Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation

Commun. Math. Plays. 183, 253 266 (1997) Communications in Mathematical

Physk �9 Springer-Verlag 1997

Nonlinear Stability of Solitary Waves of a Generalized Kadomtsev-Petviashvili Equation

Yue Liu 1, Xiao-Ping Wang 2

1 Department of Mathematics, University of Texas at Austin Austin, TX 78712, USA 2 Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hung Kong

Received: 12 December 1995 / Accepted: 6 June 1996

Abstract: We prove that the set of solitary wave solutions of a generalized Kadomtsev-Petviashvili equation in two dimensions,

(U t + (b/m+l)x -]- Uxxx) x : Uyy

is stable for 0 < m < 4/3.

1. Introduction

The generalized Kadomtsev-Petviashvili (GKP) equation

(Ut q- (um+l )x -'1- Uxxx)x :- r (1)

is a two dimensional analog of the generalized Korteweg-de Vries (GKdV) equation. When m = 1 and a 2 = 1, (1) is known as the KPI equation while m = 1 and cr 2 = - 1 corresponds to the KPII equation. Both are universal models for the propagation of weakly nonlinear dispersive long waves which are essentially one directional, with weak transverse effects [6]. It also describes the evolution of sound waves in antiferromagnetics [9]. It is well known that both KPI and KPII can be solved by the Inverse Scattering Transformation (IST) [1, 2].

Many local existence results for KP and GKP have recently appeared for both infinite space and periodic boundary conditions (see [19, 20, 13]). There are also some global results [22]. It is shown in [9, 20] by a virial method that GKP-I

(Ut + ( um+I )x q- l'lxxx)x = Uyy (2)

has blow-up solutions for m > 4 while arguments in [14] indicate that blow up should occur for much lower m, namely m > 4 3"

An important question is the stability and instability of solitary waves for GKP, that is, solutions of form u(x, y , t ) = qg(x - ct, y ) . Existence of solitary waves is shown in [14] for 1 < m < 4 and also in [21] by a different method. For GKP-I, instability is shown in [14] for 4 < m < 4 using a method of Shatah and Strauss [3] and a completed proof is provided by de Bouard and Saut [24]. In this paper, we shall prove that the solitary waves are nonlinearly stable for 0 < m < 4. After this

254 Y. Liu, X.-P. Wang

paper was completed, we learned that de Bouard and Saut [24] have a similar result by a different method.

The paper is organized as follows. In Sect. 2, we give the detailed proof of the existence of solitary waves for GKP-1 with 0 < m < 4. The solutions are obtained by using the variational method and the techniques developed by Lieb [18] to solve a constrained minimization problem. In Sect. 3, we prove the set of solitary waves of

4 KP-I is nonlinearly stable for 0 < m < g. The proof is based on the idea of Shatah [4] and Levandosky [5]. We use the variational properties of the minimizer and a convexity lemma of Shatah to estabilish the key inequality for stability theorem.

We shall use the following notations. I" Ip (resp. II " IIs) will stand for the norm in the Lebesque space LP(R 2) (resp. the Sobolev space HS(R2)) .

Because of the structure of Eq. (1), we introduce the following function space:

g( (2) = {ulu @ L2(O),Ux G L 2 ( O ) , D x l u y E L2(/2)} (3)

equipped with a norm 1

( fo( ~ dy) ~ lu lv = u 2 + I V u l 2 ) d x ,

and an inner product

(u, v) = f (uv + ~Tu. Cv) ex@. t2

Here 12 may be R 2 or a box [a, b] • [c, d] in R 2, and

O x 1 = f : c ~ (or fXa) . We thank the referee for the vaulable comments.

