I n t ur n, s p e ctr al st a bilit y h as b e e n c h ar a ct eri z e d a n al yti c all y i n t h e w e a kl y u nst a bl e li mit F → 2 a n d n u m eri c all y f or i nt er m e di at e t o l ar g e F i n t er ms of t w o si m pl e p o w er-l a w d es cri pti o ns, i n t h e s m all-a n d l ar g e-F r e gi m es, r es p e cti v el y, of t h e b a n d of p eri o ds X f or w hi c h r oll w a v es ar e s p e ctr all y st a bl e, as f u n cti o ns of F a n d dis c h ar g e r at e q ( a n i n v ari a nt of t h e fl o w d es cri bi n g t h e fl u x of fl ui d t hr o u g h a gi v e n r ef er e n c e p oi nt) [ B J N + 1 5].

a n d t h e tr a v eli n g-w a v e s ol uti o n t o a st ati o n ar y s ol uti o n U (x, t ) = (τ (x, t ), u(x, t )) = ( τ (x ), ū (x )) c o n v e ni e nt f or st a bilit y a n al yis.

We n ot e f or l at er t h at t h e tr a v eli n g-w a v e O D E b e c o m es

yi el di n g t h e k e y f a ct t h at We s h all s h o w i n t h e r est of t h e p a p er t h at t his a v er a g e d c o n diti o n is i n f a ct s u ffi ci e nt f or t h e n o nli n e ar a n al ysis of [ J Z N 1 1, J N R Z 1 4].

T h e ori gi n al g a u g el ess str at e g y t h at w or ks w h e n α is p ositi v e m a y b e a c hi e v e d b y c h o osi n g φ 1 ≡ 1, φ 2 = φ 1 / α , a n d 0 < φ 3 ≡ c o nst 1. T h e p ossi bilit y of c h o osi n g φ 2 = φ 1 / α w hil e k e e pi n g b ot h φ 1 a n d φ 2 p ositi v e is a dir e ct m a nif est ati o n of t h e f a ct t h at i n t his c as e t h e first-or d er p art of s yst e m ( 2. 4) is s y m m etri z a bl e. F or t h e g e n er al c as e, of i nt er est h er e, w e i nst e a d t a k e

t o o bt ai n t h e f oll o wi n g l e m m a. L e m m a 3. 1. T h e r e e xi st p o siti v e θ a n d C s u c h t h at a n y U s ol vi n g ( 2. 4) s ati s fi e s f o r a n y t ≥ 0 T h e o p er at or ∂ x its elf h as Bl o c h s y m b ols ∂ x + i ξ. As a r es ult, w h e n d e ali n g wit h L ξ , t h e ( e q ui v al e nt) n or m of i nt er est o n H s p e r ( 0 2) wit h t h e e as y esti m at e

.

s o l o n g as V H 1 a n d (ψ x , ψ t ) H k r e m ai n s u ffi ci e ntl y s m all. A p pl yi n g Gr o n w all's i n e q u alit y a n d r e c alli n g t h at E ψ (∂ k (V, ψ t , ψ x )(t) H s ≤ ε, t h e n, f o r all 0 ≤ t ≤ T , ( 4. 7) Tr a v eli n g w a v es ( τ, u )(x, t ) = ( τ, ū )(x ) s atisf y t h e pr o fil e O D E -c 2 τ -p ( τ ) + q = c ν τ / τ, q = c o nst .

We n ot e as i n t h e p eri o di c c as e t h at c = 0, els e ¯u, p ( τ ) ≡ c o nst, yi el di n g τ ≡ c o nst, a tri vi al s ol uti o n. Ass u m e t h at t h e s h o c k is n o n c h ar a ct eristi c, i. e., -p (τ ± ) = c 2 ; h e n c e τ ± ar e n o n d e g e n er at e e q uili bri a a n d t h e s h o c k pr o fil e d e c a ys e x p o n e nti all y t o its e n dst at es as x → ± ∞ . I n t h e pr es e nt c as e cl assi c al g a u g el ess esti m at es w o ul d r e q uir e s o m e q u a ntit yi d e nti fi e d as α τ / ν b el o w -t o b e p ositi v e e v e r y w h e r e . W h er e as i n t h e p eri o di c c o nt e xt t h e a n al o g o us q u a ntit y w as k n o w n t o b e p ositi v e o n a v er a g e, h er e w e k n o w r at h er t h at t h e li mits at pl us a n d mi n us i n fi nit y of t his k e y q u a ntit y ar e p ositi v e, h e n c e t h e str at e g y t o i ntr o d u c e w ei g hts t o r e pl a c e i n t h e dissi p ati o n t h e i n d e fi nit e q u a ntit y wit h a s m o ot h p o siti v e f u n cti o n i nt er p ol ati n g p ositi v e li mits.

T h e li n e ari z e d e q u ati o ns ar e

( 5. 2)

T a ki n g c 2 (φ 1 ) x + ( α τ ν -I ( α τ ν ))φ 1 = 0, φ 1 -α φ 2 -ν τ φ 3 = 0, φ 1 ( 0) > 0, a n d 0 < φ 2 ≡ c o nst a nt 1, w e t h us h a v e

( 5. 5)

f or s o m e p ositi v e C a n d η , a n d t h er e b y t h e s a m e li n e ar d a m pi n g esti m at e as i n t h e p eri o di c-c o e ffi ci e nt c as e:

( 5. 6)