Stability and Decay properties of Solitary wave solutions for the generalized BO-ZK equation
aa r X i v : . [ m a t h . A P ] O c t STABILITY AND DECAY PROPERTIES OF SOLITARY-WAVESOLUTIONS TO THE GENERALIZED BO–ZK EQUATION
Amin Esfahani
School of Mathematics and Computer ScienceDamghan UniversityDamghan, Postal Code 36716–41167, IranE-mail: [email protected], [email protected],
Ademir Pastor
IMECC–UNICAMPRua S´ergio Buarque de Holanda, 651Cidade Universit´aria, 13083-859, Campinas–SP, Brazil.E-mail: [email protected]
Jerry L. Bona
Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at Chicago851 S. Morgan Street MC 249 Chicago, Illinois 60601, USA.E-mail: [email protected]
Abstract.
Studied here is the generalized Benjamin-Ono–Zakharov-Kuznetsovequation u t + u p u x + α H u xx + εu xyy = 0 , ( x, y ) ∈ R , t ∈ R + , (1)in two space dimensions. Here, H is the Hilbert transform and subscriptsdenote partial differentiation. We classify when equation (1) possesses solitary-wave solutions in terms of the signs of the constants α and ε appearing in thedispersive terms and the strength of the nonlinearity. Regularity and decayproperties of these solitary wave are determined and their stability is studied. Introduction
This paper is concerned with existence and non-existence, stability and some de-cay properties of solitary-wave solutions of the two-dimensional generalized Benjamin-Ono–Zakharov-Kuznetsov equation (BO–ZK equation henceforth), u t + u p u x + α H u xx + εu xyy = 0 , ( x, y ) ∈ R , t ∈ R + . (1.1)Here p > , α and ε are non-zero real constants with ε normalized to ± y -variable while H is the Hilbert transform H u ( x, y, t ) = p . v . π Z R u ( z, y, t ) x − z dz, in the x -variable, where p . v . denotes the Cauchy principal value.When p = 1, this equation arises as a model for electromigration in thin nanocon-ductors on a dielectric substrate (see [27, 33]). Equation (1.1) may also be viewed as Mathematics Subject Classification.
Key words and phrases.
Nonlinear PDE, Solitary Wave Solution, Stability, Decay. one of the natural, two-dimensional generalizations of the one-dimensional Benjamin-Ono equation in much the same way that the Kadomtsev-Petviashvili equation andthe Zakharov-Kuznetsov equation generalize the Koreteweg-de Vries equation.The generalized Benjamin-Ono equation u t + u p u x + α H u xx = 0 , x ∈ R , t ∈ R + , and its counterpart u t + u p u x + α H u xx + βu xxx = 0 , x ∈ R , t ∈ R + , taking into account surface tension effects between the two layers of fluid, havebeen considered by many authors. Well-posedness issues for the pure initial-valueproblem have attracted a lot of interest recently (see, e.g. [11, 30, 31, 42, 43, 46]).Questions about the existence and stability of solitary traveling-waves have beeninvestigated in [1]–[7] and [28].Theory for the generalized Zakharov-Kuznetsov equation u t + u p u x + αu xxx + εu xyy = 0 , ( x, y ) ∈ R , t ∈ R + , is less abundant. Well-posedness was studied in [23, 24, 25, 41, 36, 45]. As far as weknow, the only results concerning existence and nonlinear stability of solitary-wavesolutions of this equation was provided in [16].The solitary-wave solutions of interest here have the form u ( x, y, t ) = ϕ ( x − ct, y ),where c = 0 is the speed of propagation and u belongs to a natural function spacedenoted Z and introduced presently. Substituting this form into (1.1), integratingonce with respect to the variable z = x − ct and assuming ϕ ( z, y ) decays suitablyfor large values of | z | , it transpires that ϕ must satisfy − cϕ + 1 p + 1 ϕ p +1 + α H ϕ x + εϕ yy = 0 , (1.2)where we have replaced the variable z by x . Remark 1.1.
When it is convenient, it may be assumed that (1.2) has the norma-lized form − ϕ + 1 p + 1 ϕ p +1 + H ϕ x ± ϕ yy = 0 (1.3) by scaling the independent and dependent variables, viz. u ( x, y, t ) = av ( bx, dy, et ) where a p = c , e = b = c/α and d = ε/c . If instead, we insist that d > , so ε = +1 , then equation (1.2) may be taken in the form − ϕ + 1 p + 1 ϕ p +1 ± H ϕ x + ϕ yy = 0 . (1.4) Of course, throughout, it will be presumed that the power p appearing in the non-linearity is rational and has the form k/m where k and m are relatively prime and m is odd. This restriction allows us to define a branch of the mapping w w m that is real on the real axis. Attention is now turned to the structure of the paper. The theory begins byexamining when solitary-wave solutions of (1.1) exist. As pointed out in [33], noexact formulas are known for solitary-wave solutions to (1.1), so an existence theoryis needed before questions of stability can be addressed. Pohojaev-type identitiesare used to show that solitary-wave solutions do not exist for certain values of p and TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 3 signs of ε and α . In some of the cases where such solutions are not prohibited byelementary inequalities, a suitable minimization problem can be solved using Lions’concentration-compactness principle [37, 38] (see Theorem 2.1). For example, ourresults imply there are solitary-wave solutions when c > α < ε > < p <
4. Moreover, these solutions are shown to be ground states.With solitary waves in hand, their orbital stability is at issue. The variationalapproach of Cazenave and Lions [13] comes to the fore in Section 3 in establishingstability for the case αε < cα <
0, and 0 < p < /
3. Complementary instabilityresults appeared in [20] for the same conditions on c, α and ε , but with 4 / < p < Remark 1.2.
The scale-invariant Sobolev spaces for the BO–ZK equation (1.1) are ˙ H s ,s ( R ) , where s + s = − p (see the definitions below). Hence a reasonableframework for studying local well-posedness of the BO–ZK equation (1.1) is thefamily of spaces H s ,s ( R ) , s + s ≥ − p . Remark 1.3.
The n -dimensional version of (1.1) is u t + u p u x + α H u x x + n X i =2 ε i u x x i x i = 0 , (1.5) where t ∈ R + , ( x , x , . . . , x n ) ∈ R n and α, ε i ∈ R , i = 2 , . . . , n . The theorydeveloped here has natural analogs for (1.5) which will be developed later. Notation and Preliminaries.
As already mentioned, the exponent p in (1.1) istaken to be a rational number of the form p = k/m , where m and k are relativelyprime and m is odd. This allows the nonlinearity to be given a definition that isreal-valued. The notation b f = b f ( ξ, η ) means the Fourier transform, b f ( ξ, η ) = Z R e − i( xξ + yη ) f ( x, y ) dxdy of f = f ( x, y ). For any s ∈ R , the space H s := H s ( R ) denotes the usualisotropic, L ( R )-based, Sobolev space. For s , s ∈ R , the anisotropic Sobolevspace H s ,s := H s ,s (cid:0) R (cid:1) is the set of all distributions f such that k f k H s ,s = Z R (cid:0) ξ (cid:1) s (cid:0) η (cid:1) s | b f ( ξ, η ) | dξdη < ∞ . The fractional Sobolev-Liouville spaces H ( s ,s ) p := H ( s ,s ) p (cid:0) R (cid:1) , 1 ≤ p < ∞ , arethe set of all functions f ∈ L p ( R ) such that k f k H ( s ,s p = k f k L p ( R ) + X i =1 (cid:13)(cid:13) D s i x i f (cid:13)(cid:13) L p ( R ) < ∞ , STABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION where D s i x i f denotes the Bessel derivative of order s i with respect to x i (see e.g. [32], [39]). For short, H ( k ) p ( R ) denotes the space H ( k,k ) p ( R ).The particular space Z := H , (cid:0) R (cid:1) ∩ H , (cid:0) R (cid:1) = H ( , ) (cid:0) R (cid:1) arises naturallyin the analysis to follow. It can be characterized alternatively as the closure of C ∞ ( R ) with respect to the norm k ϕ k Z = k ϕ k L ( R ) + k ϕ y k L ( R ) + (cid:13)(cid:13)(cid:13) D / x ϕ (cid:13)(cid:13)(cid:13) L ( R ) , (1.6)where D / x ϕ denotes the fractional derivative of order 1 / x , definedvia its Fourier transform by \ D / x ϕ ( ξ, η ) = | ξ | / b ϕ ( ξ, η ). Remark 1.4.