~U = ( u x , D x l U y ) T,

2. Existence of Solitary Waves

In this section, we give a more detailed proof of the existence of localized solitary waves for GKP-I given in [14]. By solitary waves, we mean traveling wave solutions of the form u(x, y, t) = q%,(x- cot, y). Substituting in (1), q~o~ satisfies the following equation:

(D(p d-Dx2fpyy - (Pxx = q)m+l . (4)

For m --- 1, this equation has an explicit solution (lump soliton [1])

o) - x2/3 + y2/(3o9) l)(x, y ) 8(09 + x2/3 + y2/(3~o))2 (5) o

We shall prove the existence of decaying solutions for 0 < m < 4 in space V = V(R 2) by a variational method. We first introduce the following functionals on V(R2),

Q(u) = I f u2 dxdy, (6) 2 rtz

(~ 1 urn+2 ~ dxdy (7) E(u) = f ((Dxluy)2 + u2) (m +-~ ] ' R 2

Nonlinear Stability of Solitary Waves of Generalized KP Equation 255

g ( u ) = f u m+2 dxdy , ( 8 ) R 2

I,o(u) = f (eou 2 + l~TulZ)dxdy. (9) R 2

Remark 1. Here and in the following, we always assume that m = ml/m2, where ml is any even integer and m2 any odd integer. This guarantees that K(u) is non- negative.

Let's also define the following minimization problem:

M(oJ) = inf Lo(u) uEV(R2) (K(u))~-~ "

I~(2u) I~(u) -- for 4 + 0 . 2

( K ( 2 u ) ) ~ (K(u))--~

Similarly, we have M(eo) = inf {/,o(u)IK(u) = 1}.

uEV

By change of variable u(x, y ) = w l V(x/wx, wy), one easily obtains that

4--m M ( w ) = w ~ M ( 1 ) w > 0 . (10)

Note that Eq. (4) is the Euler-Lagrange equation of the functional

L ~ ~ (m+l)(ml +2 )u m +2 )d x d y " (11)

Therefore, if there is a function ~o E V(R 2), such that

/,o(q~) = M(eo) with K(q~) = 1,

then q~ is a solution of

(Dq) -- (Pxx + Df2(Pyy = 2(P m+l ,

where 2 is the Lagrange multiplier. Hence 0 = 41~o is the solution of (4). We call such a solution a ground state.

Theorem 1 (Existence o f solitary waves). Let 0 < m < 4, ~o > 0 and m = ml/m2, where ml it is any even integer and rn2 any odd integer. Then there exists a minimizer ~Ow E N such that

Io~(q~o,) = inf Ir M(o~), uEN

where N = {u]u E V(R2), K(u) = 1} and K(u) 9iven by (8).

To prove this theorem, we use the techniques used in [15, 18,23 and 17]. The following lemmas are needed for the proof of Theorem 1. We shall prove the lemmas later.

It is easy to see that


L e m m a 1. Let V(~2) be the space defined in (3) and f2 may be R 2 or a box in R 2. Then f o r any 2 < n < 6, there exists a constant C, such that f o r any u E V(~2),

1

luln < C ICul = + u 2 . (12)

Lemma 2. Let u E V(B), where B C R 2 is an arbitary box. Then V8 > 0, there exist integers N~, Me, s.t.

f u2 <= 2 ~ UWn,,n2 + ~ / l ~ u l =, B n l = l n2=l B

where w,~,n 2 are orthonomal basis funct ions in V(B).

Lemma 3. Let {u J} denote a minimizin 9 sequence o f Ia =Io~=l. That is, limj__,~ I1 (u J) = infueu I1 (u). Then there exist e, 6 > 0 such that f o r all j ,

~(EluJ I > el) __> 6,

where # ( . ) denotes the Lebesoue measure.

L e m m a 4. Let u be a funct ion such that lulv < c and ~([lul > d ) > a > o. Then there exists a shift Ts, t U ( x , y ) = u(x + s, y + t), such that f o r some constant

= ~(C, a, e) > 0, u ( e n [Irs,,ul > e/23) > ~ ,

where B is the unit box (i.e. box centered at ori#in and has lenoth 1 x 1) in R 2.

Lemma 2 is used in [11]. Lemmas 3 and 4 are similar to the results in [15 and 18].

