Semiclassical limit for generalized KdV equations before the gradient catastrophe
aa r X i v : . [ m a t h . A P ] J u l Semiclassical limit for generalized KdV equationsbefore the gradient catastrophe
D. Masoero ∗ and A. Raimondo † Abstract
We study the semiclassical limit of the (generalised) KdV equation,for initial data with Sobolev regularity, before the time of the gradientcatastrophe of the limit conservation law. In particular, we show thatin the semiclassical limit the solution of the KdV equation: i) convergesin H s to the solution of the Hopf equation, provided the initial databelongs to H s , ii) admits an asymptotic expansion in powers of thesemiclassical parameter, if the initial data belongs to the Schwartzclass. The result is also generalized to KdV equations with higherorder linearities. Introduction
We consider the following class of partial differential equations ∂ t u = a ( u ) ∂ x u + n X i =1 ε i ∂ i +1 x u, u ( x,
0) = ϕ ( x ) , (G)depending on a family of parameters ε = ( ε , . . . , ε n ) ∈ R n . Here a ( u ) is a smooth function, x ∈ R , and the initial data is independent of ε .We call the above family of equations generalized KdV equations , forit contains as a particular example the KdV equation itself: ∂ t u = u ∂ x u + ε ∂ x u . Other examples are given by the Kawahara equation [15], which isobtained by choosing a ( u ) = u and n = 2 , as well as nonlinear gener-alizations of KdV, for n = 1 and a arbitrary.We are interested in the behaviour of the solutions of (G) as theparameters ε vary. In particular, we consider the behaviour when ε → , in which limit, formally, (G) becomes a quasilinear conservationlaw, of the form: ∂ t u = a ( u ) ∂ x u, u ( x,
0) = ϕ ( x ) . (H) ∗ Grupo de Física Matemática, Complexo Interdisciplinar da Universidade de Lisboa,Av. Prof. Gama Pinto, 2 PT-1649-003 Lisboa, Portugal. [email protected] † SISSA, Via Bonomea 265, 34136 Trieste, Italy. [email protected] he case a ( u ) = u is known as Hopf equation. The problem of study-ing solutions of equation (G) as ε → is known in the literature as thesemiclassical (or singular, or dispersionless) limit. Since solutions ofequation (H) may develop a singularity at a critical time + ≤ t c < ∞ ,we study local-in-time solutions of (G) for those classes of initial data,to be specified below, for which the Cauchy problem for the generalizedKdV equation is locally well-posed uniformly with respect to ε ∈ R n .There is an extensive literature on the initial value problem forgeneralized KdV equations, which is based essentially on two distinctapproaches: the first makes use of the inverse scattering and Riemann-Hilbert methods, and it applies to those equations of the class (G)which are integrable, such as KdV. The second applies to a more gen-eral class of equations, and makes use of fix-point arguments for theassociated integral equation u ( t ; ε ) = W ( t ; ε ) u (0; ε ) + Z t W ( t − s ; ε ) a ( u ( s ; ε )) u x ( s ; ε ) ds , (W) W ( t ; ε ) = exp (cid:0) t n X i =1 ε i ∂ i +1 x (cid:1) . Since the seminal paper of Lax and Levermore [18], the method ofinverse scattering has been successfully used to study the semiclassicallimit of the KdV equation, both before and after the critical time t c of the Hopf equation. For time smaller than t c , rigorous results havebeen obtained for those initial data whose scattering transform canbe computed in the semiclassical limit using the WKB analysis. Inthis case, the corresponding solutions are proven to converge in L tothe solutions of the Hopf equation [18, 22]. If, in addition, the initialdata satisfy some analyticity assumptions, then the powerful nonlinearsteepest-descent analysis [7] applies to the study of the semiclassicallimit. For these initial data, the solutions are known to converge uni-formly - see [6, 4].The integral equation (W) has been a major topic of investigationsince it was used by Kenig, Ponce and Vega [16] to establish localwell-posedness of (G) for polynomial nonlinearity a ( u ) and dispersion ε = 0 . In the particular case of the KdV equation, the authors of[16] have been able to prove local well-posedness in H s , s > . Theirresults were then refined by many authors to obtain local and globalwell-posedness for low-regular initial data, see for instance [3, 17, 5].The theory of equation (W) relies heavily on the dispersive charac-ter of equation (G), being based on the smoothing effects of the linearevolution operator W . Hence, it may be very difficult to apply it to thestudy of the semiclassical limit. Moreover, since the equation (H) maybe ill-posed for s ≤ , it seems unreasonable to study the semiclassicallimit for low-regular solutions.Due to the limitations of the inverse scattering transform and ofthe integral equation (W), in order to deal with the semiclassical limit f the general equation (G) we choose a different approach, namelyKato’s theory of quasi-linear equations [13]. This approach turns outto be particularly suitable to our problem because it allows to treatequation (G) for all values of ε ∈ R n on the same footing, and it isvery robust under perturbations. Following Kato, we consider (G) as a quasilinear equations on aBanach space X , of the form du ( ε ) dt = A ( u ( ε ); ε ) u + f ( u ( ε ); ε ) , ≤ t ≤ T, u (0) = ϕ. (Q)Here A ( y ; ε ) is a linear operator, depending on ε ∈ R n and on someelement y ∈ X . In addition, for any fixed ε and y the operator A ( y ; ε ) generates a C − semigroup on X . In our case, A ( y ; ε ) = a ( y ) ∂ x + n X i =1 ε i ∂ i +1 x , f = 0 , and we choose X = L ( R ) . Kato himself used his theory to constructlocal–in–time solutions of equation (G). In particular, he establishedlocal well-posedness for the KdV equation in H s , s > , both when ε = 0 [14] and when ε = 0 [13]. However, he did not consider thesemiclassical limit.We establish simple conditions, under which the local-in-time solu-tion of the Cauchy problem (G) with initial datum in H s is continuousand N − differentiable with respect to ε .Essentially, we show that: • If s ≥ n + 1 , the local-in-time solution u ( ε ) of (G) is continuouswith respect to ε ∈ R n . • Let K = P ni =1 N i (2 i + 1) and N = P ni =1 N i . If s − K ≥ n + 1 ,then the partial derivative ∂ N u ( ε ) ∂ε N . . . ∂ε N n n exists in H s − K and it is continuous with respect to ε ∈ R n .The above results can be applied to the project, proposed by Du-brovin and Zhang [10, 8], of Hamiltonian perturbations of quasilinearconservation laws (H). Indeed, any equation the form (H) can be writ-ten as an infinite dimensional Hamiltonian system; within this theory,one looks for suitable deformations – depending on arbitrary functionsof u and its derivatives – such that the equation remains Hamiltonian.One simple example is given by KdV, which can be obtained as