By combining fractional Gagliardo-Nirenberg and H¨older’s inequalityone can deduce the existence of a positive constant C such that k u k p +2 L p +2 ≤ C k u k (4 − p ) / L k D / x u k pL k u y k p/ L , ≤ p < . (1.7) This in turn implies the continuous embedding Z ֒ → L p (cid:0) R (cid:1) , ≤ p < . (1.8)2. Solitary waves
This section is devoted to establishing existence and non-existence results forsolitary-wave solutions of the BO-ZK equations. We begin with a non-existenceresult.
Theorem 2.1.
Equation (1.2) cannot have a non-trivial solitary-wave solutionunless either(i) ε = 1 , c > , α < , p < ,(ii) ε = − , c < , α > , p < ,(iii) ε = 1 , c < , α < , p > , or(iv) ε = − , c > , α > , p > .Proof. This follows from some Pohojaev-type identities. If (1.2) is multiplied by ϕ , xϕ x and yϕ y and the results integrated over R , then the identities Z R (cid:18) − cϕ + αϕ H ϕ x − εϕ y + 1 p + 1 ϕ p +2 (cid:19) dxdy = 0 , (2.1) Z R (cid:18) cϕ + εϕ y − p + 1)( p + 2) ϕ p +2 (cid:19) dxdy = 0 , (2.2) Z R (cid:18) cϕ − αϕ H ϕ x − εϕ y − p + 1)( p + 2) ϕ p +2 (cid:19) dxdy = 0 , (2.3)emerge. These formulas follow from the elementary properties of the Hilbert trans-form together with suitably chosen formal integrations by parts. The identitiescan be justified for functions of the minimal regularity required for them to makesense by first establishing them for smooth solutions and then using a standardtruncation argument as in [17].Summing (2.1) and (2.2) leads to Z R (cid:18) αϕ H ϕ x + p ( p + 1)( p + 2) ϕ p +2 (cid:19) dxdy = 0 , (2.4) TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 5 whilst adding (2.2) and (2.3) yields Z R (cid:18) cϕ − α ϕ H ϕ x − p + 1 ϕ p +2 (cid:19) dxdy = 0 . (2.5)If the integral of ϕ p +2 is eliminated between (2.4) and (2.5), there appears Z R (cid:0) pcϕ + α (4 − p ) ϕ H ϕ x (cid:1) dxdy = 0 . (2.6)On the other hand, adding (2.1) and (2.3) gives Z R (cid:18) εϕ y − p ( p + 1)( p + 2) ϕ p +2 (cid:19) dxdy = 0 . (2.7)Finally, substituting (2.2) into (2.7), there obtains Z R (cid:0) pcϕ + ε ( p − ϕ y (cid:1) dxdy = 0 . (2.8)The advertised results follow immediately from (2.6) and (2.8). (cid:3) For cases (i) and (ii) from Theorem 2.1, the existence of solitary-wave solutionsof (1.1) is established in the next result.
Theorem 2.2.
Let αε, cα < and p = km < , where m ∈ N is odd and m and k are relatively prime. Then equation (1.2) admits a non-trivial solution ϕ ∈ Z .Proof. The proof is based on the concentration-compactness principle [37, 38]. Sup-pose that α < α > α = − c = 1 so that ε = +1 (see Remark 1.1) and consider the mini-mization problem I λ = inf (cid:26) I ( ϕ ) ; ϕ ∈ Z , J ( ϕ ) = Z R ϕ p +2 dxdy = λ (cid:27) (2.9)where λ = 0 and I ( ϕ ) = 12 Z R (cid:0) ϕ + ϕ H ϕ x + ϕ y (cid:1) dxdy = 12 k ϕ k Z . Clearly, I λ < ∞ if there are elements ϕ ∈ Z such that R R ϕ p +2 dxdy = λ. Theembedding (1.8) allows us to adduce a positive constant C such that0 < | λ | = (cid:12)(cid:12)(cid:12)(cid:12)Z R ϕ p +2 dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k ϕ k p +2 Z = CI ( ϕ ) p +22 , from which one concludes that I λ ≥ (cid:16) | λ | C (cid:17) p +2 > λ let { ϕ n } n ∈ N be a minimizing sequence for I λ . For n = 1 , , · · · and r >
0, define the concentration function Q n ( r ) associated to ϕ n by Q n ( r ) = sup ( e x, e y ) ∈ R Z B r ( e x, e y ) ρ n dxdy Depending on p , this might require that λ >
0. Of course, I λ is a number, but we willsometimes refer to it as the minimization problem. For example, the phrase “ { φ n } is a minimizingsequence for the problem I λ ” means that J ( φ n ) = λ for all n and I ( φ n ) → I λ as n → ∞ . STABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION where ρ n = | ϕ n | + (cid:12)(cid:12)(cid:12) D / x ϕ n (cid:12)(cid:12)(cid:12) + | ∂ y ϕ n | and B r ( x, y ) denotes the ball of radius r > x, y ) ∈ R . If evanescence of the sequence { ϕ n } n ∈ N occurs,which is to say, for any r > n → + ∞ sup ( e x, e y ) ∈ R Z B r ( e x, e y ) ρ n dxdy = 0 , then embedding (1.8) implies that lim n →∞ k ϕ n k L p +2 = 0, which contradicts theconstraint imposed for the minimization problem. Thus, according to the concentration-compactness theorem, either dichotomy or compactness must occur for the sequence { ϕ n } n ∈ N .The occurrence of dichotomy is ruled out next. Suppose that γ ∈ (0 , I λ ), whereit is assumed that γ = lim r → + ∞ lim n → + ∞ sup ( e x, e y ) ∈ R Z B r ( e x, e y ) ρ n dxdy. By the definition of γ , for a given ǫ >
0, there exist r ∈ R and N ∈ N such that γ − ǫ < Q n ( r ) ≤ Q n (2 r ) < γ + ǫ, for any r ≥ r and n ≥ N . Hence, there is a sequence { ( e x n , e y n ) } n ∈ N ⊂ R for which Z B r ( e x n , e y n ) ρ n dxdy > γ − ǫ and Z B r ( e x n , e y n ) ρ n dxdy < γ + ǫ. Let φ, ψ lie in C ∞ ( R ) and suppose • supp φ ⊂ B (0 , φ ≡ B (0 ,
0) and 0 ≤ φ ≤ • supp ψ ⊂ R \ B (0 , ψ ≡ R \ B (0 ,
0) and 0 ≤ ψ ≤ { g n } n ∈ N and { h n } n ∈ N by g n ( x, y ) = φ r (( x, y ) − ( e x n , e y n )) ϕ n and h n ( x, y ) = ψ r (( x, y ) − ( e x n , e y n )) ϕ n , where φ r ( x, y ) = φ (cid:18) ( x, y ) r (cid:19) and ψ r ( x, y ) = ψ (cid:18) ( x, y ) r (cid:19) . It is clear that g n , h n ∈ Z .The following commutator estimate is helpful in obtaining the splitting lemmato follow. Lemma 2.3 ([12, 14]) . Let g ∈ C ∞ ( R ) with g ′ ∈ L ∞ ( R ) . Then [ H , g ] ∂ x is abounded linear operator from L ( R ) into L ( R ) with k [ H , g ] ∂ x f k L ( R ) ≤ C k g ′ k L ∞ ( R ) k f k L ( R ) . The splitting lemma proved next enables us to rule out the possibility of di-chotomy occuring in the present context.
Lemma 2.4.
Let { g n } n ∈ N and { h n } n ∈ N be as just defined. Then, for every ǫ > ,there exists δ = δ ( ǫ ) > with lim ǫ → δ ( ǫ ) = 0 , µ ∈ (0 , I λ ) , n ∈ N and ρ ∈ (0 , λ ) TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 7 such that for all n ≥ n , | I ( ϕ n ) − I ( g n ) − I ( h n ) | ≤ δ, (2.10) | I ( g n ) − µ | ≤ δ, | I ( h n ) − I λ + µ | ≤ δ, (2.11) | J ( ϕ n ) − J ( g n ) − J ( h n ) | ≤ δ, (2.12) | J ( g n ) − ρ | ≤ δ, | J ( h n ) − λ + ρ | ≤ δ. (2.13) Proof.