P r o o f o f Theorem 1. By (10), it suffices to show that there exists a minimizer u0 E N such that I i (uo) = M(1). We denote I (u ) = I i (u) and I ~ = infueu I (u) . Let {uj} be a minimizing sequence, i.e I ( u j ) - + I ~ with I (u j ) < C, [Uj]m+2 = 1. We then have uj ~ uo weakly in V, uj ~ uo weakly in ~L m+2. It follows from Lemma 2, uj ~ uo strongly in L 2 on any bounded domain. And uj ~ uo a.e. in R 2.

We next show uj ~ uo strongly in L m+2. To do that it suffices to prove ]u0]tm+2 = 1; i.e. that u0 E N. Since uj ~ uo weakly in L m+2, this implies [uo]Lm+2 < 1. Next, we want to show that [u0[tm+2 4=0, i.e. u0 ~ 0. From Lemmas 3 and 4, there exists ~, such that

~(B n [IT~,,~usl > e]) > a . Let T~j, tjuj be the new sequence denoted also by uj, we have

~(B n [lujl > d) > ~ .

Since uj ~ uo a.e., it follows that # ( B A [luol > ~ ] ) > ~ , therefore uo ~ 0 and Ilu01km+= + 0 .

Next, we show that i f 0 < 2 = f u~ +2 < 1, we have a contradiction. Denote vj = u j - Uo, so that vj ~ 0 weakly in L m+2. Observe that due to a theorem of Brezis and Lieb (a refined Fatou lemma [16])

f]vj]m+2-'+ 1 - - f u ~ + 2 = 1 - - 2 . (13) Note that

I (u j ) = I (v j + uo) = I (v j ) + I(uo) + 2 f VjxUO~ + 2 f (Dx 1 l)jy ) (D x I UOy) _~_ 2 f VjUo.

The last three terms converge to zero by weak convergence o f vj ~ 0 in V(R 2).


Hence i 0 = l im I ( v j ) + I ( u o ) . (14 )

j--*o~

Let fi0 = u? , then we have f [rio[ m+2 = 1. B y homogene i ty , 2 ~

l(uo) = 2 ~ I ~ , ~ ) = = 2 ~ I ~ , ~ I ( ~ 0 ) > . ( 1 5 )

Let ~j - - vj F r o m (13) , we have f la i l 'n+2 f IvJlm+~ (1-x)~-~" - 1-4 ~ 1. S imi la r ly ,

I(vj) (1 2 = (1 - 2)~-~I(vj) ( 1 - 2 ) ~ ~

l im I(vj) = (1 - 2)~-~ .lim I ( ~ j ) > (1 _ 2 ) ~ i 0 2 . j-~oo j--~oo

Therefore 2

I ~ - I ( u o ) >= (1 - 2 ) ~ I ~ . (16 )

Equat ions (15) and (16) give

I ~ > (2~+2 + (1 - 2)~-~+~ ) I ~ > I ~ ,

which is a contradict ion. Therefore

2 = f [U0[ m+2 = 1 ,

i.e. uj ~ uo s t rongly in L m+2. Moreover , because uj ~ u0 w e a k l y in V, I(uo) < infI (uj) and uo min imizes I

in N. Hence I(uj) ~ l(uo). Therefore uj ~ uo s t rongly in V(R2) , and this es tab- l ishes Theo rem 1.

W e n o w prove the lemmas.

Proof o f Lemma 1. W e shall p rove (12 ) for f2 = 112 only. The case wi th f2 be ing a box can be p roved similar ly. Cons ide r the Four ie r t ransform representa t ions o f U, Ux and Oxluy,

u = f f~(p, q)eipx+iqYdpdq, Ux = f ff'~(p, q)eipx+iqYdpdq,

Oxluy = f O~y(p, q)eipx+iqYdp dq.

Then, we have

f lul2 dxdy = f lfil2dpdq, f l~Tul2 dxdy = f [~ul2dpdq.

For some 0 __< e __< 1 and p , q4=O, we have

f i(p, q) = f e-ipx-iqYu(x, y) dxdy

= c~fe-ipx-iqYu(x,y)dxdy + (1 - cOfe-ipx-iqYu(x,y)dxdy

( 1 - ~ ) p �9 . _ ~ fe- ipx- iqyux(x ,y)dxdy + _ _ fe-~px- 'qyDx]uydxdy.