aHamiltonian perturbation of the Hopf equation. An alternative method, which may be suitable for the study of the semiclassical limit,is given by the Bona-Smith energy method [2]. n addition to the above problem, the project includes a classifica-tion of integrable perturbations and a characterization of the solutionsof the perturbed equations, both before the critical time and in a neigh-borhood of it. It should also be noted that the project applies not onlyto single equations, but also to systems of first oder quasilinear PDEs,[8, 9, 10] .Before critical time, the Dubrovin-Zhang theory provides a way toconstruct solutions of the perturbed equation in terms of solutions ofthe unperturbed one. Without going into detail, one looks for solutionsof the perturbed equations as formal power series u = v + v ε + . . . , and argues that v is a solution of the unperturbed equation, whilethe subsequent coefficients can by obtained from v by a recursive pro-cedure. This construction, although extremely powerful in predictingbehaviour of the solutions, is based on formal identities, and requiresrigorous justification.The paper is organized as follows: after recalling in Section 1 somebasic elements of the theory of C − semigroups, we consider in Section2 the results obtained by Kato in [13], which we will use in order toprove our results.Sections 3 and 4 are the core of the paper. We first prove the exis-tence of a positive time T for which the solutions of the problem (G),with initial data in H s , s ≥ n + 1 , are continuous functions of theparameters ε . In particular, such solutions are continuous as ε → ,implying H s − convergence to the solution of (H) in this limit. Thisresult generalizes the one obtained by Lax and Levermore (before thecritical time) to equations of type (G) – which are not necessarily in-tegrable – and to initial data in the Sobolev space H s .In Section 4 we consider differentiability of solutions of (G) withrespect to ε . Although our result – with suitable modifications – holdsfor every equation of type (G), for simplicity we consider in detail theKdV example only. We show that if the initial datum of the KdVequation lies in the Sobolev space H s , then the solution of the Cauchyproblem is N − times differentiable with respect to ε , for N = ⌊ s/ − ⌋ .In Section 5 we present the Dubrovin-Zhang theory of Hamiltonianperturbations of equation (H), considering those aspects of the theorywhich are directly related to the results of the present paper: the clas-sification results of Hamiltonian perturbations and the construction ofthe solutions of the perturbed equations before critical time.In the last Section we apply the results obtained in Sections 3 and4 to the Dubrovin’s theory. Since equations of type (G) can be seenas Hamiltonian perturbations of (H), we provide – for this class ofequations – a rigorous justification to the heuristic results of Section5. In addition, we find an explicit formula for the coefficient v of the olution of a generic Hamiltonian perturbation of (H) in terms of thesolution v of the unperturbed equation. This is of the form: v = t ∂∂x (cid:18) ( c a ′ ) ′ ( v x ) + 2 c a ′ v xx + t c ( a ′ ) v x v xx + t c ′ ( a ′ ) ( v x ) (1 + t a ′ v x ) (cid:19) , where a = a ( v ) is the non-linearity of (H), and c = c ( v ) is a functioncharacterizing the Hamiltonian perturbation. Notation
Given the real Banach spaces
X, Y, . . . , we let k k X , k k Y , . . . de-note the corresponding norms. L ( Y, X ) denotes the Banach space ofbounded linear operators from Y to X , with norm k k Y,X , while L ( X ) denotes the Banach space of bounded linear operator from X to it-self with the norm k k X . We call D ( A ) the domain of an operator A . L := L ( R ) denotes the Hilbert space of square integrable realfunctions and H s := H s ( R ) , s ≥ denotes the Sobolev space of order s . The symbol ∂ nx denotes the n -th derivative with respect to x or thecorresponding operator on L with domain H n .In the present paper we consider real Banach spaces and real func-tions only. Acknowledgments
We are grateful to Boris Dubrovin, Percy De-ift, Ken McLaughlin and Tamara Grava for encouraging us in ourresearch. A.R. and D.M. thank, respectively, the Grupo de FísicaMatemática da Universidade de Lisboa and the SISSA MathematicalPhysics sector for the kind hospitality.The research was partially supported by the INDAM–GNFM ‘Pro-getto Giovani ’. D.M. is supported by a Postdoc scholarship of theFundação para a Ciência e a Tecnologia, project PTDC/MAT/104173-/2008 (Probabilistic approach to finite and infinite dimensional dynam-ical systems). C − semigroups The theory of C − semigroup is a standard tool in analysis. Following[13] and [20], we recall the elements of the theory we will use in therest of the paper.A one parameter family of linear operators { T ( t ) , ≤ t < ∞} ona Banach space X is a C − semigroup if it is a strongly continuoussemigroup of bounded linear operators, namely, it satisfies: • T (0) = I, T ( t ) T ( s ) = T ( t + s ) , t, s ≥ , • lim t ↓ T ( t ) x = x, ∀ x ∈ X .Here I is the identity operator on X . The linear operator defined by D ( A ) = (cid:26) x ∈ X : lim t ↓ T ( t ) x − xt exists (cid:27) nd Ax = lim t ↓ T ( t ) x − xt , ∀ x ∈ D ( A ) , is the infinitesimal generator of the C − semigroup { T ( t ) } . The oper-ator A is closed and densely defined.A standard theorem shows that for any C − semigroup there existtwo positive constants M ≥ and β ≥ such that k T ( t ) k X ≤ M e βt , ≤ t < ∞ . (1)In particular, a C − semigroup with constants M = 1 , β = 0 is calleda semigroup of contractions . We denote by G ( X, M, β ) the set of in-finitesimal generators of C − semigroups with constants M , β . Remark 1.
Note that if A ∈ G ( X, , β ) , then A − βI ∈ G ( X, , . Later on we will need a perturbation theorem for generators ofsemigroups of contractions; for this purpose we introduce the followingnotions.An operator A on a Hilbert space X is said to be dissipative if forevery x ∈ D ( A ) , we have Re ( Ax, x ) ≤ .If A and B are operators on a Banach space X , we say that B is relatively bounded with respect to A with relative bound ρ ≥ if D ( A ) ⊂ D ( B ) and there exists a σ ≥ such that k Bx k X ≤ ρ k Ax k X + σ k x k X , ∀ x ∈ D ( A ) . (2) Theorem 1.
Let A ∈ G ( X, , be the generator of a C − semigroupof contractions on a Hilbert space X . Let B be dissipative and relativelybounded with respect to A with relative bound ρ < . Then A + B isthe generator of a semigroup of contractions.Proof. See [20], Corollary 3.3The definition of dissipative operator and Theorem 1 can be gener-alised to any Banach space, with slight modifications. However, suchgeneralisation is not necessary in our study. The following exampleswill be useful in the rest of the paper.
Example 1.
