stability of the sum of N decoupled solitons). t 0, u(t) Q c 0 ( x (t)) < A 0 (α + e γ0l ). (ii) (Asymptotic stability of the sum of N solitons). c + 1,..., c + N > 0 with c+ c 0 < A 0 (α + e γ0l ) s.t. = 1,..., N, u(t, + x (t)) u(t) as t +. Q c + k=1 Q c + ( (x k (t) x (t))) 0 in, ẋ (t) c +, (1) k ( x (t)) 0 () L (x>c 0 1 t/10)

3 Stability of sum of N solitons 3 After this paper, Martel and Merle [4] improved Theorem 1 (ii), that is, they showed that u(t) Q c + ( x (t)) 0 (x>c 0 1 t/10) as t +. In this talk, we do not prove (), only treat Theorem 1 (i) and (ii) (1). Outline of Proof of Theorem 1 Let 0 < c 0 1 < < c 0 N, and let For α, L > 0, define U(α, L) := σ 0 := 1 min{c0 1, c 0 c 0 1, c 0 3 c 0,..., c 0 N c 0 N 1}. { v inf y >y 1 +L v Q c 0 ( y ) < α The following lemma is a very useful tool to examine the behavior of solutions close to the sum of N solitons. Lemma 1 (Decomposition of the solution). L 1, α 1, K 1 > 0 s.t. the following is true: If L > L 1, 0 < α < α 1, and t 0 > 0 satisfy }. t [0, t 0 ], u(t) U(α, L), then!c 1 -functions c 1,..., c N : [0, t 0 ] (0, + ), x 1,..., x N : [0, t 0 ] R s.t. ε(t, x) := u(t, x) R (t, x), where R (t, x) := Q c (t)(x x (t)), satisfies t [0, t 0 ], = 1,..., N, R (t)ε(t) = (R ) x (t)ε(t) = 0, ε(t) + c (t) c 0 K 1 α, x (t) > x 1 (t) + L/, ( ) 1/ ċ (t) + ẋ (t) c (t) K 1 e σ 0 x x (t) / ε(t) + K 1 e σ 0 (L+σ 0 t)/4. (3)

4 4 Presenter: Noriyoshi Fukaya.1 Proof of Stability in the Energy Space Let γ 0 := σ 0 /16. For A 0, L, α > 0, we define ( V A0 (L, α) := U A 0 (α + e γ0l/ ), L ) { = v } inf v Q c 0 ( y ) < A 0 (α + e γ0l/ ). y >y 1 +L/ Theorem 1 follows from the following proposition. Proposition 1 (A priori estimate). A 0, L 0, α 0 > 0 s.t. if u 0, L > L 0, α < α 0, and x 0 1,..., x 0 N satisfy u 0 Q c 0 ( x 0 ) < α, and =,..., N, x 0 > x L, (4) and if t > 0 satisfies then t [0, t ], u(t) V A0 (L, α), (5) t [0, t ], u(t) V A0 /(L, α). Assuming this proposition, we prove Theorem 1 (i). Proof of Theorem 1. Let A 0, L 0, α 0 > 0 be chosen as in Proposition 1, and let u 0, L, α, and x 0 1,..., x 0 N satisfy the assumptions of Theorem 1. Then by continuity of u(t) in H1, τ 0 > 0 s.t. t (0, τ 0 ), u(t) V A0 (L, α). Let t := sup{ t 0 t [0, t], u(t ) V A0 (L, α) }. Assume t < +. Then by Proposition 1, we have u(t) V A0 /(L, α). Therefore, by continuity of u(t) in, τ > 0 s.t. t [0, t +τ], u(t) V A0 /3(L, α), which contradicts the definition of t. Hence, the stability result follows. Next, we prove Proposition 1. Proof of Proposition 1. Let A 0 > 0 to be fixed later. First, note that α (A 0 ) > α 1, L (A 0 ) > L 1 s.t. α (0, α (A 0 )), L > L (A 0 ), A 0 (α + e γ 0L/ ) α 1, where α 1 and L 1 are defined in Lemma 1. Therefore, by (5) and Lemma 1, there exist c : [0, t ] (0, + ), x : [0, t ] R s.t. ε(t, x) = u(t, x) R (t, x), where R (t, x) = Q c (t)(x x (t)),