Obviously, supp g n ∩ supp h n = ∅ . Write g n = φ r ϕ n and h n = ψ r ϕ n so that2 I ( g n ) = Z R φ r h ϕ n + ϕ n ∂ x H ϕ n + (cid:0) ∂ y ϕ n (cid:1) i dxdy + 2 Z R φ r ϕ n ( ∂ y φ r )( ∂ y ϕ n ) dxdy + Z R (cid:2) ( ∂ y φ r ) ϕ n + ϕ n φ r H ( ϕ n ∂ x φ r ) (cid:3) dxdy + Z R ϕ n φ r [ H , φ r ] ∂ x ϕ n dxdy and2 I ( h n ) = Z R ψ r h ϕ n + ϕ n ∂ x H ϕ n + (cid:0) ∂ y ϕ n (cid:1) i dxdy + 2 Z R ψ r ϕ n ( ∂ y ψ r )( ∂ y ϕ n ) dxdy + Z R (cid:2) ( ∂ y ψ r ) ϕ n + ϕ n ψ r H ( ϕ n ∂ x ψ r ) (cid:3) dxdy + Z R ϕ n ψ r [ H , ψ r ] ∂ x ϕ n dxdy. Since k φ r k L ∞ = k ψ r k L ∞ = 1, k∇ φ r k L ∞ ≤ r k∇ φ k L ∞ and k∇ ψ r k L ∞ ≤ r k∇ ψ k L ∞ ,it follows from Lemma 2.3 that (cid:12)(cid:12)(cid:12)(cid:12) I ( g n ) − Z R φ r h ϕ n + ϕ n ∂ x H ϕ n + (cid:0) ∂ y ϕ n (cid:1) i dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ ( ǫ )and (cid:12)(cid:12)(cid:12)(cid:12) I ( h n ) − Z R ψ r h ϕ n + ϕ n ∂ x H ϕ n + (cid:0) ∂ y ϕ n (cid:1) i dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ ( ǫ ) . These inequalities imply (2.10), from which, one infers (taking subsequences ifnecessary) that there exists µ = µ ( ǫ ) ∈ [0 , I λ ] such that lim n →∞ I ( g n ) = µ . Inconsequence, we see that | I ( g n ) − I λ + µ | ≤ δ ( ǫ ) . From (2.10) again, the fact that supp g n ∩ supp h n = ∅ and the embedding (1.8),one obtains | J ( ϕ n ) − J ( g n ) − J ( h n ) | ≤ Cδ ( ε )for some constant C . It may therefore be presumed that there is a ρ = ρ ( ǫ ) and e ρ = e ρ ( ǫ ) such that lim n → + ∞ J ( g n ) = ρ ( ǫ ) , lim n → + ∞ J ( h n ) = e ρ ( ǫ )with | λ − ρ ( ǫ ) − e ρ ( ǫ ) | ≤ δ ( ǫ ). If lim ǫ → ρ ( ǫ ) = 0, then for ǫ sufficiently small, it mustbe that J ( h n ) > n large enough. Hence, by considering ( e ρ ( ǫ ) J ( h n )) p +2 h n ,and noting that J (cid:16) ( e ρ ( ǫ ) J ( h n )) p +2 h n (cid:17) = e ρ ( ǫ ), it transpires that I e ρ ( ǫ ) ≤ lim inf n → + ∞ I ( h n ) ≤ I λ − γ + δ ( ǫ ) , which leads to a contradiction since lim ǫ → e ρ ( ǫ ) = λ . Thus ρ = lim ǫ → ρ ( ǫ ) > ρ < λ , because the case ρ = λ is ruled out in the same manner as justused to rule out ρ = 0, but with h n replacing g n in the argument. Since ρ ∈ (0 , λ ),one infers that necessarily µ = lim ǫ → + ∞ µ ( ǫ ) ∈ (0 , I λ ). This completes the proof ofthe lemma. (cid:3) STABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION
Now, attention is returned to the proof that dichotomy cannot happen. Theprevious lemma implies that I λ ≥ I ρ + I λ − ρ , (2.14)which contradicts the subadditivity of I λ coming from the fact that I λ = λ / ( p +2) I .Hence dichotomy is ruled out.The remaining case in the concentration-compactness principle is local compact-ness. Thus, there exists a sequence { ( x n , y n ) } n ∈ N ⊂ R such that for all ǫ > R > n > Z B R ( x n ,y n ) ρ n dxdy ≥ ι λ − ǫ, for all n ≥ n , where ι λ = lim n → + ∞ Z R ρ n dxdy. This implies that for n large enough, Z B R ( x n ,y n ) | ϕ n | dxdy ≥ Z R | ϕ n | dxdy − ǫ. Since ϕ n is bounded in the Hilbert space Z , there exists ϕ ∈ Z such that asubsequence of { ϕ n ( · − ( x n , y n )) } n ∈ N (denoted again by { ϕ n ( · − ( x n , y n )) } n ∈ N )converges weakly in Z to ϕ . It follows that Z R | ϕ | dxdy ≤ lim inf n → + ∞ Z R | ϕ n | dxdy ≤ lim inf n → + ∞ Z B R ( x n ,y n ) | ϕ n | dxdy + 2 ǫ = lim inf n → + ∞ Z B R (0 , | ϕ n (( x, y ) − ( x n , y n )) | dxdy + 2 ǫ. But, when restricted to bounded sets in R , Z is compactly embedded into L .Consequently, { ϕ n ( · − ( x n , y n )) } n ∈ N may be presumed to converges strongly inthe Fr´echet space L loc ( R ). The last inequality above implies that this strongconvergence also takes place in L ( R ) by what are, by now, standard arguments.Thus, because of the embedding (1.8), { ϕ n ( · − ( x n , y n )) } n ∈ N also converges to ϕ strongly in L p +2 ( R ), whence J ( ϕ ) = λ and I λ = lim n → + ∞ I ( ϕ n ) = I ( ϕ ) , which is to say, ϕ is a solution of I λ .The Lagrange multiplier theorem now implies there exists θ ∈ R such that ϕ + H ϕ x − ϕ yy = θ ( p + 2) ϕ p +1 (2.15)as an equation in Z ′ (the dual space of Z in L − duality). A change of scale yieldsa e ϕ which satisfies (1.2). (cid:3) Remark 2.5.
Theorem 2.2 shows the existence of solitary-wave solutions of (1.1) in the cases (i) and (ii) in Theorem 2.1. The question of existence or nonexistenceof solitary waves in cases (iii) and (iv) is currently open.
TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 9
Definition 2.6.
A solution ϕ of equation (1.2) is called a ground state, if ϕ min-imizes the action S ( u ) = E ( u ) + c F ( u ) among all solutions of (1.2), where F ( u ) = 12 Z R u dxdy and E ( u ) = 12 Z R (cid:18) εu y − αu H u x − p + 1)( p + 2) u p +2 (cid:19) dxdy. Next, it is established that the minima obtained in Theorem 2.2 are preciselythe ground-state solutions of (1.2). The proof is inspired by that of Lemma 2.1 in[18].
Theorem 2.7.