Let X be a Banach space and A be an anti-self-adjointoperator. Due to the Stone Theorem, A generates a C − group of uni-tary operators, hence a C − semigroup of contractions. For instance,the derivative operator ∂ n +1 x , n ∈ N , with domain D ( ∂ n +1 x ) = H n +1 ,is an anti-self-adjoint operator on the space L . More generally, theoperator D n +1 ε = n X i =0 ε i ∂ i +1 x , ε = ( ε , . . . , ε n ) ∈ R n , (3) with domain H n +1 is anti-self-adjoint on L for any value of the pa-rameter ε . xample 2. [13] Let f ( x ) be a bounded differentiable function withbounded derivative on the whole real axis, take X = L ( R ) , B = f ( x ) ∂ x and D ( B ) = H . The operator B can be decomposed into an anti-self-adjoint part and a bounded self-adjoint part, B = B + B with: B = ( f ∂ x + 12 f x ) , B = − f x . In particular, B has domain H and is anti-self-adjoint, thus, it gen-erates a C − semigroup of contractions. On the other hand, B is abounded self-adjoint operator with norm k B k L = sup x ∈ R | f x ( x ) | ,and a simple computation shows that B − k B k L I is dissipative. Dueto Theorem 1, we have that B − k B k L I generates a C − semigroupof contractions. Therefore, B ∈ G ( X, , β ) with β = k B k L . In ad-dition, B is relatively bounded with respect to ∂ n +1 x , n ≥ with anyrelative bound ρ > . Due to Theorem 1 and the above discussion, B + ε ∂ n +1 x ∈ G ( X, , β ) , for any ε ∈ R . Example 3.
Let g ( x ) be a continuous bounded function on R , take X = L ( R ) , and let C = g ( x ) be the operator of multiplication by g , with D ( C ) = L . We have that C is a bounded operator, withnorm β ′ = sup x ∈ R | g ( x ) | . Due to Theorem 1, if A ∈ G ( X, , β ) then A + C ∈ G ( X, , β + β ′ ) . In this section we review Kato’s results on quasilinear equations on aBanach space X , of the form dudt = A ( t, u ) u + f ( t, u ) , ≤ t ≤ T, u (0) = ϕ. (4)Here A ( t ; y ) is a linear operator, depending on the time t and on someelement y ∈ X , and such that for any fixed t and y the operator A ( t ; y ) generates a C − semigroup on X . It should be noted that we do notpresent these theorems in their strongest form, but in a form adequateto our purpose. For the reader’s convenience, we follow – as much aspossible – the notation of the original paper [13].First, we consider the linear case, when the operator A and theforcing term f do not depend on u . Theorem 2.
The linear non-homogeneous Cauchy problem dudt = A ( t ) u + f ( t ) , ≤ t ≤ T, u (0) ∈ Y, (5) has a unique solution u ( t ) ∈ C ([0 , T ]; Y ) ∩ C ([0 , T ]; X ) , provided the following assumptions are satisfied: i) X is a Banach space and Y ⊂ X is another Banach space, con-tinuously and densely embedded in X . Moreover, there exists anisomorphism S of Y into X .(ii) There exists a positive β such that A ( t ) ∈ G ( X, , β ) ∀ t ∈ [0 , T ] .(iii) SA ( t ) S − − A ( t ) = B ( t ) ∈ L ( X ) , and t B ( t ) is a continuousoperator valued function.(iv) Y ⊂ D ( A ( t )) so that A ( t ) ∈ L ( Y, X ) , and t A ( t ) is a continu-ous operator valued function.(v) u (0) ∈ Y and f ∈ C ([0 , T ]; Y ) .Proof. See [12], Theorem I and II.The following is a perturbation theorem for the linear equation (5).
Theorem 3.
In addition to the assumptions of Theorem 2, considerthe sequence of Cauchy problems du n dt = A n ( t ) u n + f n ( t ) , ≤ t ≤ T, u n (0) ∈ Y, (6) and suppose that, for any fixed n , the operator A n satisfies conditions(i) through (v) of Theorem 2. Moreover, suppose that(vi) A n ( t ) → A ( t ) strongly in L ( Y, X ) , and sup t ∈ [0 ,T ] k A n ( t ) k Y,X isuniformly bounded in n.(vii) B n ( t ) → B ( t ) strongly in L ( X ) , and sup t ∈ [0 ,T ] k B n ( t ) k X is uni-formly bounded in n.(viii) u n (0) → u (0) in Y and f n → f in C ([0 , T ] , Y ) .Then u n ( t ) → u ( t ) in C ([0 , T ] , Y ) ∩ C ([0 , T ] , X ) , where u n ( t ) is theunique solution of (6) and u ( t ) is the unique solution of (5).Proof. See [12] Theorem V-VI.We now move to the analogue results for quasilinear equations ofthe form (4). For our purposes, it is sufficient to consider only thehomogeneous case, when f = 0 , namely: dudt = A ( t, u ) u, ≤ t ≤ T, u (0) ∈ W ⊂ Y . (7)Here the set W is a bounded subset of Y . Due to the nonlinearity,existence of the solutions is not guaranteed on the whole time interval [0 , T ] , but – in general – only for a smaller time T ′ , with < T ′ ≤ T .The reason we restrict to a bounded subset W is because we expectthe time of existence to depend on the norm of the initial data.Following Kato, we make the following assumptions:(X) X is a reflexive Banach space and Y ⊂ X is another reflexiveBanach space, continuously and densely embedded in X . Thereis an isometric isomorphism S of Y into X . Moreover, we fix aball W ⊂ Y of radius R and centered in . A1) There exists a positive β such that A ( t, y ) ∈ G ( X, , β ) , for all t ∈ [0 , T ] and y ∈ W .(A2) For any t, y ∈ [0 , T ] × W , we have SA ( t ) S − − A ( t ) = B ( t ) ∈ L ( X ) , and k B ( t ) k X ≤ λ . (A3) For all t, y ∈ [0 , T ] × W , we have A ( t ) ∈ L ( Y, X ) . Fixed y ∈ W ,the function t A ( t, y ) is a continuous operator-valued function,and fixed t ∈ [0 , T ] , y → A ( t, y ) is Lipschitz continuous, in thesense that there exists a µ such that k A ( t, y ) − A ( t, z ) k Y,X ≤ µ k y − z k X . Theorem 4.
Suppose conditions (X),(A1),(A2),(A3) are satisfied. Then,the quasilinear homogeneous Cauchy problem (7) has a unique solution u ( t ) ∈ C ([0 , T ′ ]; W ) ∩ C ([0 , T ′ ]; X ) , for some < T ′ ≤ T. In addition, T ′ has a lower bound uniquely de-pending on β, λ , µ , R and monotonically decreasing in each variable.Proof. See [13] Theorem 6.We now state the perturbation theorem in the case of quasilinearequations of type (7).
Theorem 5.