5 Stability of sum of N solitons 5 satisfies C(A 0 ) > 0 s.t., t [0, t ], R (t)ε(t) = (R ) x (t)ε(t) = 0, (6) c (t) c 0 + ċ (t) + ẋ (t) c 0 + ε(t) C(A 0 )K 1 (α 0 + e γ 0L 0 / ). (7) Note that by (4) and Lemma 1 at t = 0, ε(0) + c (0) c 0 K 1 α, x (0) x 1 (0) L. (8) From (7) and (8), α 0 (A 0 ) (0, α (A 0 )), L 3 (A 0 ) > L (A 0 ) s.t. α (0, α 0 ), L > L 3, t [0, t ], c 1 (t) σ 0, ẋ 1 (t) σ 0, c (t) c 1 (t) σ 0, ẋ (t) ẋ 1 (t) σ 0, (9) x (t) x 1 (t) L/ + σ 0 t, ε(t) 1 ( σ0 ) 1 p 1. (10) 8 Throughout this talk, we assume (9) and (10). Now, we give a uniform upper bound on c (t) c (t) and ε(t) on [0, t ] improving (7) for A 0 large enough. Lemma (Quadratic control of the variation of c (t)). K > 0 independent of A 0, L 4 > L 3 s.t. L > L 4, t [0, t ], c (t) c (0) K (ε(t) H + 1 ε(0) H + 1 e γ 0L ). Lemma 3 (Control of ε(t) ). K 3 > 0 independent of A 0, L 0 > L 4 s.t. L > L 0, t [0, t ], ε(t) K 3(ε(0) + e γ 0L ). We give proofs of Lemmas and 3 in Section 3.1. Now, we finish the proof of Proposition 1. By the regularity of c Q c, (8), and Lemmas and 3, we have u(t) Q c 0 (x x (t)) u(t) ε(t) + C R (t) + R (t) c (t) c 0 Q c 0 (x x (t))

6 6 Presenter: Noriyoshi Fukaya ε(t) + C c (t) c (0) + C c (0) c 0 ε(t) + CK (ε(0) + e γ 0L ) + CK 1 α K 4 (α + e γ 0L/ ), where K 4 > 0 is a constant independent of A 0. Choosing A 0 = K 4, we complete the proof of Proposition 1 and thus the proof of Theorem 1 (i).. Proof of Asymptotic Stability Result In this subsection, we prove the following asymptotic result on ε(t) as t +. Proposition (Convergence around solitons, p =, 3, or 4). Under the assumptions of Theorem 1, the following is true: (i) {1,..., N}, ε(t, + x (t)) 0 weakly in (R) as t +. (11) (ii) There exist 0 < c + 1 < < c + N s.t. c (t) c +, ẋ (t) c + as t +. The asymptotic stability of sum of N solitons follows from the Liouville property close to solitons. Theorem (Liouville property close to a soliton for p =, 3, or 4 []). Let p =, 3, or 4, and let c 0 > 0 and x 0 R. Then α 0 > 0 s.t. if u 0 Q c0 ( x 0 ) < α 0, and if there exists y : R R s.t. δ 0 > 0, A 0 > 0 s.t. t R, u(t, x + y(t)) δ 0, (L -compactness) x >A 0 then c > 0, x R s.t. (t, x) R, u(t, x) = Q c (x x c t). Proof of Proposition (i). We prove by contradiction. Assume that (11) does not hold for some. Since ε(t) and c (t) is bounded by the stability results, ε 0 \ {0}, c 0 > 0, (t n ) with t n + s.t. ε(t,. + x (t)) ε 0 weakly in (R), c (t n ) c 0 as t +. Moreover, by weak convergence and the stability result, ε 0 sup t 0 ε(t) A 0 (α 0 + e γ 0L 0 ), and therefore ε 0 is as small as we want by taking α 0 small and L 0 large.