- i p q

258

It follows that

I ~(p,q)[ < ~ l u x l

p2 Then Let e -- iql+p2.

Y. Liu, X.-P. Wang

( )2 < + (1 - )p . V/[if. t2 + [D-'~Uy[2 "

v~lpl ~/I t"- I~,l <-_ Iql + p~ ~x12 + IDx~uyl2 '

I ~ I m __< k l ~ 7 p 2) ( IGI 2 + IDT'uyl2) -~ .

From H61der's inequality, we have

f [~lmdpdq < f {l~xl 2+lo;luyl2}~dpdq p2+q2>= l p2+q2- ->1 \ ]q[ + p2J

2--m

<= f dpdq pZ+q2>_l [ql T p2 j

{ ; • f IGI 2 + I D ~ u y l 2 d p d q . p2 + q 2 > 1

It is easy to see that fp2+q2__>l(~)=L-&'-dpdq is convergent if 6 < m < 2. Hence

f [gtl"dpdq ~ C1 f [~12 + iD~uyl2dpdq p2+q2 > 1 p2+q2 => 1

m

= < c l [~xl 2 + yl 2 R 2

2ha w h e r e C 1 = r ( ~ ~ doda On the other hand, we have 3 p Z + q 2 ~ l k [ q [ + p 2 l "IF ~1"

f V, Vdpdq <-<_ C2 V,12dpdq <- C2 ( f V,12dpdq) -' . p2+q2<l p2+ <:1

Again, f denotes the integral over 112. It follows then, m

i~fo~dq __< c, ( s I~xf 2 + I"7-u,12a~aq): + c2 ( f Ic,?apaqy m

m ( )- _-< c0 f 1412 + IO;a-'-~yl 2 + la[Zdpdq : ,

By the theorem of Hausdorff-Young (p. 72 in [7]), we have

( f lul"dxdy) 5 < ( f laV) ~ ,

Nonlinear Stability of Solitary Waves of Generalized KP Equation

1 = 6 where ~ + m 1. Therefore, if < m < 2 , we have 2 < n < 6 , and

259

lulndx < CO U 2 -'[- ( V x l u y ) 2 "-[- u2dx �9 []

Proof o f Lemma 2. Let us assume that the box is o f length 2~z • 2r~. The general case can be obtained by a scaling.

Consider the Fourier series representations o f u, Ux and Dxluy

U =- E a m , n eimx+iny Ux = E h imx+iny , ~m,n e ,

O • l u y ~ E Cm, n eimx+iny �9

Then, f lul2dxdy E 2 2 = am, n , f lgul2dxdy = E b2,n + Cm, n"

m,n m,n

For some 0 < c~ _< 1 and p, q4=0, we have

am,n = f e--imx--inYu(x, y) dxdy

= c~ f e-imx-inYu(x, y) dxdy + ( 1 - o 0 f e--imx--inYu(x, y) dxdy

__ ~ f e_ipx_iqyux(x,y)dxdy + (1 - oOm f e_imx_inyDxluydxdy " - i m n

It follows that

lam, nl <= iTslbm, nl § ICm, nl

< ~ ( ~ ) 2 + ( ( 1 - - n ~ n l2+[Cm, nl2 .

m 2

Let ~ = inl+m2. Then

la,.,.I

am,n [2

< Inllm[+ m 2 ~lbm,nl2 + [Cm,,i 2

< [nl+m2j ([bm, n[2+lc'~,.I 2)"

Ve > O, qN~, M,,s.t. for n >N~, m >M~, we have (-~m=) 2 < e . Hence

f u 2 = 2 < ~ am, n § § am, n ~ m,n m,n m,n m,n \ m , n

= E E blWm, n --~-g f U 2 x § 2 . m=l n=l 12

[]