In the context of equation (1.2) for solitary-wave solutions of theBO-ZK equation, let K ( u ) = 12 Z R ( cu + u y ) dxdy − p + 1)( p + 2) J ( u ) with J ( u ) = R u p +2 dxdy as in (2.9) . Up to a change of scale, the following asser-tions about a function u ∗ ∈ Z are equivalent:(i) If J ( u ∗ ) = λ ∗ then u ∗ is a minimizer of I λ ∗ ,(ii) K ( u ∗ ) = 0 and inf (cid:26)Z R u H u x dxdy, u ∈ Z , u = 0 , K ( u ) = 0 (cid:27) = Z R u ∗ H u ∗ x dxdy, (iii) u ∗ is a ground state,(iv) K ( u ∗ ) = 0 and inf (cid:26) K ( u ) , u ∈ Z , u = 0 , Z R u H u x dxdy = Z R u ∗ H u ∗ x dxdy (cid:27) = 0 . Proof . We set λ ∗ = (2( p + 1) I ) p +2 p and proceed with the proof.(i) ⇛ (ii) : Assume that u ∗ satisfies (i). Let u ∈ Z with u = 0 and K ( u ) = 0,from which it follows that J ( u ) >
0. Define u µ ( x, y ) = u (cid:18) xµ , y (cid:19) , with µ = J ( u ∗ ) J ( u ) , so that J ( u µ ) = J ( u ∗ ) and K ( u µ ) = 0 . Since u ∗ is a minimum of I λ ∗ , it must bethe case that K ( u ∗ ) = 0 and K ( u ∗ )+ C p J ( u ∗ )+ 12 Z R u ∗ H u ∗ x dxdy ≤ K ( u µ )+ C p J ( u µ )+ 12 Z R u µ H ( u µ ) x dxdy, where C p = p +1)( p +2) . This in turn implies that Z R u ∗ H u ∗ x dxdy ≤ Z R u H u x dxdy, and (ii) holds. (ii) ⇛ (iii) : If u ∗ satisfies (ii), then there is a Lagrange multiplier θ such that cu ∗ − u ∗ yy + θ H u ∗ x − p + 1 ( u ∗ ) p +1 = 0 . By multiplying the above equation by u ∗ , integrating by parts and using that K ( u ∗ ) = 0, we can see that θ is positive. Hence the scale change u ∗ ( x, y ) = u ∗ ( x/θ, y ) satisfies equation (1.2).On the other hand, the identity S ( u ) = K ( u ) + R R u H u x dxdy shows that if u is a solution of (1.2), then S ( u ) = 12 Z R u H u x dxdy ≥ Z R u ∗ H u ∗ x dxdy = 12 Z R u ∗ H ( u ∗ ) x dxdy = S ( u ∗ ) , whence u ∗ is a ground state.(iii) ⇛ (i) : From the proof of Theorem 2.1, one sees that if u is a solution of(1.2), then K ( u ) = 0 and I ( u ) = 12 (cid:18) p (cid:19) Z R u H u x dxdy. (2.16)Hence if u ∗ is a ground state, then u ∗ minimizes both I ( u ) and R R u H u x dxdy among all solutions of (1.2). Let λ = J ( u ) and e u be a minimum of I λ . Then I λ = I ( e u ) ≤ I ( u ∗ ) (2.17)and there is a positive number θ such that c e u − e u yy + H e u x = θp + 1 e u p +1 . Using the equations satisfied by e u and u ∗ , inequality (2.17) is written as I λ = λθp + 1 ≤ λp + 1 , from which it is deduced immediately that θ ≤
1. On the other hand, u ∗ = θ p e u satisfies equation (1.2), and since u ∗ is a ground state, it must be the case that I ( u ∗ ) ≤ I ( u ∗ ) ≤ θ p I ( e u ) , so that θ ≥
1. In consequence, u ∗ = e u is a minimum of I λ with λ = λ ∗ .(ii) ⇛ (iv) : Let u ∈ Z with R R u H u x dxdy = R R u ∗ H u ∗ x dxdy . Suppose that K ( u ) <
0. Since K ( τ u ) > τ > τ ∈ (0 , K ( τ u ) = 0. Thus by setting e u = τ u , one has e u ∈ Z , K ( e u ) = 0 and Z R e u H e u x dxdy < Z R u H u x dxdy = Z R u ∗ H u ∗ x dxdy, which contradicts (ii) and shows that u ∗ satisfies (iv) because K ( u ∗ ) = 0.(iv) ⇛ (ii) : Let u ∈ Z with K ( u ) = 0 and u = 0. Suppose that Z R u H u x dxdy < Z R u ∗ H u ∗ x dxdy. Since K ( τ u ) < τ >
1, there is a τ > Z R ( τ u ) H ( τ u ) x dxdy = Z R u ∗ H u ∗ x dxdy TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 11 and K ( τ u ) < . This contradicts (iv). Hence R R u H u x dxdy ≥ R R u ∗ H u ∗ x dxdy and (ii) holds. Remark 2.8.
Note that the proof of the above theorem shows that, indeed, (i) and (iii) are equivalent and imply (ii) and (iv) , which are also equivalent. The converseholds modulo a scale change. Stability
The notion of orbital stability employed here is the standard one.
Definition 3.1.
Let ϕ c be a solitary-wave solution of (1.1). We say that ϕ c isorbitally stable if for all η > , there is a δ > such that for any u ∈ H s (cid:0) R (cid:1) , s > , with k u − ϕ c k Z < δ , the corresponding solution u ( t ) of (1.1) with u (0) = u satisfies sup t ≥ inf r ∈ R k u ( t ) − ϕ c ( · − r ) k Z < η. Some of the arguments below can be found in [3] where the stability of solitarywaves for the generalized BO equation has been established. Hereafter, withoutloss of generality, we take α = − ε = +1, and c > Theorem 3.2.
Let λ = 0 .(i) Every minimizing sequence for the problem I λ converges, up to translations,in Z to an element in the set M λ = { ϕ ∈ Z ; I ( ϕ ) = I λ , J ( ϕ ) = λ } of minimizers for I λ .(ii) Let { ϕ n } be a minimizing sequence for I λ . Then, it must be the case that lim n → + ∞ inf ψ ∈ M λ , z ∈ R k ϕ n ( · + z ) − ψ k Z = 0 , (3.1)lim n → + ∞ inf ψ ∈ M λ k ϕ n − ψ k Z = 0 . (3.2) Proof.
Part (i) follows immediately from the proof of Theorem 2.2. The equality(3.1) is proved by contradiction. Indeed, if (3.1) does not hold, then there exists asubsequence of the sequence { ϕ n } , denoted again by { ϕ n } , and an ǫ > ̟ = inf ψ ∈ M λ ,r ∈ R k ϕ n ( · + r ) − ψ k Z ≥ ǫ, for all n sufficiently large. On the other hand, since { ϕ n } is a minimizing sequencefor I λ , part (i) implies that there exists a sequence { r n } ⊂ R such that, up to asubsequence, ϕ n ( · + r n ) → ϕ in Z , as n → ∞ . Hence, for n large enough, it isinferred that ǫ ≥ k ϕ n ( · + r n ) − ϕ k Z ≥ ̟ ≥ ǫ, which is a contradiction.The proof of (3.2) follows from (3.1), the fact that if ψ ∈ M λ then ψ ( · + r ) ∈ M λ for all r ∈ R , and the equalityinf ψ ∈ M λ k ϕ n − ψ k Z = inf ψ ∈ M λ ,r ∈ R k ϕ n − ψ ( · − r ) k Z = inf ψ ∈ M λ ,r ∈ R k ϕ n ( · + r ) − ψ k Z . This completes the proof of the theorem. (cid:3)
The next lemma shows that there exists a λ >
Lemma 3.3. If λ = (cid:0) p + 1) I (cid:1) p +2 p in the minimization problem (2.9) , then any ϕ ∈ M λ is a solitary-wave solution of (1.2). For λ as in the preceding lemma, define the set N c = { ϕ ∈ Z ; J ( ϕ ) = 2( p + 1) I ( ϕ ) = λ } . It is clear that M λ = N c ; the latter notation simply emphasizes the dependenceupon the wave speed c . Next, for any c > ϕ ∈ N c , define the function d : R → R by d ( c ) = E ( ϕ ) + c F ( ϕ ) . (3.3) Lemma 3.4.
The function d in (3.3) is constant on N c and differentiable andstrictly increasing for c > . Moreover, d ′′ ( c ) > if and only if < p < .Proof. It is straightforward to check that d ( c ) = I ( ϕ ) − p + 1)( p + 2) J ( ϕ ) = p p + 1)( p + 2) J ( ϕ ) = p (2( p + 1)) p p + 2 I p +2 p . (3.4)It is plain that d is constant on N c . From the second equality in (3.4) and thedefinition of J , one obtains d ( c ) = p p + 1)( p + 2) c p − J ( ψ ) , (3.5)where ψ ( x, y ) = c − p ϕ (cid:18) xc , y √ c (cid:19) . Note that ψ satisfies (1.2), with c = 1. But, from(2.4) and (2.6), one infers that1( p + 1)( p + 2) J ( ϕ ) = 2 c − p F ( ϕ ) . Thus, from (3.5) follows the formula d ′ ( c ) = c ( p − ) F ( ψ ) , whence d ′′ ( c ) = (cid:18) p − (cid:19) c ( p − ) F ( ψ ) . This proves the lemma. (cid:3)
A study is initiated of the behavior of d in a neighborhood of the set N c . Lemma 3.5.
Let c > . Then, there exists a positive number ǫ and a C -map v : B ǫ ( N c ) → (0 , + ∞ ) defined by v ( u ) = d − (cid:18) p p + 1)( p + 2) J ( u ) (cid:19) , such that v ( ϕ ) = c for every ϕ ∈ N c , where B ǫ ( N c ) = (cid:26) ϕ ∈ Z ; inf ψ ∈ N c k ϕ − ψ k Z < ǫ (cid:27) . TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 13
Proof.
By definition, N c is a bounded set in Z . Moreover, N c ⊂ B (0 , r ) ⊂ Z , where r = (2( p + 1)) p I p +2 p and B (0 , r ) is the ball of radius r > Z . Let ρ > N c ⊂ B (0 , ρ ) ⊂ Z . Since thefunction u J ( u ) is uniformly continuous on bounded sets, there exists ǫ > u, v ∈ B (0 , ρ ) and k u − v k Z < ǫ then | J ( u ) − J ( v ) | < ρ . Considering theneighborhoods I = ( d ( c ) − ρ, d ( c ) + ρ ) and B ǫ ( N c ) of d ( c ) and N c , respectively,we have that if u ∈ B ǫ ( N c ) then J ( u ) ∈ I . Therefore v is well defined on B ǫ ( N c )and satisfies v ( ϕ ) = c , for all ϕ ∈ N c . (cid:3) Here is the crucial inequality in the study of stability.