7 Stability of sum of N solitons 7 Let ũ(t) be the global solution of (gkdv) with ũ(0) = Q c0 + ε 0. Let x(t) and c(t) be the geometrical parameters associated to the solution ũ(t) (apply the modulation theory for a solution close to a single soliton Q c0 ). Note that x(t) and c(t) is defined in R by the stability of the single soliton Q c0. We claim that the solution ũ(t) is L -compact. Lemma 4 (L -compactness of the asymptotic solution). δ 0 > 0, A 0 > 0 s.t. t R, ũ(t, x + x(t)) δ 0. x >A 0 Assuming this lemma, we finish the proof of Proposition (i). Indeed, by choosing α 0 small enough and L 0 large enough, we can apply the Liouville theorem to ũ(t). Therefore, c > 0, x R s.t. ũ(t) = Q c ( x c t). In particular, Q c0 + ε 0 = ũ(0) = Q c ( x ). Since by (6) and weak convergence, 0 = ε 0 (Q c0 ) x = Q c (x x )(Q c0 ) x, we have x = 0. Next, since 0 = ε 0 Q c0 = (Q c Q c0 )Q c0, we have c = c 0, and so ε 0 = 0. This is a contradiction. We prove Lemma 4 in Section 3.. We define ϕ(x) := cq( σ 0 x/), ψ(x) := x ϕ(y) dy, where c := ( σ0 Q) 1. Note that x R, ψ > 0, 0 < ψ(x) < 1, and lim x ψ(x) = 0, lim x + ψ(x) = 1. To prove Lemma 4 and Proposition (ii), we introduce for y 0 > 0, J L (t) := (1 ψ( (x (t) y 0 )))u(t), J R (t) := ψ( (x (t) + y 0 ))u(t). Lemma 5 (Monotonicity on the right and on the left of a soliton). C 1, y 1 > 0 s.t. the following is true: y 0 > y 1, T = T (y 0 ) > 0 s.t. if T < t < t, then J L (t) J L (t ) C 1 e γ 0y 0, J R (t) J R (t ) + C 1 e γ 0y 0. Proof of Proposition (ii). Let δ > 0 be arbitrary. Since R(t) (p 1) = c (t) Q and ε(t,. + x (t)) 0 in L loc as t +, T 1(δ) > 0, y 1 (δ) s.t. t > T 1 (δ), y 0 > y 1 (δ), (ψ(x (x (t) y 0 )) ψ(x (x (t) + y 0 )))u(t, x) c (t) 5 p (p 1) Q δ. By Lemma 5, y (δ) > y 1 (δ), T (δ) > T 1 (δ) s.t. if T (δ) < t < t, then J L (t) J L (t ) δ, J R (t) J R (t ) + δ. 5 p

8 8 Presenter: Noriyoshi Fukaya It follows that T 3 (δ) > T (δ), J + L 0, J + R 0 s.t. t T 3 (δ), J L (t) J + L δ, J R (t) J + R δ. Therefore, by conservation of L -norm, if T 3 < t < t, then c (t) 5 p (p 1) c (t ) 5 p (p 1) C δ. Since δ is arbitrary, it follows that c (t) 5 p (p 1) has a limit as t +, that is, c + > 0 s.t. c (t) c + as t +. The fact that ẋ (t) c + is a direct consequence of (3). 3 Proof of Lemmas 3.1 Proof of Lemmas and 3. First, we prove Lemma. Put R(t, x) := R (t, x) = Q c (t)(x x (t)). The following lemma follows from the decay properties of Q and the conservation of the energy E. Lemma 6 (Energy bounds). K 4 > 0 s.t. L > L, t [0, t ], [E(R (t)) E(R (0))] + 1 (ε x pr p 1 ε )(t) Recall that ϕ(x) = cq( σ 0 x/), ψ(x) = For, let I (t) := K 4 (ε(0) + ε(t)3 + e σ 0 L/4 ). x u(t, x) ψ(x m (t)), ϕ(y) dy, where c = ( σ0 m (t) := x 1(t) + x (t). Q) 1. The following lemma is essential for the proof of the stability of sum of N solitons. Lemma 7 (Almost monotonicity of the mass on the right of each soliton). K 5 > 0, L 4 > L 3 s.t., L > L 4, t [0, t ], I (t) I (0) K 5 e σ 0 L/16.