Proof of Lemma 3. Since {uj} is a minimizing sequence, we have

1 = f [Ujl m+2 = f [Uj[ m+2 -]- f [Uj] m+2 + f [Uj[ m+2

[lujl > } ] [lujt--<~] [=< lujl < }1

< f [Ujlm+2+7 + 8 m f l u / = + ~( [ lu j l > 8]) [lujt> �89 [UJ[ 7 [tujl__<~]

< e 7 f ]uj] m+2§ + S m f lujl 2 + c~([ luj l > 83), [lusl_-> }3 [Iwl--<~]

w h e r e 0 < 7 < 4 - m. F r o m L e m m a 1, w e have

f lujl m+2§ < f luj] m+2§ < C,, [lujl _-> }1 R2

and f lujl ~ ~ f lujl ~ ~ c2.

[lujl>__~] R2

By choosing e small enough, we have

1 - gTC 1 -- 8mc2 ~([lujl > 8]) > = a . [] = c~

Proof of Lemma 4. For simplicity, we assume [ulv < 1. Consider any function v, such that IVlv < 1 and Iv[: 4=0. Let k = 1 + io-~2. We first prove that there is some x,

such that f (v 2 + [ V v l Z ) d x d y < m k f v 2 d x d y , (17)

Bx Bx

where Bx is the unit box centered at x and m is a certain integer. I f (17) is not true, then we can cover R 2 with unit boxes {Bx~} so that each

point x is covered by at most m unit boxes and we have

m f (v 2 + I•vl2)dxdy >= E f ( v2 + lgvl2)dxdy > rake f v2dxdy R2 i gxi i Bxi

> m k R2fv2dxdy=m( 1+ R2fv2) >m"

Therefore Ivl~ = f (~2 + I r > 1,

R 2

which is a contradiction. By Lemma 1, we have, for x satisfies (17) and some p > 2 ,

2

IvlPdxdy < Cl f (lvl2 + lXYvl2)dxdy < Clmk f lvl2dxdy B:, Bx

2

<= Clink IvlPdx (~(Bx N supp(v))) - / - (18)


and p--2

#(Bx N supp(v)) > (19)

Equation (19) is true for any v satisfying Ivlv 1 and 1f12+0. Now let's take v = max([u I - ~,0), we have [vlv < 1 and fn2 vZdxdy > (~)2~ and we have k < 1 + 4 " From (19), there exists x, s.t.

ezz

ll(Bx N supp(v)) > ~, ~ :

i.e. (Bx n [lul > e/21) >

The conclusion of the lemma follows with a shift. []

3. Stability of the Solitary Wave

In this section, we show that the solitary wave o f GKP-I is nonlinearly stable if 0 < m < 4. In order to study the stability of solitary wave of GKP-I, we need to consider the local existence for GKP-I. There are many results on local existence for GKP-I (see [19, 20, 13]). For our purpose, we state here the local existence results by Saut [20]. Let

equipped with the norm

1 ^ E LZ(R2) / = { f E S'(R2), ~ f ( r 1 6 2

1 1 / l l - 2 , x = ,

and

with

�9 - 2

Theorem 2. Let c~ E Xs, s > 3, such that ~)yy E H x . There exists T > 0 such that (1) has a unique solution u with u( O ) = ~) satisfying

u E C( [ -T , T];HS(R2)) U C I ( [ - T , T];HS-3(R2)),

Dxluy E C([-T,T];HS-I(R2)) .

Moreover, Q(u) and E(u) are well defined and independent of t�9

We next give our definition of stability of the solitary waves�9


Definition 1. A set S C X is X-stable with respect to GKP-I i f Ve > O, 3~ > 0 2 . - -2

such that f o r any uo C X n X s and Oyuo E H x , s > 3 with

inf I l u 0 - Vllx < fi, (20) yES

the solution u(t) o f (1) with u (O)= v can be extended to a 9lobal solution in C([O, o o ) ; X NXs), s > 3 and

sup inf I l u ( t ) - v l l x < e . (21) O < t < ~ vCS

Otherwise S is called X-unstable.