Lemma 3.6.
Let c > and suppose that d ′′ ( c ) > . Then for all u ∈ B ǫ ( N c ) andany ϕ ∈ N c , E ( u ) − E ( ϕ ) + v ( u ) ( F ( u ) − F ( ϕ )) ≥ d ′′ ( c ) | v ( u ) − c | . Proof.
For ω >
0, let I ω be the functional I ω ( ϕ ) = 12 Z R (cid:0) ωϕ + ϕ H ϕ x + ϕ y (cid:1) dxdy. It follows that E ( u ) + v ( u ) F ( u ) = I v ( u ) ( u ) − p + 1)( p + 2) J ( u ) . Let ϕ ω denote any element of N ω . It is easy to see that J ( u ) = J (cid:0) ϕ v ( u ) (cid:1) because d ( v ( u )) = p p +1)( p +2) J ( u ) for u ∈ B ǫ ( N c ) and d ( v ( u )) = p p +1)( p +2) J (cid:0) ϕ v ( u ) (cid:1) . Itthus transpires that I v ( u ) ( u ) ≥ I v ( u ) (cid:0) ϕ v ( u ) (cid:1) . A Taylor expansion of d around the value c yields E ( u ) + v ( u ) F ( u ) ≥ I v ( u ) (cid:0) ϕ v ( u ) (cid:1) − p + 1)( p + 2) J (cid:0) ϕ v ( u ) (cid:1) = d ( v ( u )) ≥ d ( c ) + F ( ϕ )( v ( u ) − c ) + 14 d ′′ ( c ) | v ( u ) − c | = E ( ϕ ) + v ( u ) F ( ϕ ) + 14 d ′′ ( c ) | v ( u ) − c | , and the lemma follows. (cid:3) Before proving stability, we state a well-posedness result for (1.1). This can beproved in several standard ways, for example by using a parabolic regularization(see [26] and [15]).
Theorem 3.7.
Let s > . Then for any u ∈ H s ( R ) , there exist T = T ( k u k H s ) > and a unique solution u ∈ C ([0 , T ]; H s ( R )) of equation (1.1) with u (0) = u . Inaddition, u ( t ) depends continuously on u in the H s − norm and satisfies E ( u ( t )) = E ( u ) , F ( u ( t )) = F ( u ) , for all t ∈ [0 , T ) . When 0 < p < , the stability in Z of the set of minimizers N c is establishednext. Theorem 3.8.
Let c > , s > , < p < and λ = (cid:0) p + 1) I (cid:1) p +2 p . Thenthe set N c = M λ is Z -stable with regard to the flow of the BO-ZK equation.That is, for any positive ǫ , there is a positive δ = δ ( ǫ ) such that if u ∈ H s and inf ϕ ∈ N c k u − ϕ k H s ≤ δ , then the solution u ( t ) of (1.1) with u (0) = u satisfies sup t ≥ inf ψ ∈ N c k u ( t ) − ψ k Z ≤ ǫ. Proof.
Assume that N c is Z -unstable with regard to the flow of the BO-ZK equa-tion. Then, there is a sequence of initial data u k (0) ∈ H s (cid:0) R (cid:1) such thatinf ϕ ∈ N c k u k (0) − ϕ k H s ≤ k and sup t ∈ [0 ,T ) inf ψ ∈ N c k u k ( t ) − ψ k Z ≥ ǫ, (3.6)where u k ( t ) is the solution of (1.1) with initial data u k (0). By continuity in t , forall k large enough, there are times t k such thatinf ϕ ∈ N c k u k ( t k ) − ϕ k Z = ǫ . (3.7)Since E and F are conserved quantities, it follows from (3.6) that | E ( u k ( t k )) − E ( ϕ k ) | = | E ( u k (0)) − E ( ϕ k ) | → , (3.8) | F ( u k ( t k )) − F ( ϕ k ) | = | F ( u k (0)) − F ( ϕ k ) | → , (3.9)as k → + ∞ . In this circumstance, Lemma 3.6 implies that E ( u k ( t k )) − E ( ϕ k ) + v ( u k ( t k )) (cid:0) F ( u k ( t k )) − F ( ϕ k ) (cid:1) ≥ d ′′ ( c ) | v ( u k ( t k )) − c | , for all k large enough. Since { u k ( t k ) } is uniformly bounded in k , the right-handside of the last inequality goes to zero as k → + ∞ on account of (3.8) and (3.9).This in turn implies that v ( u k ( t k )) → c as k → + ∞ . Hence, by the definition of v and continuity of d , we must havelim k → + ∞ J ( u k ( t k )) = 2( p + 1)( p + 2) p d ( c ) . (3.10)On the other hand, Lemma 3.4 implies that I ( u k ( t k )) = E ( u k ( t k )) + c F ( u k ( t k )) + 1( p + 1)( p + 2) J ( u k ( t k ))= d ( c ) + E ( u k ( t k )) − E ( ϕ k ) + c ( F ( u k ( t k )) − F ( ϕ k ))+ 1( p + 1)( p + 2) J ( u k ( t k )) . The limit (3.10) then yieldslim k → + ∞ I ( u k ( t k )) = p + 2 p d ( c ) = (cid:0) p + 1) (cid:1) p I p +2 p . (3.11)Defining ϑ k ( t k ) = (cid:16) J (cid:0) u k ( t k ) (cid:1)(cid:17) − p +2 u k ( t k ) , it is seen that J ( ϑ k ( t k )) = 1. Combining (3.10), (3.11) and Lemma 3.4 leads tolim k → + ∞ I ( ϑ k ( t k )) = I . (3.12) TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 15
Hence { ϑ k ( t k ) } is a minimizing sequence for I . Thus, from Theorem 3.2, thereexists a sequence { ψ k } ⊂ M such thatlim k → + ∞ k ϑ k ( t k ) − ψ k k Z = 0 . (3.13)The Lagrange multiplier theorem then implies there is a sequence { θ k } ⊂ R suchthat H ( ψ k ) x + cψ k − ( ψ k ) yy = θ k ( p + 2) ψ p +1 k . (3.14)In other words, 2 I = θ k ( p + 2), which implies θ k = θ for all k . Write ϕ k = µψ k with µ p = θ ( p + 1)( p + 2) = 2( p + 1) I . Then the ϕ k satisfy (1.2) and 2( p + 1) I ( ϕ k ) = J ( ϕ k ) = µ p +2 so that ϕ k ∈ N c forall k . Additionally, (3.10)-(3.13) and Lemma 3.4 together allow the conclusion k u k ( t k ) − ϕ k k H s = J (cid:0) u k ( t k ) (cid:1) p +2 (cid:13)(cid:13)(cid:13) J (cid:0) u k ( t k ) (cid:1) − p +2 (cid:0) u k ( t k ) − ϕ k (cid:1)(cid:13)(cid:13)(cid:13) H s ≤ J (cid:0) u k ( t k ) (cid:1) p +2 (cid:16)(cid:13)(cid:13) ϑ k ( t k ) − µ − ϕ k (cid:13)(cid:13) H s + µ − k ϕ k k H s − J (cid:0) u k ( t k ) (cid:1) − p +2 (cid:17) . This in turn implies that lim k → + ∞ k u k ( t k ) − ϕ k k Z = 0 , which contradicts (3.7) and completes the proof of the Theorem. (cid:3) Decay and Regularity
To investigate the regularity and the spatial asymptotics of the solitary-wavesolutions of (1.1), it is convenient to take the Fourier transform of equation (1.2)for the solitary-wave in both x and y . If ( ξ, η ) are the variables dual to ( x, y ) byway of the Fourier transform, then (1.2) implies that b ϕ = b g c − α | ξ | + εη , where g = − p + 1 ϕ p +1 . (4.1)Taking the inverse Fourier transform then yields ϕ = − p + 1 Z R K (cid:0) x − s, y − t (cid:1) ϕ p +1 ( s, t ) dsdt. (4.2)Properties of the integral kernel K in (4.2) will be central in the analysis tofollow. Here are few standard properties of anisotropic Sobolev spaces that will behelpful in expressing useful aspects of K . Lemma 4.1. If s i > / , for i = 1 , , then H s ,s is an algebra. Lemma 4.2.
Let s ij , ≤ i, j ≤ and θ ∈ [0 , be given real numbers with s j ≤ s j , j = 1 , . Define ̺ j = θs j + (1 − θ ) s j for j = 1 , . Then, H ̺ ,̺ is an interpolationspace between H s ,s and it’s subspace H s ,s . Moreover, if f ∈ H s ,s , then k f k H ̺ ,̺ ≤ k f k θH s ,s k f k − θH s ,s . (4.3) Remark 4.3.