In addition to the assumptions of Theorem 4, considerthe sequence of quasilinear homogeneous Cauchy problems du n dt = A n ( t, u n ) u n , ≤ t ≤ T, u n (0) ∈ W, (8) and assume that conditions (X), (A1), (A2) and (A3) of Theorem 4are satisfied for every n , with constants β, λ , µ independent on n .Moreover, suppose that(A4) k B n ( t, y ) − B n ( t, z ) k X ≤ µ k y − z k Y , uniformly in n .(C1) A n ( t ) → A ( t ) strongly in L ( Y, X ) .(C2) B n ( t ) → B ( t ) strongly in L ( X ) .(C3) u n (0) ∈ W and u n (0) → u (0) in Y , as n
7→ ∞ .Then, there exists a positive time < T ′′ ≤ T , such that there is aunique solution u n ∈ C ([0 , T ′′ ] , W ) ∩ C ([0 , T ′′ ] , X ) of (8), for every n , and a unique solution u of (7) in the same class.Moreover u n ( t ) → u ( t ) in C ([0 , T ′′ ] , W ) ∩ C ([0 , T ′′ ] , X ) , as n
7→ ∞ . The time T ′′ has a lower bound uniquely depending on β, λ , µ , µ , R, and monotonically decreasing in each variable.Proof. See [13], Theorem 7. Continuity of solutions for generalised KdVequations
Here we apply Kato’s theory to study local-in-time solutions of thefollowing family of Cauchy problems: dudt = A ( u ; ε ) u, u (0) ∈ H s , (9) A ( u ; ε ) = a ( u ) ∂ x + n X i =1 ε i ∂ i +1 x , depending on a family of parameters ε = ( ε , . . . , ε n ) ∈ R n . Here a ( u ) is a smooth function, the initial value u (0) is independent on the ε i forall i , and s ≥ n + 1 . We call the above equations generalized KdVequations . We are interested in the behaviour of the solutions of (9) asthe parameters ε vary; in particular, we want to prove the continuityof the solutions with respect to ε in a suitable Banach space.To apply the results of the previous section we choose X = L and Y = H s . Before proving the main result, we collect some known factsabout Sobolev spaces (in one space dimension). Lemma 1. (0) The map Λ s = (1 − ∂ x ) s is an isometric isomorphismof H s , s ≥ into L . The inverse of Λ s is Λ − s .(i) Sobolev embedding (particular case): if u ∈ H s , s > + n , then u is n-times differentiable and there exists a constant c such that k ∂ nx u k L ∞ ≤ c k u k H s . (10) (ii) Algebra property: if u, v ∈ H s , s > , then there exists a constant c ( s ) such that k u v k H s ≤ c ( s ) k u k H s k v k H s . (11) (iii) Schauder estimate: if a : R → R is a smooth function and s > ,then there exists a constant c ( s, a, k u k H s , k v k H s ) such that k a ( u ) − a ( v ) k H s ≤ c ( s, a, k u k H s , k v k H s ) k u − v k H s . (12) (iv) Commutator estimates: let u ∈ H s , s > . Then the operator T u = (Λ s u∂ x − u∂ x Λ s )Λ − s is bounded on L and there exists aconstant c ( s ) such that k T u k L ≤ c ( s ) k u k H s . (13) Proof.
For ( iv ) see [13], Lemma A.2. For all other statements, see [21]Appendix A.We prove the following heorem 6. Let W be a ball of radius R in H s , such that u (0) ∈ W .There is a time T such that for any ε ∈ R n the Cauchy problem (9)has a unique solution u ( t ; ε ) ∈ C ([0 , T ] , W ) ∩ C ([0 , T ] , H s − (2 n +1) ) . The map ε u ( t ; ε ) from R n to C ([0 , T ] , W ) ∩ C ([0 , T ] , H s − (2 n +1) ) is continuous.Proof. As a first step we prove that u ( t ; ε ) exists and is continuous asa map from R n to C ([0 , T ] , H s ) ∩ C ([0 , T ] , L ) . To prove this, it issufficient to show that all conditions of Theorems 4 and 5 are satisfieduniformly in R n , assuming X = L ( R ) and Y = H s ( R ) .(X) It is trivially satisfied since X , Y are Hilbert spaces. The requiredisometry can be chosen to be Λ s .(A1) Following Example 2 above, we see that A ( u ; ε ) ∈ G ( X, , β ( R )) ,with β ( R ) = 12 sup u ∈ W sup x ∈ R | ∂ x a ( u ) | ≤ R sup | x |≤ R a ′ ( x ) . (A2) Since Λ s commutes with the derivative operator, we have B ( u, ε ) := (Λ s A ( ε ) − A ( ε )Λ s )Λ − s = T a ( u ) , where T is defined as in Lemma 1 (iv). Due to the commutatorestimate and the Schauder estimate, we have that k B ( u ; ε ) k L ( R ) ≤ c ( s, a, R ) R = λ ( R ) . (A3) A ( u ; ε ) is a continuous operator from H s to H l , for ≤ l ≤ s − (2 n + 1) . Indeed, P ni =1 ε i ∂ i +1 x is continuous from H s to H l and a ( u ) ∂ x is continuous from H s to H s − due to the Schauderestimate and the algebra property of Sobolev spaces. A simplecomputation shows that k ( A ( u ; ε ) − A ( v ; ε )) k H S ,L ≤ c ( s ) sup | x |≤ R | a ′ ( x ) | k u − v k L , for some constant c ( s ) depending on s only. We choose µ ( R ) = c ( s ) sup | x |≤ R | a ′ ( x ) | . (A4) The same reasoning as in (A2) above, shows that property A (4) is satisfied with µ ( R ) = c ( s, a, R ) for some constant c ( s, a, R ) .(C1) The operator A ( u ; ε ) depends continuously (in norm) on ε . In-deed, if ε ′ = ( ε ′ , . . . , ε ′ n ) , we have k ( A ( u ; ε ) − A ( u ; ε ′ )) k H S ,L ≤ n X i =1 | ε i − ε ′ i | . C2) The operator B ( u ; ε ) does not depend on ε .(C3) The initial data do not depend on ε .We have thus proved that there exists a T > such that u ( t ; ε ) existsand is continuous as a map from R n to C ([0 , T ] , H s ) ∩ C ([0 , T ] , L ) , we want to prove that it is continuous also as a map from R n to C ([0 , T ] , H s ) ∩ C ([0 , T ] , H s − (2 n +1) ) . Now the time derivative of u sat-isfies u t ( t ; ε ) = A ( u ( t ); ε ) u ( t ; ε ) , and A ( u ( t ); ε ) , for fixed t, ε is a continuous operator from H s to H s − (2 n +1) .To complete the proof, it is enough to prove that the map A ( , ) : W × R n → L ( H s , H s − (2 n +1) ) is continuous. Indeed, k A ( v, ε ′ ) y − A ( u, ε ) y k H s − (2 n +1) ≤ k a ( v ) ∂ x y − a ( u ) ∂ x y k H s − (2 n +1) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X i =1 ( ε ′ i − ε i ) ∂ i +1 x y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s − (2 n +1) . By Cauchy-Schwarz and the Schauder estimate we have k a ( v ) ∂ x y − a ( u ) ∂ x y k H s − (2 n +1) ≤ c ( s, a, R ) k u − v k H s k y k H s . Moreover (cid:13)(cid:13)P ni =1 ( ε ′ i − ε i ) ∂ i +1 x y (cid:13)(cid:13) H s − (2 n +1) ≤ P ni =1 | ε ′ i − ε i | k y k H s . Remark 2.