9 Stability of sum of N solitons 9 Proof of Lemma. Let β =. First, we show that C > 0 s.t. p 1 c (0)[c (t) β 1/ c (0) β 1/ ] C(ε(t) H + 1 ε(0) H + 1 e γ 0L ) + C [c (t) c (0)]. (1) Let us prove (1). By linearization, we have c β+1/ (t) c β+1/ (0) = β+1 (0)] + O([c (t) c (0)] ). Since E(Q c ) = κ cβ+1/ Q, where κ = 5 p c β 1/ [E(R (t)) E(R (0))] ( = κ = 1 ( where we used β+1 Let ) N Q [c (t) β+1/ c (0) β+1/ ] c β 1 (0)[c β 1/ (t), we have ) N ( N Q c (0)[c (t) β 1/ c (0) β 1/ ] + O [c (t) c (0)] ), (13) β 1 = 1 κ We claim, ( ) ( Q d (t) d (0). Therefore, by Lemma 6, we obtain (1). d (t) := c k (t) β 1/. k= ) [ Q (d (t) d (0)) + C p+3 ] ε(0) + e γ 0L. (14) Let us prove (14). Recall that by Lemma 7, we have, I (t) I (0) + K 5 e γ0l, where I (t) = ψ(x m (t))u(t, x). Since R(t) = c (t) β 1/ Q, R (t)ε(t) = 0, we have ( ) I (t) Q d (t) ψ(. m (t))ε(t) Ce γ0l. Therefore, ( ) Q (d (t) d (0)) ψ(. m (0))ε(0) ψ(. m (t))ε(t) + Ce γ 0L (15)

10 10 Presenter: Noriyoshi Fukaya ψ(. m (0))ε(0) + Ce γ 0L. (16) Note that by u(t) = u(0) and u(t) = R(t) + ε(t) + ( = R(t) + ε(t) = R(t)ε(t) ) Q d 1 (t) + ε(t) + O(e γ 0L ), we obtain ( ) Q (d 1 (t) d 1 (0)) ε(0) ε(t) + Ce γ 0L (17) ε(0) + Ce γ0l. (18) Therefore, (16) and (18) implies (14). By Abel transform, we have c (0)[c (t) β 1/ c (0) β 1/ ] = N 1 c (0)[d (t) d +1 (t) (d (0) d +1 (0))] + c N (0)[d N (t) d N (0)] = c 1 (0)[d 1 (t) d 1 (0)] + Therefore, by (1), c 1(0)[d 1 (t) d 1 (0)] + (c (0) c 1 (0))(d (t) d (0)). (19) = (c (0) c 1 (0))(d (t) d (0)) = C(ε(t) + ε(0) + e γ 0L ) + C Since c 1 (0) σ 0, c (0) c 1 (0) σ 0, by (14) and (0), we have σ 0 N d (t) d (0) c 1 (0) d 1 (t) d 1 (0) + [ [c (t) c (0)]. (0) (c (0) c 1 (0)) d (t) d (0) = c 1 (0)[d 1 (t) d 1 (0)] + ] (c (0) c 1 (0))(d (t) d (0)) =

