Presenter: Noriyoshi Fukaya

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), 347-373. 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), 347-373. 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

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 0 1 + 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)

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 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 0 1 + 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)),

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 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.

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 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.

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 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)) =

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)]

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 )))

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 Presenter: Noriyoshi Fukaya Fix y 0 > 0 large enough so that (ψ(x + y 0 ) ψ(x y 0 ))ũ(0, x) m 0 1 10 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 0 9 10 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 0 1 5 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 0 4 5 δ 0. By Lemma 5 and the choice of y 0, we have J R (t n + t 0 ) J R (t n ) + 1 10 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) 153 183.

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), 19 54. [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), 35 80. [4] Y. Martel and F. Merle, Asymptotic stability of solitons of the subcritical gkdv equations revisited, Nonlinearity 18 (005), 55 80. [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), 347 373. [6] M.I. Weinstein, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure. Appl. Math. 39, (1986) 51 68. [7] M.I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16, (1985) 47 491. Department of Mathematics, Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinuku-ku, Tokyo 16-8601, Japan E-mail address: 111670@ed.tus.ac.p