Since b K ( ξ, η ) = 1 c − α | ξ | + η , the Residue Theorem allows us towrite the kernel K as an integral, namely K ( x, y ) = K c ( x, y ) = C Z + ∞ | α |√ tα t + x e − (cid:16) ct + y t (cid:17) dt, (4.4) where C > is independent of α , x and y . Fubini’s theorem can then be used toshow that k K k L = C Z + ∞ Z R | α |√ tα t + x e − (cid:16) ct + y t (cid:17) dxdydt = C ( α ) Z + ∞ e − ct dt. In consequence of representation (4.4), the following facts about K become clear. Lemma 4.4.
The kernel K is positive, an even function of both x and y , monotonedecreasing in both | x | and | y | , tends to zero as | ( x, y ) | → ∞ and is C ∞ away fromthe origin. Furthermore, b K ∈ L p ( R ) for any p ∈ (3 / , + ∞ ] and K ∈ L p ( R ) , forany p ∈ [1 , . (However, while K ( x, y ) is symmetric in both x and y , it is notradially symmetric.) Lemma 4.5. K ∈ H s , (cid:0) R (cid:1) ∩ H ,s (cid:0) R (cid:1) for any s < and s < . Moreover, K ∈ H r,s (cid:0) R (cid:1) ∩ H s ,s (cid:0) R (cid:1) , where rs + ss = s s and r ∈ [0 , . Lemma 4.6. (i) b K ∈ H s , (cid:0) R (cid:1) ∩ H ,s (cid:0) R (cid:1) , for any s < and s ∈ R .Moreover, b K ∈ H r,s (cid:0) R (cid:1) ∩ H ( s ,s ) (cid:0) R (cid:1) , where rs + ss = s s and r ∈ [0 , .(ii) b K ∈ H ( s ,s ) p (cid:0) R (cid:1) , for any s < p , p ≥ and s ∈ R .(iii) | x | s | y | s K ∈ L p (cid:0) R (cid:1) , for any s , s ≥ and ≤ p ≤ ∞ such that s < − p and s + s > − p . With these facts about K in hand, the solitary-wave solutions of the BO-ZKequation (1.1) now become the focus of attention. Theorem 4.7.
Let p be a positive integer. Any solitary-wave solution ϕ of (1.1)belongs to H ( k ) r , for all k ∈ N and all r ∈ [1 , + ∞ ] . In particular, the solitary-wave solutions of the BO-ZK equation are continuous, bounded and tend to zero atinfinity.Proof. Formula (4.1) implies that ϕ ∈ H , ( R ) ∩ H , ( R ) ∩ H , ( R ). Lemma 4.2and the embedding (1.8) then imply that ϕ ∈ H s, − s ) ( R ), for any s ∈ [0 , (cid:3) More detailed aspects of the solitary-wave solutions of (1.1) are now addressed.Interest will focus first upon their symmetry properties. For u : R → R + , u ♯ willdenote the Steiner symmetrization of u with respect to { x = 0 } and u ∗ the Steinersymmetrization of u with respect to { y = 0 } (see, for example, [10, 29, 44]). Noticethat u ♯ ∗ = u ∗ ♯ is a function symmetric with respect to both the x - and y -axis. Lemma 4.8. If f ∈ Z , then | f | lies in Z and I ( | f | ) ≤ I ( f ) .Proof. If g = | f | , then for any c > h f, K ∗ f i ≤ h g, K ∗ g i , where K = K c . It thus transpires that Z R b K ( ξ, η ) (cid:12)(cid:12)(cid:12) b f ( ξ, η ) (cid:12)(cid:12)(cid:12) dξdη = h f, K ∗ f i ≤ h g, K ∗ g i = Z R b K ( ξ, η ) | b g ( ξ, η ) | dξdη. Since (cid:13)(cid:13) b f (cid:13)(cid:13) L = (cid:13)(cid:13)b g (cid:13)(cid:13) L , it follows that Z R c (cid:16) − c b K (cid:17) | b g ( ξ, η ) | dξdη ≤ Z R c (cid:16) − c b K (cid:17) (cid:12)(cid:12)(cid:12) b f ( ξ, η ) (cid:12)(cid:12)(cid:12) dξdη. (4.5) TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 17
Taking the limit as c → + ∞ on both sides of (4.5), the Monotone ConvergenceTheorem yields Z R (cid:0) | ξ | + η (cid:1) | b g ( ξ, η ) | dξdη ≤ Z R (cid:0) | ξ | + η (cid:1) (cid:12)(cid:12)(cid:12) b f ( ξ, η ) (cid:12)(cid:12)(cid:12) dξdη, (4.6)which shows that | f | ∈ Z and that I ( | f | ) ≤ I ( f ). (cid:3) Corollary 4.9.
For c > , there is always a non-negative solitary-wave solution ϕ c of the BO-ZK equation.Proof. Theorem 2.7 assures that there are solitary-wave solutions ψ , say. The lastresult shows that if ψ ∈ M λ , then so is ϕ = | ψ | . (cid:3) If p = km where m is odd and k and m relatively prime it follows from the formula ϕ = 1 p + 1 K ∗ ϕ p +1 (4.7)that if k is odd, then necessarily all solitary-wave solutions are non-negative. Thisis false if k is even, however. Indeed, in this case, if ϕ is a solitary wave, then so is − ϕ . Hence, when k is even, there are always at least two solitary-wave solutions,one positive and one negative. Of course, when k is even, it is also the case that J ( | f | ) = J ( f ). Lemma 4.10. If f ∈ Z is non-negative, it’s Steiner symmetrizations f ♯ and f ∗ also lie in Z . Moreover, I ( f ♯ ) ≤ I ( f ) and I ( f ∗ ) ≤ I ( f ) .Proof. Remark first that K ♯ = K = K ∗ . The Reisz-Sobolev rearrangement in-equality (see [10, 29, 44]) implies that Z R f ( x, y ) f ( s, t ) K ( x − s, y − t ) ds dt dx dy ≤ Z R f ♯ ( x, y ) f ♯ ( s, t ) K ( x − s, y − t ) ds dt dx dy. In the Fourier transformed variables, this amounts to Z R b K ( ξ, η ) (cid:12)(cid:12)(cid:12) b f ( ξ, η ) (cid:12)(cid:12)(cid:12) dξdη ≤ Z R b K ( ξ, η ) (cid:12)(cid:12)(cid:12) b f ♯ ( ξ, η ) (cid:12)(cid:12)(cid:12) dξdη. On the other hand, the fact that symmetrization does not change the measuretheoretic properties of f implies that (cid:13)(cid:13) b f (cid:13)(cid:13) L ( R ) = (cid:13)(cid:13) f (cid:13)(cid:13) L ( R ) = (cid:13)(cid:13) f ♯ (cid:13)(cid:13) L ( R ) = (cid:13)(cid:13) b f ♯ (cid:13)(cid:13) L ( R ) . This together with the analysis in Lemma 4.8 shows that f ♯ ∈ Z and that I ( f ♯ ) ≤ I ( f ). The same argument applies to f ∗ . (cid:3) Since Steiner symmetrization preserves the L p +2 − norm, it follows that J ( ϕ ) = J ( ϕ ♯ ). In consequence of Lemma 4.10, I λ ≤ I (cid:0) ϕ ♯ (cid:1) ≤ I ( ϕ ) = I λ . Therefore ϕ ♯ ∈ M λ . The same argument shows that ϕ ∗ ∈ M λ . Corollary 4.11.
There are non-negative, solitary-wave solutions of the BO-ZKequation (1.1) that are symmetric with respect to both the propagation directionand the transverse direction and are monotone decreasing in both | x | and | y | . Proof.
By Theorems 2.2 and 4.7, there is a non-negative function ϕ satisfying(1.2). Since Steiner symmetrization preserves the L p +2 − norm, it follows that J ( ϕ ) = J ( ϕ ♯ ) = J ( ϕ ♯ ∗ ). On the other hand, because of Lemma 4.10, the dou-ble rearrangement ϕ ♯ ∗ has the property that I λ ≤ I (cid:0) ϕ ♯ ∗ (cid:1) ≤ I ( ϕ ♯ ) ≤ I ( ϕ ) = I λ . Therefore, ϕ ♯ ∗ is a non-negative solitary-wave solution of equation (1.1) which issymmetric with respect to both { x = 0 } and { y = 0 } and which is monotonedecreasing with respect to both | x | and | y | . (cid:3) Remark 4.12.