Following [14], it is possible to prove a slightly strongerversion of this theorem. Indeed, we can prove that the Cauchy problem (9) is uniformly locally well-posed in H s with s > . However, thisstronger result is unnecessary for the purpose of studying the deriva-tives (with respect to ε i ) of the solution of the Cauchy problem (9) . Theorem 6 establishes that, fixed the nonlinearity a ( u ) and the ini-tial datum ϕ ∈ H s , there exists a time T > such that the Cauchyproblem (9) is locally well-posed in the time interval [0 , T ] , continu-ously with respect to ε ∈ R n . In particular the life-span of the solutioncan be chosen independently on ε .The natural problem is to find the supremum of all the positivetimes such that the Cauchy problem is locally well-posed and continu-ous with respect to ε ∈ U for some open U ⊂ R n . We denote this time T U . Suppose that for some a ( u ) and some ϕ , there exists a U ⊂ R n ,such that the Cauchy problem is globally well-posed for all ε ∈ U . Inthis case, it follows that T U = ∞ . For example, if n = 1 , ε = 0 , and lim c →∞ a ( c ) c = 0 , then the Cauchy problem is globally well-posed in H s , for s > [14]. On the other hand, if for instance a ( u ) = u , somesolutions do blow-up at a finite time [19].Since the general pattern is unknown, here we analyze the KdVCauchy problem u t = uu x + εu xxx , u | t =0 = ϕ ∈ H s , s ≥ . (14) f ε = 0 , the solution of above Cauchy problem develops a gradientcatastrophe singularity at a finite time t = t c > . Here t c coincideswith the supremum of the positive time t for which the solution of theCauchy problem can be continued. Conversely, if ε = 0 the Cauchyproblem is globally well-posed [14].We now show that the solution of the Cauchy problem of KdV iscontinuous with respect to ε in any time interval [0 , T ] , with T strictlysmaller than the critical time t c . This is a simple corollary of Theorem6: Theorem 7.
Let T be any positive time smaller than the critical time T < t c and let u ( t ; ε ) ∈ C ([0 , T ] , H s ) ∩ C ([0 , T ] , H s − ) be the uniquesolution of the KdV Cauchy problem (14) . Then ε → u ( t ; ε ) is acontinuous map from R to C ([0 , T ] , H s ) ∩ C ([0 , T ] , H s − ) .Proof. The proof follows from Theorem 6 and a standard continuationargument.
Corollary 1.
Let s ≥ , T be any positive time smaller than thecritical time T < t c and let u ( t ; ε ) be the unique solution of the KdVCauchy problem (14) . Then lim ε → (cid:13)(cid:13) u ( t ; ε ) − v ( t ) (cid:13)(cid:13) H s , uniformly in t ∈ [0 , T ] . (15) Here, v is the unique solution of the Cauchy problem for the Hopfequation v t = v v x , v | t =0 = ϕ. ε -expansion of KdV solutions This section is devoted to study the differentiability of solutions ofequation (9) with respect to ε . For simplicity, we consider in detail thecase of KdV only; as explained in Theorem 9 below, there is no troublein extending the results to the whole class (9).We show that if s ≥ N + 3 , then the solution of the Cauchyproblem of KdV (14) is N -times differentiable with respect to ε in anytime interval [0 , T ] , with T strictly smaller than the critical time t c .Before our main theorem, state a technical lemma. Lemma 2.
Let f : R → C ([0 , T ] , H ) and g : R → C ([0 , T ] , H ) betwo functions, and take X = L , Y = H s , with s ≥ . Then, thefamily of linear operators A ( ε , ε , ε ) = f ( ε ) ∂ x + g ( ε ) + ε ∂ x (16) satisfy the conditions (i) through (iv) of Theorem 2. Moreover, if ( ε n , ε n , ε n ) → ( ε , ε , ε ) is any converging sequence, then the sequenceof operators A n = A ( ε n , ε n , ε n ) satisfy conditions (vi) and (vii) of The-orem 3. roof. The operator A ( ε , ε , ε ) was considered in Examples 2 and3 above, where it was shown that it belongs to G ( X, , β + β ′ ) with β = sup x ∈ R | f x | and β ′ = sup x ∈ R | g | . The verification of conditions(i) through (vii) follows the same steps as the proof of Theorem 6. Theorem 8.
Let N = ⌊ s/ − ⌋ and [0 , T ] be a time interval such that < T < t c . Let u : R → C ([0 , T ] , H s ) ∩ C ([0 , T ] , H s − ) be the map that associates to ε ∈ R the unique solution of the KdVCauchy problem (14). Then, there exist and are continuous the maps u ( k ) : R → C ([0 , T ] , H s − k ) ∩ C ([0 , T ] , H s − k +1) ) , for k = 1 , . . . , N, defined as u ( k ) ( ε ) = d k u ( ε ) dε k . Fixed ε ∈ R and k ∈ N , ≤ k ≤ N , then the function u ( k ) ( ε ) satisfiesthe following Cauchy problem u ( k ) t = A ε u ( k ) + k − X j =1 (cid:18) kj (cid:19) u ( j ) u ( k − j ) x + k u ( k − xxx , (17) u ( k ) | t =0 ( ε ) = 0 , (18) where the linear operator A ε is defined as A ε = u ( ε ) ∂ x + u x ( ε ) + ε∂ x , (19) and we use the convention u (0) ( ε ) = u ( ε ) . Note that (17) is a linear non-homogeneous differential equation.