11 Stability of sum of N solitons 11 + C ε(0) + Ce γ 0L C(ε(t) + ε(0) + e γ 0L ) + C [c (t) c (0)]. Since c (t) c (0) C c (t) β 1/ c (0) β 1/ C( d (t) d (0) + d +1 (t) d +1 (0) ), we obtain, c (t) c (0) C(ε(t) H + 1 ε(0) H + 1 e γ 0L ) + C [c (t) c (0)]. Choosing a smaller α 0 (A 0 ) and a larger L 0 (A 0 ), by (7), we assume C c (t) c (0) 1/, and thus, Lemma is proved. Next, we prove Lemma 3. It is well known that λ 1 > 0 s.t. if v (R) satisfies Qv = Qx v = 0, then (v x pq p 1 v + v ) λ 1 v (1) (See proof of Proposition.9 in Weinstein [7]). By using the local version of (1) (see [3]), we can obtain the following lemma. Lemma 8 (Positivity of the quadratic form). L 0 > L 4, λ 0 > 0 s.t. if, c (t) σ 0, x (t) x 1 (t) + L 0, then t [0, t 0 ], (ε x (t) pr(t) p 1 ε(t) + c(t, x)ε(t) ) λ 0 ε(t), where c(t, x) = c 1 (t) + N = (c (t) c 1 (t))ψ(x m (t)). Proof of Lemma 3. By Lemma 6, (13), (19), and Lemma, we have 1 (ε x (t) pr(t) p 1 ε(t) ) [E(R (t)) E(R (0))] + K 5 (ε(0) H + 1 ε(t)3 H + 1 e γ 0L ) 1 ( ) N Q c (0)[c (t) β 1/ c (0) β 1/ ] + C [c (t) c (0)]

12 1 Presenter: Noriyoshi Fukaya + K 5 (ε(0) H + 1 ε(t)3 H + 1 e γ 0L ) 1 ( ) Q [c 1 (0)(d 1 (t) d 1 (0)) + (c (0) c 1 (0))(d (t) d (0))] + C(ε(0) + ε(t)3 + e γ 0L ). = = Therefore, by (15),(17), and Lemma, we have (ε x (t) pr(t) p 1 ε(t) ) ( c 1 (0) ε(t) + (c (0) c 1 (0)) + C(ε(0) H + 1 ε(t)3 H + 1 e γ 0L ) ( c 1 (t) ε(t) + (c (t) c 1 (t)) ( + C = ψ(x m (t))ε(t) ) ψ(x m (t))ε(t) ) ) N ε(t) c (t) c (0) + C(ε(0) H + 1 ε(t)3 H + 1 e γ 0L ) c(t, x)ε(t) + C(ε(0) + ε(t)3 + e γ 0L ) () where c(t, x) = c 1 (t) + N = (c (t) c 1 (t))ψ(x m (t)). By Lemma 8 and (), we obtain ε(t) C(ε(0) + ε(t)3 + e γ 0L ). Choosing a smaller α 0 (A 0 ) and a larger L 0 (A 0 ), by (7), we assume Cε(t) 1/, and thus, Lemma 3 is proved. 3. Proof of Lemmas 4 and 5 We prove Lemmas 4 and 5. Proof of Lemma 5. Since ψ is monotonically increasing, we have u(t) J L (t) = ψ( (x (t) y 0 ))u(t) ψ( (x (t) y 0 σ 0 (t t )))u(t). (3) By using (gkdv), since ψ σ 0 4 ψ and ẋ (t) σ 0, we have d ψ( (x (t) y 0 σ 0 dt (t t )))u(t) = ( 3u x (t) (ẋ (t) σ 0 )u(t) p p + 1 u(t)p+1 ) ψ ( (x (t) y 0 σ 0 (t t )))