Now define the set o f all ground state with speed o9 > 0 as

So, = {q~ E V(R2); K(qQ = Io,(~p)= (M(og))-~-m+-2 } .

Let q~o, be a ground state of GKP-I. For simplicity, we denote ~oo, by ~p. Then

ogfp + D x 2 c p y y - ~Oxx - (/9 m+l : 0 .

It is well known in [8] that the stability of the solitary waves depends on the behavior of the following functional:

d(og) = E(~oo,) + mQ(q~o,) (po, E So,. (22)

It follows that

11 1 K d(og) = ~ o,(q~o,) m Sr 2 (~oo,)

m m 2(m + 2) Io,(q~o,) -- 2(m + 2) K(q~o,) " (23) I

4 with m = ml/m2, where ml is any Theorem 3 (Nonlinear stability). Le t 0 < m < ] even inteyer and m2 any odd integer and w > O. Then So, is V-stable, where V is defined in (3).

Remark 2. It is easy to calculate that

( 4 - - m ) ( 4 - - 3 m ~ 4 m 2 d"(o9) = ~ k. 2m j o9-~-,,- A ,

1 where A = f f 2 + (Dxl oy)2)dxdy. Hence

4 d"(o9) > 0 ~ 0 < m < - .

3

In order to prove Theorem 3, we need several lemmas.

L e m m a 5. d(o9) is differentiable and strictly increasing for o9 > 0, 0 < m < 4 with m = m l / m 2 , where ml is any even integer and m2 any odd integer.


Proof

and

In fact, from (5), we have

d ( o g ) m 4 . . . . 2 - - - - O9-~--" ( M ( 1 ) ) - ~ -

2 ( m + 2 )

4 - m 4--3m

d'(w) = 4(m + 2) w 2m ( M ( I ) ) - ~ > 0

for O < m < 4 .

Lemma 6. Let d"(og) > 0 with o9 > O. Then 3~ > O, such that for 091 > 0 with [(D 1 --O91 "~ g we have

d(ogl) > d(og) + d'(o9)(o91 - co) + 4d"(o9)]o9 - o9112 �9 (24)

Proof This follows by Taylor's expansion at o91 = co. []

Define

U~o~=, { u E V(R2); ~o~s~inf ,[u-r < e} .

Since d(og) is differentiable and strictly increasing for co > 0, it follows that for u near q~ and ~o E S~,

og(u) = d - ' 2(---. 7vK(u) (25)

is a C 1 map: o9(.): Uo~,~+R + for smalle > 0 ,

and og(q~o~) = co for any q~o E &o. The next lemma uses the variational characterization of ground states to establish

the key inequality in the proof of stability.

L e m m a 7. Suppose d"(og) > O for o9 > O. Then there exists e > 0 such that for all u E Uo,,~ and qho E S~,

E(u) - E(q~o~) + o9(u)(Q(u) - Q(qgo~)) > ld"(og)log(u) - o912, (26)

where o9(u) is defined by

o g ( u ) = d _ l ( m ) ) 2 ( m 7 - ~ K(u

Proof First of all, we have

Since

and

for u E Uo~,~ .

1 B 1 . E(u) + o9(u)Q(u) = ~ ~(u)(U) - ~ - ~ K ( u )

2(m + 2)d(o9(u) ) = K ( u ) , m

2(m + 2)d(o9(u) ) = K(~oo~(u)), q~o~(u) E So(u), m

(27)

264

then

This implies that

K ( u ) = K(qo~o(u)) .

Y. Liu , X . -P . W a n g

(28)

Since q~o)(u) is a minimizer of lo~(u) subject to the constraint K ( u ) = K(q),o~u)) and e)(u) C C 1, then by (27) and Lemma 6 we have

11 1 E ( u ) + o9(u)Q(u) > ~ o~(u)(~o~(u)) - -~ -~K(qg ,o (u ) ) = d(oo(u))

> d(og) + a'(o~)(~o(u) - o~) + 4d"(~o)la~(u) - col 2

= E(cpo~) + co(u)Q(q~o) + ld"(eo)l~o(u) - r 2 , (29) or

where we use the fact d'(o~) = Q(~p~o) �9 (30)

Now we can prove Theorem 3.