Proof. If N = 0 , then the first part of the theorem follows from Theo-rem 7, while the second part is empty. Assume N ≥ , we now provedifferentiability. To this aim we introduce the difference quotient u (1) ( ε, h ) = u ( ε + h ) − u ( ε ) h , and a simple computation shows that u (1) ( ε, h ) satisfies the equation u (1) t ( ε, h ) = A ( ε, h ) u (1) ( ε, h ) + u xxx ( ε + h ) , u (1) ( ε, h ) | t =0 = 0 , where A ( ε, h ) = u ( ε ) ∂ x + u x ( ε + h ) + ε ∂ x . In the limit h → ,the above equation converges to (17), with k = 1 . This is a linearnon-homogeneous equation, with forcing term u xxx ( ε ) . From The-orem 7, we have that u xxx ( ε ) is a continuous function from R to C ([0 , T ] , H s − ) .Hence, we can prove the convergence of lim h → u ( ε, h ) using theperturbation Theorem 3, provided that: i) we look for solutions of (17) ying in the same space of the forcing term, namely H s − , and ii) theforcing term belongs to D ( A ε ) . The latter condition holds since s ≥ by hypothesis.Due to Theorems 2 and 3 and to Lemma 2, we conclude that u (1) ( ε ) := lim h → u (1) ( ε, h ) solves (17) and maps R continuously to C ([0 , T ] , H s − ) ∩ C ([0 , T ] , L ) .Moreover, we have that the function u (1) maps R continuously into C ([0 , T ] , H s − ) ∩ C ([0 , T ] , H s − ) . Indeed, u (1) t ( ε ) equals A ε u (1) + u xxx ( ε ) , and the operator A ε mapsany continuous function R → C ([0 , T ] , H s − ) to a continuous function R → C ([0 , T ] , H s − ) . The last statement can be proved in a similarway as in the proof of Theorem 6.We continue the proof by induction on the order of the derivative.Suppose the thesis is valid for i = 1 , . . . , k < N . As before, we definethe difference quotient u ( k +1) ( ε, h ) = u ( k ) ( ε + h ) − u ( k ) ( ε ) h , that satisfies the non-homogeneous linear equation u ( k +1) t ( ε, h ) = A ( ε, h ) u ( k +1) ( ε, h ) + u ( k ) xxx ( ε + h ) + f k +1 ( ε, h )+ u (1) ( ε ) u ( k ) x ( ε ) + u (1) x ( ε ) u ( k ) ( ε ) , (20) u ( k +1) ( ε, h ) | t =0 = 0 , where f k +1 ( ε, h ) = 1 h k − X j =1 (cid:18) kj (cid:19) u ( j ) ( ε + h ) u ( k − j ) x ( ε + h ) + k u ( k − xxx ( ε + h ) − k − X j =1 (cid:18) kj (cid:19) u ( j ) ( ε ) u ( k − j ) x ( ε ) + k u ( k − xxx ( ε ) . (21)The non-homogeneous term of equation (20) belongs to C ([0 , T ] , H s − k +1) ) , and it continuously depends on ε . In the limit h → , the quantity f k +1 ( ε, h ) converges in C ([0 , T ] , H s − (3 k +3) ) , continuously with respectto ε , to f k +1 ε ( ε ) := ddε k − X j =1 (cid:18) kj (cid:19) u ( j ) ( ε ) u ( k − j ) x ( ε ) + k u ( k − xxx ( ε ) . Hence, the same reasoning as in the case of u (1) ( ε ) shows that u ( k +1) ( ε ) := lim h → u ( k ) ( ε, h ) olves the equation u ( k +1) t ( ε ) = A ( ε ) u ( k +1) ( ε ) + u ( k ) xxx ( ε ) + f k +1 ε ( ε )+ u (1) ( ε ) u ( k ) x ( ε ) + u (1) x ( ε ) u ( k ) ( ε ) , (22) u ( k +1) ( ε ) | t =0 = 0 , and it maps R continuously to C ([0 , T ] , H s − k +1) ) ∩ C ([0 , T ] , H s − k +2) ) .It is a simple computation to show that (22) coincides with (17).Theorem 8 shows that if the initial datum of the KdV equation liesin the Sobolev space H s , then the solution of the Cauchy problem is N − times differentiable with respect to ε , for N = ⌊ s/ − ⌋ . Con-sequently, if the initial datum lies in all the Sobolev space – e.g. itbelongs to the Schwartz class – then the solution of the Cauchy prob-lem is smooth with respect to ε . In particular, the solution admits anasymptotic expansion in power series of ε . More precisely, we have thefollowing corollary of Theorem 8. Corollary 2.
Let ϕ ∈ H ∞ = ∩ s ≥ H s , T > be any positive timesmaller than the critical time T < t c and u : R → C ([0 , T ] , H ∞ ) be thesolution of the Cauchy problem (14). Then(i) u : R → C ([0 , T ] , H ∞ ) is a smooth function (of ε ).(ii) In ε = 0 , u admits an asymptotic expansion in power series of ε : u ( ε ) ∼ ∞ X k =0 v k ε k . (23) where v = u (0) is the solution of the Cauchy problem v t = v v x (24) v | t =0 = ϕ, (25) for the Hopf equation, and v k = u ( k ) (0) is the solution of the k -thlinear Cauchy problem (17) when ε = 0 , that is: v kt = k X j =0 (cid:18) kj (cid:19) v j v k − jx + k v k − xxx , (26) v k | t =0 = 0 , k ≥ . (27)Note that a similar Theorem was proven for the defocusing Non-linear Schrödinger equation [11].Below, we state the analogue of Theorem 8 for the general equation(9) and we give a sketch the proof. The details of the full proof, whichis rather long, will be given elsewhere. Theorem 9.
Let U be an open subset of R n , [0 , T ] be a time intervalsuch that the Cauchy problem is locally well-posed and continuous withrespect to ε ∈ U and let u : R → C ([0 , T ] , H s ) ∩ C ([0 , T ] , H s − (2 n +1) ) e the map that associates to ε ∈ U the unique solution of the Cauchyproblem for (9). Moreover, let K = P ni =1 N i (2 i +1) and N = P ni =1 N i .If s − K ≥ n + 1 , then the partial derivative ∂ N u ( ε ) ∂ε N . . . ∂ε N n n exists in C ([0 , T ] , H s − K ) ∩ C ([0 , T ] , H s − K − (2 n +1) ) and is continuouswith respect to ε ∈ U .Proof. The Theorem can be proven along the very same lines of theproof of the analogue Theorem 8 for KdV. More precisely, it is possibleto prove existence and continuity of the partial derivative, by showingthat it satisfies a linear non-homogeneous equation. Note that thecondition s − K ≥ n + 1 implies that the forcing term belongs to thedomain of ∂ n +1 x . We now consider the results of the previous sections in the setting ofthe general construction, proposed by Dubrovin and Zhang [8, 9, 10],of Hamiltonian regularization of the quasilinear conservation law: u t = a ( u ) u x , u | t =0 = ϕ, (28)where a and the initial value ϕ are assumed to be smooth functions,and ϕ is either periodic or rapidly decreasing at infinity. We discussthe aspects of the Dubrovin-Zhang construction which are more relatedwith the present paper; in the next section we will show how the resultsobtained in Section 3 and 4 provide a rigorous justification to thismethod for a particular class of equations of type (9).Let us consider equation (28). It is well known that this equationcan formally be written as a Hamiltonian system u t = { u, H } = ∂ x δHδu ( x ) , (29)with Hamiltonian H = Z R h ( u ( x )) dx, h ′′ ( u ) = a ( u ) , and where the Euler-Lagrange operator is defined as δHδu ( x ) = X k ≥ ( − k d k dx k ∂ h∂u ( k ) x , for any local functional H = R h ( u, u x , u xx , . . . ) dx . The above Poissonbracket is given by { H , H } = Z R δH δu ( x ) ∂ x δH δu ( x ) dx, or any pair of functionals H i = R h i ( u, u x , u xx , . . . ) dx , i = 1 , .Following Dubrovin, by Hamiltonian regularization (or perturba-tion ) of the quasilinear conservation law (28) we mean an expression u t = n u, ˜ H o = ∂ x δ ˜ Hδu ( x ) , u | t =0 = ϕ, (30)where the Hamiltonian is given by a formal series ˜ H = H + X k ≥ H k ε k , H k = Z h k ( u ; u x , . . . , u ( k ) x ) dx, k ≥ , for some h k , which are assumed to be differential polynomials in thederivatives. In addition, the solutions of equation (30) are sought tobe of the form u ( x, t ) = X i ≥ v i ( x, t ) ε i , (31)with coefficients v i smooth functions of x and t . Within this setting,all identities are understood in the sense of formal power series in ε - they are assumed to hold identically at every order in ε . Therefore,the perturbed Hamiltonian and the solutions are not required to beconvergent (neither asymptotic) series. Note, however, that the initialvalues of the perturbed and the unperturbed equations are assumedto be the same. In particular, the function ϕ is independent of ε , andfrom the expansion (31) we deduce the identities v ( x,
0) = ϕ ( x ) , v i ( x,
0) = 0 , ∀ i ≥ . (32)A primary task in the Dubrovin-Zhang approach is the classifica-tion of Hamiltonian perturbations, which is performed by consideringequations (30) modulo quasi-Miura transformation . These are trans-formations of the form: u v = X k ≥ ε k F k ( u ; u x , . . . , u ( k ) x ) , (33)where the functions F k are rational with respect to the derivatives. Apartial result is given by the following Theorem 10. [8] Any Hamiltonian perturbation of equation (28) oforder ε can be reduced by a Miura-type transformation to an equationof the form (30) , with Hamiltonian ˜ H = Z ˜ h ( u ; u x , u xx , ε ) dx, h ′′ = a, (34) ˜ h = h − ε c h ′′′ u x + ε (cid:20)(cid:18) p h ′′′ + 310 c h (4) (cid:19) u xx − (cid:18) c c ′′ h (4) + c c ′ h (5) + c h (6) + p ′ h (4) + p h (5) − s h ′′′ (cid:19) u x (cid:21) , for arbitrary functions c ( u ) , p ( u ) , s ( u ) . et us now consider in more detail the solutions of the perturbedequation (30), which – after Theorem 10 – we will consider togetherwith a Hamiltonian of the form (34). By expanding both sides ofequation (30) according to the Ansatz (31), in first approximation oneobtains v t = a ( v ) v x , (35)which says that v must be a solution of the unperturbed equation (28).Accordingly, from the higher order coefficients one obtains an infiniteset of linear non-homogeneous equations (or transport equations ) forthe coefficients v k ( x, t ) , which can be solved recursively starting fromthe solution of (35). For instance, the equation for v turns out to be v t = ∂ x (cid:18) a v + c a ′ v xx + 12 ( c a ′′ + c ′ a ′ ) (cid:0) v x (cid:1) (cid:19) , v | t =0 = 0 (36)where a = a (cid:0) v (cid:1) , c = c (cid:0) v (cid:1) . Remark 3.
In the class of initial data considered in the present paper,any solution of equation (35) develops a singularity at a finite time t = t c > , known as time of gradient catastrophe. An importantaspect of Dubrovin’s theory is concerned with the study of the solutionof (30) in a neighborhood of the critical time t c , in order to show howthe singularity is regularized by the dispersive perturbations [8].This, however, is out of the scope of our present method of investi-gation. In what follows we study the series (31) for any time interval [0 , T ] strictly smaller than the critical time: T < t c . Example 4.
The KdV equation u t = u u x + ε u xxx is obtained from (30) , (34) by choosing h ( u ) = u , c ( u ) = 1 , and p ( u ) = s ( u ) = 0 . In this case, a simple computation shows that theCauchy problems for the transport equations are given by v kt = k X j =0 (cid:18) kj (cid:19) v j v k − jx + k v k − xxx , k ≥ ,v | t =0 = ϕ, v k | t =0 = 0 . One can – in principle – solve these equations recursively. Note thatthese Cauchy problems coincide with (24) , (26) . From the above discussion it follows that – at least in principle –all coefficients of the expansion (31) can be obtained once a solution v of equation (35) is known. The method followed in [8, 9] (see also theolder result [1]) to find solutions of the perturbed equation (30) is toconstruct a quasi-Miura transformation, relating the solution v ( x, t ) ofthe unperturbed equation (35) to the the solution u ( x, t ) of perturbedequation (30). The required transformation has been suggested in [8]to be of the form: v u = v − ε (cid:8) v ( x ) , K (cid:9) + ε (cid:8)(cid:8) v ( x ) , K (cid:9) , K (cid:9) + . . . (37) here the functional K , up to order in ε , is given by K = − Z " ε c ( v )2 v x log v x + ε c ( v ) (cid:18) v xx v x (cid:19) − p ( v )4 (cid:0) v xx (cid:1) v x ! dx. (38)This in particular implies v = 12 ∂ x (cid:18) c ( v ) v xx v x + c ′ ( v ) v x (cid:19) , (39)and a direct substitution shows that the above function satisfies equa-tion (36), provided v satisfies (35). Note, however, that the function(39) is bounded only for monotone solutions of equation (35). Further-more, (39) does not satisfy the required initial condition (32). The classification problem of Hamiltonian perturbations, together withthe quasi-Miura transformation discussed above, are main ingredientsof the Dubrovin–Zhang constructions before the critical time, which isthe time-span we are interest in the present paper. As already noticed,this approach is mainly based on identities of formal power series, andindeed, if for a given Hamiltonian the corresponding equation (30) isjust a formal series, then the only way to construct a (formal) solutionseems to be through the use of a formal series, for instance like (31).However, if the Hamiltonian perturbation (30) is well-defined – forexample if the Hamiltonian ˜ H is given by (34) – then the resultingequation may happen to be locally well-posed in some function space,say H s , for s big enough. At the same time, the solution may be N − differentiable with respect to ε and the formal series (30) may justbe – up to order N – the Taylor expansion of the actual solution.Using the results of Section 3 and 4 we can show that this is trueif the Hamiltonian perturbation (30) coincides with a generalised KdVequation (9). This is the case for the general equation (9), provided n ≤ . Indeed, the following Lemma holds: Lemma 3.