One may also obtain symmetry properties of the solitary-wave so-lutions of (1.1) by using the reflection method and a unique continuation argument(see [40] and [21] ). Spatial Asymptotics
Attention is now turned to the spatial decay properties of the solitary-wavesolutions of (1.1). In this analysis, we follow the lead of [9].
Lemma 5.1.
Let j ∈ N . Suppose also that ℓ and m are two constants satisfying < ℓ < m − j . Then there exists C > , depending only on ℓ and m , such that forall ǫ ∈ (0 , , we have Z R j | a | ℓ (1 + ǫ | a | ) m (1 + | b − a | ) m d a ≤ C | b | ℓ (1 + ǫ | b | ) m , ∀ b ∈ R j , | b | ≥ , (5.1) and Z R j d a (1 + ǫ | a | ) m (1 + | b − a | ) m ≤ C (1 + ǫ | b | ) m , ∀ b ∈ R j . (5.2)The proof of this elementary lemma is essentially the same as the proof of Lemma3.1.1 in [9] (see [19]). Theorem 5.2.
Let ϕ be a solitary-wave solution of (1.2).(i) For all q ∈ (3 / , + ∞ ) , ℓ ∈ [0 , ̺ ≥ , | x | ℓ | y | ̺ ϕ ( x, y ) ∈ L q (cid:0) R (cid:1) .(ii) For all q ∈ (3 / , + ∞ ) and any θ ∈ [0 , , | ( x, y ) | θ ϕ ( x, y ) ∈ L q (cid:0) R (cid:1) .(iii) And finally, ϕ ∈ L (cid:0) R (cid:1) .Proof. (i) For q ∈ (1 ,
3) and 1 − q < s < − q , let ℓ ∈ h , s − q (cid:17) . Also, for s > − q , choose ̺ ∈ h , s − q (cid:17) . For 0 < ǫ <
1, define h ǫ by h ǫ ( x, y ) = A ( x, y ) ϕ ( x, y ) , where A ( x, y ) = | x | ℓ | y | ̺ (1 + ǫ | x | ) s (1 + ǫ | y | ) s , Then, by using the explicit representation of h ǫ , it is easy to check that h ǫ ∈ L q ′ (cid:0) R (cid:1) , where q ′ = qq − . H¨older’s inequality and (4.2) then implies that | ϕ ( x, y ) | ≤ C ( s , s , q ) (cid:18)Z R | G x,y ( z, w ) | q ′ dzdw (cid:19) q ′ , where G x,y ( z, w ) = g ( ϕ )( z, w ) (cid:0) | x − z | (cid:1) s (cid:0) | y − w | (cid:1) s , TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 19 g ( t ) = t p +1 p +1 and C := C ( s , s , p ) = (cid:13)(cid:13) (1 + | x | ) s (1 + | y | ) s K (cid:13)(cid:13) L q ( R ) < ∞ . This last constant is finite thanks to Lemma 4.6. Since the solitary wave ϕ convergesto the rest state as | ( x, y ) | → + ∞ , it follows that for every δ >
0, there exists R δ > | ( x, y ) | ≥ R δ , then (cid:12)(cid:12) g ( ϕ )( x, y ) (cid:12)(cid:12) ≤ δ | ϕ ( x, y ) | . An application of H¨older’s inequality yields Z R \ B (0 ,R δ ) | h ǫ ( x, y ) | q ′ dxdy = Z R \ B (0 ,R δ ) | h ǫ ( x, y ) | q ′ − r A r ( x, y ) | ϕ ( x, y ) | r dxdy ≤ C r Z R \ B (0 ,R δ ) | h ǫ ( x, y ) | q ′ − r A r ( x, y ) k G x,y k rL q ′ ( R ) ( x, y ) dxdy ≤ C r k h ǫ k q ′ − rL q ′ ( R \ B (0 ,R δ )) (cid:13)(cid:13)(cid:13) A k G x,y k L q ′ ( R ) (cid:13)(cid:13)(cid:13) rL q ′ ( R \ B (0 ,R δ )) . Because h ǫ ∈ L q ′ (cid:0) R (cid:1) , the latter inequality implies k h ǫ k rL q ′ ( R \ B (0 ,R δ )) ≤ C r (cid:13)(cid:13)(cid:13) A k G x,y k L q ′ ( R ) (cid:13)(cid:13)(cid:13) rL q ′ ( R \ B (0 ,R δ )) , which is to say, Z R \ B (0 ,R δ ) | h ǫ ( x, y ) | q ′ dxdy ≤ C q ′ Z R \ B (0 ,R δ ) A q ′ ( x, y ) k G x,y k q ′ L q ′ ( R ) dxdy. Fubini’s theorem and Lemma 5.1 combine to show that Z R \ B (0 ,R δ ) A q ′ ( x, y ) k G x,y k q ′ L q ′ ( R ) ( x, y ) dxdy = Z R (cid:12)(cid:12) g ( ϕ )( z, w ) (cid:12)(cid:12) q ′ Z R \ B (0 ,R δ ) A q ′ ( x, y )(1 + | x − z | ) q ′ s (1 + | y − w | ) q ′ s dxdy ! dzdw ≤ C Z R \ B (0 ,R δ ) (cid:12)(cid:12) g ( ϕ )( z, w ) (cid:12)(cid:12) q ′ A q ′ ( z, w ) dzdw + Z B (0 ,R δ ) (cid:12)(cid:12) g ( ϕ )( z, w ) (cid:12)(cid:12) q ′ Z R \ B (0 ,R δ ) A q ′ ( x, y )(1 + | x − z | ) q ′ s (1 + | y − w | ) q ′ s dxdy ! dzdw, (5.3)where we used (5.1) (with j = 1) to show that for | ( z, w ) | large, Z R \ B (0 ,R δ ) A q ′ ( x, y )(1 + | x − z | ) q ′ s (1 + | y − w | ) q ′ s dxdy ≤ C A q ′ ( z, w ) . The second integral on the right-hand side of (5.3) is bounded by a constant, say C ′ , depending on ϕ and R δ , but independent of ǫ . Therefore, by using the fact that (cid:12)(cid:12) g ( ϕ )( x, y ) (cid:12)(cid:12) ≤ δ | ϕ ( x, y ) | on R \ B (0 , R δ ), there obtains Z R \ B (0 ,R δ ) | h ǫ ( x, y ) | q ′ dxdy ≤ C q ′ Cδ q ′ Z R \ B (0 ,R δ ) | h ǫ ( x, y ) | q ′ dxdy + C ′ ! . Choosing δ such that CδC q ′ <
1, the last inequality entails that Z R \ B (0 ,R δ ) | h ǫ ( x, y ) | q ′ dxdy ≤ C ′′ , (5.4)where C ′′ is a constant independent of ǫ . Letting ǫ → Z R \ B (0 ,R δ ) | x | ℓq ′ | y | ̺q ′ | ϕ ( x, y ) | q ′ dxdy ≤ C. Hence | x | ℓ | y | ̺ ϕ ( x, y ) ∈ L q ′ (cid:0) R (cid:1) , for q ′ = qq − .In the limits q → q →
3, we have ℓ → q ′ ∈ (3 / , + ∞ ). This provespart (i) of the theorem.(ii) This follows directly from (i).(iii) Let s > g , δ and R δ be as defined in the proof of (i). For ǫ > A be A ǫ ( x, y ) = 1(1 + ǫ | ( x, y ) | ) s . Fubini’s Theorem, Lemma 5.1 and the fact that ϕ , A ǫ ∈ L (cid:0) R (cid:1) so that the product ϕ A ǫ ∈ L (cid:0) R (cid:1) allow us to adduce the inequalities Z R \ B (0 ,R δ ) | ϕ ( x, y ) | A ǫ ( x, y ) dxdy ≤ Z R (cid:12)(cid:12) g ( ϕ )( z, w ) (cid:12)(cid:12) Z R \ B (0 ,R δ ) A ǫ ( x, y ) K ( x − z, y − w ) dxdy ! dzdw ≤ Z R (cid:12)(cid:12) g ( ϕ )( z, w ) (cid:12)(cid:12) Z R \ B (0 ,R δ ) A − ( x − z, y − w ) K ( x − z, y − w ) dxdy ! × Z R \ B (0 ,R δ ) A ( x − z, y − w ) A ǫ ( x, y ) dxdy ! dzdw ≤ C ( s ) C Z R (cid:12)(cid:12) g ( ϕ )( z, w ) (cid:12)(cid:12) A ǫ ( z, w ) dzdw ≤ C ( s ) C δ Z R \ B (0 ,R δ ) | ϕ ( z, w ) | A ǫ ( z, w ) dzdw + C ( s ) C Z B (0 ,R δ ) (cid:12)(cid:12) g ( ϕ )( z, w ) (cid:12)(cid:12) dzdw. Letting ǫ →
0, Fatou’s lemma together with the restriction on δ leads to the con-clusion that ϕ ∈ L (cid:0) R (cid:1) . (cid:3) Theorem 5.2, identity (4.7) and the elementary inequality | t | θ ≤ C (cid:0) | t − s | θ + | s | θ (cid:1) , for θ ≥ . (5.5)imply the following. Corollary 5.3.