P r o o f Assume that So) is V-unstable. Then by the definition of stability, 36 > 0 and initial data uk(0) E U,o,~ such that

sup inf I luk( t ) -~o l lv > 6 (31) t > O r o =

where uk( t ) is the solution of GKP-I with initial data uk(O). By continuity in t, we can pick the first time tk so that

inf Iluk(tk) - ~ollv : 6 , (32) ~pES, o

Since E ( u ) and Q ( u ) are conserved at t and continuous for u, we can find (Pk E S~ such that

[E(uk(tk)) -- E(q~k)l = [E(uk(O)) -- E(~Pk)I ~ 0 (33)

as k ~ oo and

IQ(u~(tD) - Q(q~k)l = IQ(u~(0)) - Q(~ok)l ~ 0

as k ---+ oc. Choose 6 small enough so that Lemma 7 applies,

E(uk ( t k ) ) -- E(q)k) + O9(Uk(tk))(Q(uk(tk)) -- O(q~k)) > l d"(o~)l~o(uk(tk)) - 14

By (32), there exists ~Pk c So) such that

Iluk(tk)llv <= II~okllv + 2 6

< ( l + l ) Io)(~Pk)+26

m+2 < c ( o ) ) M ( c o ) 7 + 26 < + c ~ .

(34)

0)] 2 .

(35)

(36)

(37)

Nonl inea r Stabili ty o f Sol i tary W a v e s o f General ized KP Equa t ion 2 6 5

Since co(u) is a continuous map, co(uk(tk)) is uniformly bounded for k. By (35), letting k ~ 0% we have

co(uk(tk )) ~ co. (38) Hence

lim K(uk(tk)) = lim 2(m + 2)d(co(uk(tk)) _ 2(m + 2)d(co) " (39) k---+~ k---+~ m m

On the other hand,

2 I~o(uk(tk)) = 2(E(uk(&)) + coO(uk(tk)) + ~ - - ~ K ( u k ( t k ) )

2 = 2d(co(uk(&)) § 2(e) - co(u~(&))Q(uk(tk)) § ~ K ( u k ( & ) ) .

m --t- z

Since Q(uk(tk)) = Q(uk(O)) < Iluk(tk)[Iv < C(co) < +c~ ,

then by (39)

_ _ _ _ 2 2(m § 2(m +2)d(co) ask--~cxD. Io~(uk(tk)) 2d(co) ___+ + m + 2 m m

(40)

That is Lo(uk(&)) ~ I(~oo)) = (M(co))-~ .

Let 1

vk(tk) = (K(uk( tk) ) ) - ~ uk(t,) .

Then K(vk(tk)) = 1 and

Lo(vk(tk)) = (K(uk(tk)))-~-~Lo(Uk(&))

(M(co))~ m+2 = (M(co))-7-(M(co))-; = M(co). (43) (~d(co))~

Hence, vk(&) is a minimizing sequence. Therefore, 3q~k E So~ such that

lim Ilvk(tk) - (M(co))-lq~kHv = 0 (44) k---* ~

where K((M(co))-~cpk)= 1. This implies that

lim Iluk(tk) q~kllv lim [(K(uk(tk)))~-~ - = �9 I I ( g ( u k ( t k ) ) ) - ~ ( u ~ ( t k ) - - ~ok)llv] k---*c~ k---~ c~

< M~(CO) [L k--'~lim IlVk(tk)-- M-{(co)q~kllv]

(41)

(42)

+ lim [M@(co) - (K(uk(tk)))-~--~[ Ilrpkav = 0 k---* cx3

1 1 - . M - ..~_~2 since II~okll~, _-< (1 + N)Io~(q~k) --* (1 + ~)( (co))-- < +(x~. Hence (45) contradicts with (32). []

(45)


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Preprint 1996

Communicated by A. Jaffe

Nonlinear stability of solitary waves of a generalized Kadomtsev-Petviashvili equation

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