13 Stability of sum of N solitons 13 + u(t) ψ ( (x (t) y 0 σ 0 (t t ))) σ 0 u(t) ψ ( (x (t) y 0 σ 0 4 (t t ))) (4) + u(t) p+1 ψ ( (x (t) y 0 σ 0 (t t ))). Since ẋ (t) ẋ 1 (t) σ 0, T = T (y 0 ) > 0 s.t. if T < t < t, then I x 1 (t) + y 0 x (t) y 0 σ 0 (t t ) x (t) y 0. (5) Let I := [x 1 (t) + y 0 /, x (t) y 0 /]. Then x I, since k {1,..., N}, x x k (t) y 0 /, we have u(t, x) p 1 σ 0 4 for large y 0, and so u p+1 ψ ( (x (t) y 0 σ 0 (t t ))) σ 0 u ψ ( (x (t) y 0 σ 0 4 (t t ))). (6) Moreover, since by ẋ (t) ẋ 1 (t) σ 0 and (5), x (t) x 1 (t) y 0 + σ(t t ), we have ( sup ψ ( (x (t) y 0 σ 0 (t t ))) ψ y0 I c + σ ) 0 (t t ) Ce σ 4 (y 0+ σ 0 (t t )). (7) Therefore, by (4), (6), and (7), we have ψ( (x (t) y 0 σ 0 (t t )))u(t) = I ψ( (x (t ) y 0 ))u(t ) + C 1 e γ 0y 0 u(t ) J L (t ) + C 1 e γ 0y 0, and so by u(t) = u(t ) and (3), we have the estimate for J L. Similarly, we can obtain the estimate for J R. Proof of Lemma 4. Recall from [] that we have stability of (gkdv) by weak convergence in (R) in the following sense t R, u(t + t n, + x (t + t n )) ũ(t, + x(t)) in L loc (R) as n +. (8) We prove Lemma 4 by contradiction. Let m 0 := ũ(0) = ũ(t), m 1 := u(0) = u(t). Assume that Lemma 4 does not hold. Then δ 0 > 0 s.t. y 0 > 0, t 0 (y 0 ) R, s.t. x <y 0 ũ(t 0 (y 0 ), x + x(t 0 (y 0 ))) m 0 δ 0. (9)

14 14 Presenter: Noriyoshi Fukaya Fix y 0 > 0 large enough so that (ψ(x + y 0 ) ψ(x y 0 ))ũ(0, x) m δ 0, (30) (m 0 + m 1 ) sup x >y 0 {ψ(x + y 0 ) ψ(x y 0 )} 1 10 δ 0. (31) Assume that t 0 = t 0 (y 0 ) > 0 and, by possibly considering a subsequence of (t n ), that n, t n+1 t n + t 0. Since 0 < ψ < 1 and ψ > 0, by (31) and (9), we have (ψ(x ( x(t 0 ) y 0 )) ψ(x ( x(t 0 ) + y 0 )))ũ(t 0, x) ũ(t 0, x + x(t 0 )) + m 0 sup {ψ(x + y 0 ) ψ(x y 0 )} x <y 0 x >y 0 ũ(t 0, x + x(t 0 )) + 1 x <y 0 10 δ 0 m δ 0. (3) Then, by (30), (3) and (8), N 0 > 0 s.t. n N 0, (ψ(x (x (t n ) y 0 )) ψ(x (x (t n ) + y 0 )))u(t n, x) m δ 0, (33) (ψ(x (x (t n + t 0 ) y 0 )) ψ(x (x (t n + t 0 ) + y 0 )))u(t n + t 0, x) m δ 0. By Lemma 5 and the choice of y 0, we have J R (t n + t 0 ) J R (t n ) δ 0. Therefore, by conservation of the L -norm and (34), (33), we have J L (t n + t 0 ) J L (t n ) + 1 δ 0. Since J L (t n+1 ) J L (t n + t 0 ) 1 10 δ 0 by Lemma 5, we finally obtain n N 0, J L (t n+1 ) J L (t n ) + 5 δ 0. This contradicts the fact that t > 0, J L (t) m 1. This completes the proof. (34) References [1] T.B. Benamin, The stability of solitary waves, Proc. Roy. Soc. London A 38, (197)

15 Stability of sum of N solitons 15 [] Y. Martel and F. Merle, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Rational Mech. Anal. 157, (001), [3] Y. Martel and F. Merle, Stability of the blow up profile and lower bounds on the blow up rate for the critical generalized KdV equation, Ann. of Math. () 155 (00), [4] Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gkdv equations revisited, Nonlinearity 18 (005), [5] Y. Martel, F. Merle and T-P. Tsai, Stability and asymptotic stability in the energy space of the sum of N solitons for subcritical gkdv equations, Comm. Math. Phys. 31 (00), [6] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure. Appl. Math. 39, (1986) [7] M.I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16, (1985) Department of Mathematics, Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinuku-ku, Tokyo , Japan address: @ed.tus.ac.p