Suppose that ϕ ∈ L ∞ (cid:0) R (cid:1) satisfies (1.2) and ϕ → at infinity.Then(i) | x | ℓ | y | ̺ ϕ ( x, y ) ∈ L ∞ (cid:0) R (cid:1) , for all ℓ ∈ [0 , and any ̺ ≥ ,(ii) | ( x, y ) | θ ϕ ( x, y ) ∈ L ∞ (cid:0) R (cid:1) , for all θ ∈ [0 , . TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 21
The aim now is to display even stronger decay properties in the x -variable forsolitary-wave solutions of the BO-ZK equation. These results are developed in asequence of lemmas.
Lemma 5.4. | x | | y | ̺ K ∈ L ∞ (cid:0) R (cid:1) , for any ̺ ≥ .Proof. In view of the explicit form of K , the proof is straightforward. (cid:3) Corollary 5.5. | x | ℓ | y | ̺ ϕ ( x, y ) ∈ L ∞ (cid:0) R (cid:1) , for any ≤ ℓ ≤ and any ̺ ≥ .Proof. The proof is based on a standard bootstrapping argument. Decay in the y -variable is not in question, so without loss of generality, take it that that ̺ = 0.Setting γ = min { , p + 1 } and making use of the inequality | x | γ | ϕ | . | x | γ | K | ∗ | g ( ϕ ) | + | K | ∗ || x | γ | g ( ϕ ) || , (5.6)where g ( t ) = t p +1 p +1 , we obtain from Corollary 5.3, Lemma 5.4 and Theorem 5.2that | x | γ ϕ ∈ L ∞ ( R ). The proof is compete if γ = 2. If γ <
2, then define γ = min { , ( p + 1) } and repeat the above argument to show | x | γ ϕ ∈ L ∞ ( R ).Continuing in this manner, one concludes that | x | ϕ ∈ L ∞ ( R ) after a finite numberof steps. (cid:3) The following corollary follows from (5.5), Corollary 5.3 and Theorem 5.2.
Corollary 5.6. (i) | x | ℓ | y | ̺ ϕ ( x, y ) ∈ L (cid:0) R (cid:1) , for all ℓ ∈ [0 , and any ̺ ≥ ,(ii) | ( x, y ) | θ ϕ ( x, y ) ∈ L (cid:0) R (cid:1) , for all θ ∈ [0 , . Lemma 5.7.
For any ≤ r, q < ∞ , there is σ > such that for all σ ∈ [0 , σ ) and s ∈ ( − r − q , − r ) , we have | x | s e σ | y | K ∈ L rx L qy ( R ) ∩ L qy L rx ( R ) . Proof.
It suffices to choose σ = q cq , where c is the wave velocity and use (4.4). (cid:3) The next result is a consequence of another of Young’s inequalities, namely k f ∗ g k L qy L rx ( R ) ≤ k f k L q y L r x ( R ) k g k L q y L r x ( R ) , where 1 ≤ r, q, r , q , r , q ≤ ∞ , 1 + r = r + r and 1 + q = q + q . Corollary 5.8. ϕ ∈ L rx L qy ( R ) ∩ L qy L rx ( R ) , for any ≤ r ≤ ∞ satisfying r + 12 q > . Here is the main result about the spatial decay of the solitary-wave solutions.
Theorem 5.9.
Let σ > be in Lemma 5.7. Then, for any σ ∈ [0 , σ ) and any ≤ s < / , it transpires that | x | s e σ | y | ϕ ( x, y ) ∈ L (cid:0) R (cid:1) ∩ L ∞ (cid:0) R (cid:1) .Proof. Without loss of generality, assume that s = 0. By using Lemma 5.7 and theproof of Corollary 3.14 in [9], with natural modifications, it may be seen that thereis a e σ ≥ σ such that e σ | y | ϕ ( x, y ) ∈ L (cid:0) R (cid:1) , for any σ < e σ . The inequality | ϕ ( x, y ) | e σ | y | ≤ Z R | K ( x − z, y − w ) | e σ | y − w | | ϕ ( z, w ) | e σ | w | | ϕ ( z, w ) | p dzdw (5.7)and the facts ϕ ( x, y ) e σ | y | ∈ L (cid:0) R (cid:1) , ϕ ∈ L ∞ (cid:0) R (cid:1) and K ( x, y ) e σ | y | ∈ L (cid:0) R (cid:1) , forany σ < σ , entails that ϕ ( x, y ) e σ | y | ∈ L ∞ (cid:0) R (cid:1) , for the same range of σ . (cid:3) Finally, the following theorem deals with the analyticity of the solitary-wavesolutions. Of course, for this, one needs to restrict p so that z z p is analytic ina full neighborhood of the origin in C . Theorem 5.10.
Let ≤ p < be an integer. Then, there is a σ > and aholomorphic function f of two variables z and z , defined in the domain H σ = (cid:8) ( z , z ) ∈ C ; | Im(z ) | < σ, | Im(z ) | < σ (cid:9) such that f ( x, y ) = ϕ ( x, y ) for all ( x, y ) ∈ R . Similar results are obtained by the same method for related evolution equationsin [34] and [9]. Results of this nature for dispersive equations made via Gevrey-spaceanalysis appear in [8] (and see also the reference therein).
Proof.
By the Cauchy-Schwarz inequality, Theorem 4.7 implies that b ϕ ∈ L (cid:0) R (cid:1) .Equation (1.2) implies in turn that | ξ | | b ϕ | ( ξ, η ) ≤ p +1 z }| { | b ϕ | ∗ · · · ∗ | b ϕ | ( ξ, η ) , (5.8) | η | | b ϕ | ( ξ, η ) ≤ | b ϕ | ∗ · · · ∗ | b ϕ | | {z } p +1 ( ξ, η ) . (5.9)Denote by T the correspondence T ( | b ϕ | ) = | b ϕ | and, for m ≥ T m +1 ( | b ϕ | ) = T m ( | b ϕ | ) ∗ | b ϕ | . A straightforward induction yields r m | b ϕ | ( ξ, η ) ≤ ( m − p + 1) m − T mp +1 ( | b ϕ | )( ξ, η ) , (5.10)where r = | ( ξ, η ) | . It follows that r m | b ϕ | ( ξ, η ) ≤ ( m − p + 1) m − k T mp +1 ( | b ϕ | ) k L ∞ ( R ) ≤ ( m − p + 1) m − k T mp ( | b ϕ | ) k L ( R ) k b ϕ k L ( R ) ≤ ( m − p + 1) m − k b ϕ k mpL ( R ) k b ϕ k L ( R ) . Let a m = ( p + 1) m − k b ϕ k mpL ( R ) k b ϕ k L ( R ) m , so that a m +1 a m −→ ( p + 1) k b ϕ k pL ( R ) , as m → + ∞ . In consequence, the series P ∞ m =0 t m r m | b ϕ | ( ξ, η ) /m ! converges uni-formly in L ∞ ( R ) provided 0 < t < σ = p +1 k b ϕ k − pL ( R ) . Hence e tr b ϕ ( ξ, η ) ∈ L ∞ ( R ),for t < σ . Now define the function f ( z , z ) = Z R e i ( ξz + ηz ) b ϕ ( ξ, η ) dξdη. By the Paley-Wiener Theorem, f is well defined and analytic in H σ while Plancherel’sTheorem assures that f ( x, y ) = ϕ ( x, y ) for all ( x, y ) ∈ R . This proves the theo-rem. (cid:3) TABILITY OF SOLITARY WAVES OF THE BO–ZK EQUATION 23
Acknowledgments
The financial support of the research council of Damghan university with thegrant number 93/MATH/125/227 is acknowledged by the first author. AP waspartially supported by CNPq. Both AP and JB were supported by FAPESP duringpart of this collaboration. JB, who also received support from the University ofIllinois at Chicago, thanks the Instituto de Matem´atica, Estat´ıstica e Computa¸c˜aoCient´ıfica at UNICAMP and the Instituto Nacional de Matem´atica Pura e Aplicadafor hospitality during the development of this work.
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