The equation u t = a ( u ) u x + ε α u xxx + ε β u xxxxx , (40) which is obtained by (9) in the case n = 2 and ε = α ε , ε = β ε , co-incides with the Hamiltonian perturbation (30) with Hamiltonian (34) ,provided the coefficients are chosen in the following way: c = αa ′ , p = β a ′ − α a ′′ ( a ′ ) , (41) s = α (cid:18)
25 ( a ′′ ) ( a ′ ) − a ′′ a ′′′ ( a ′ ) + 124 a ′′′′ ( a ′ ) (cid:19) − β (cid:18) ( a ′′ ) ( a ′ ) − a ′′′ ( a ′ ) (cid:19) . heorem 11. Let ϕ ∈ H s , s ≥ , U a neighborhood of ε = 0 , u ( ε ) bethe solution of the Cauchy problem (40) with initial data ϕ and [0 , T ] be a time interval such that the Cauchy problem is locally well-posedfor any ε ∈ U and continuous with respect to ε ∈ U . Moreover, let s ≥ N . Then u ( ε ) has a Taylor expansion in ε = 0 , up to order N , of the form u ( ε ) = N X k =0 v k ε k + r ( ε ) , (42) where v k ∈ C ([0 , T ] , H s − k ) , r ( ε ) ∈ C ([0 , T ] , H s − N ) . Here v = u (0) is the solution of the Cauchy problem for unperturbedequation (35) , while v satisfies the first transport equation (36) withparameters (41) , v k , k > is the solution of the higher transport equa-tions. The remainder term r is o ( ε N ) .Moreover, if α = 0 then the same is true, under weaker hypothesis.Namely, if s ≥ N , then v k = 0 for k odd, v l ∈ C ([0 , T ] , H s − l ) ,and r : R → C ([0 , T, H s − N ]) is continuous, with r = o ( ε N ) .Proof. The proof follows from Theorem 9.
Remark 4.
Due to Theorem 6, T Q > for any Q ⊂ R . If, for ( ε, α, β ) = (0 , , the Cauchy problem is globally well-posed [16], thenthe time T Q in the above theorem can be chosen to be any positive timesmaller than the critical time t c of the unperturbed equation (35). Thisis the case, for instance, if a ( u ) = u . In the following theorem we provide a correction to the formula(39) for the first coefficient v . Our formula turns out to be validfor solutions of (35) which are not monotone, and satisfies the correctinitial value. Theorem 12.
Consider the Cauchy problem for equation (35) withinitial datum v (0) = ϕ ∈ H s , s ≥ , denote by t c be the associatedcritical time, and let v ( x, t ) , with ( x, t ) ∈ R × [0 , t c ) , be its uniqueclassical solution. Then, the solution of the linear transport equation (36) , with v given above and initial datum v (0) = 0 is v ( x, t ) = ∂∂x δ ˜ K t [ u ] δu ( x ) | u = v ( x,t ) ! , ( x, t ) ∈ R × [0 , t c ) , where the family of functionals ˜ K t , with t ∈ R , is defined by ˜ K t [ u ] := − Z R c ( u ) u x log (cid:16) t a ′ ( u ) u x (cid:17) dx, for every u ∈ H s , with k u k H s small enough. roof. The explicit form of the function v stated in the theorem isgiven by v = t ∂∂x (cid:18) ( c a ′ ) ′ ( v x ) + 2 c a ′ v xx + t c ( a ′ ) v x v xx + t c ′ ( a ′ ) ( v x ) (1 + t a ′ v x ) (cid:19) , (43)where the functions a , c , and the corresponding derivatives are eval-uated at u = v ( x, t ) . A direct calculation shows that this functionsatisfies equation (36) with the correct initial value. Remark 5.
Note that the above theorem holds for any choice of thefunctions a and c . This fact suggests that similar results of the oneobtained in Theorem 11 remain true for a generic Hamiltonian pertur-bation of the quasilinear conservation law (28) . Remark 6.
Formula (43) has been obtained making use of the so called‘string equation’, introduced in the setting of Hamiltonian perturbationsof nonlinear PDEs by Dubrovin [8]. Although heuristic, the use of thestring equation turns out to be a very powerful method for describingsolutions of the perturbations both before the critical time and in aneighborhood of it.
References [1] V. A. Ba˘ıkov, R. K. Gazizov, and N. Kh. Ibragimov. Approximatesymmetries and formal linearization.
Zh. Prikl. Mekh. i Tekhn.Fiz. , 2:40–49, 1989.[2] J. L. Bona and R. Smith. The initial-value problem for theKorteweg-de Vries equation.
Philos. Trans. Roy. Soc. LondonSer. A , 278(1287):555–601, 1975.[3] J. Bourgain. Fourier transform restriction phenomena for certainlattice subsets and applications to nonlinear evolution equations.II. The KdV-equation.
Geom. Funct. Anal. , 3(3):209–262, 1993.[4] T. Claeys and T. Grava. Universality of the break-up profile forthe KdV equation in the small dispersion limit using the Riemann-Hilbert approach.
Comm. Math. Phys. , 286(3):979–1009, 2009.[5] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao.Sharp global well-posedness for KdV and modified KdVon R and T . J. Amer. Math. Soc. , 16(3):705–749 (electronic), 2003.[6] P. Deift, S. Venakides, and X. Zhou. An extension of the steepestdescent method for Riemann-Hilbert problems: the small disper-sion limit of the Korteweg-de Vries (KdV) equation.
Proc. Natl.Acad. Sci. USA , 95(2):450–454 (electronic), 1998.[7] P. Deift and X. Zhou. A steepest descent method for oscillatoryRiemann-Hilbert problems. Asymptotics for the MKdV equation.
Ann. of Math. (2) , 137(2):295–368, 1993.[8] B. Dubrovin. On Hamiltonian perturbations of hyperbolic systemsof conservation laws. II. Universality of critical behaviour.
Comm.Math. Phys. , 267(1):117–139, 2006.
9] B. Dubrovin. Hamiltonian PDEs: deformations, integrability, so-lutions.
J. Phys. A , 43(43):434002, 20, 2010.[10] B.A. Dubrovin and Y. Zhang. Normal forms of hierarchies ofintegrable PDEs, Frobenius manifolds and Gromov - Witten in-variants. arXiv:math/0108160v1.[11] E. Grenier. Semiclassical limit of the nonlinear Schrödinger equa-tion in small time.
Proc. Amer. Math. Soc. , 126(2):523–530, 1998.[12] T. Kato. Linear evolution equations of “hyperbolic” type. II.
J.Math. Soc. Japan , 25, 1973.[13] T. Kato. Quasi-linear equations of evolution, with applicationsto partial differential equations. In
Spectral theory and differen-tial equations (Proc. Sympos., Dundee, 1974; dedicated to KonradJörgens) , pages 25–70. Lecture Notes in Math., Vol. 448. Springer,Berlin, 1975.[14] T. Kato. On the Cauchy problem for the (generalized) Korteweg-de Vries equation. In
Studies in applied mathematics , volume 8of
Adv. Math. Suppl. Stud. , pages 93–128. Academic Press, NewYork, 1983.[15] T. Kawahara. Oscillatory solitary waves in dispersive media.
J.Phys. Soc. Japan , 33:260–264, 1972.[16] C. E. Kenig, G. Ponce, and L. Vega. Well-posedness of the initialvalue problem for the Korteweg-de Vries equation.
J. Amer. Math.Soc. , 4(2):323–347, 1991.[17] C. E. Kenig, G. Ponce, and L. Vega. A bilinear estimate withapplications to the KdV equation.
J. Amer. Math. Soc. , 9(2):573–603, 1996.[18] P. D. Lax and C. D. Levermore. The small dispersion limit of theKorteweg-de Vries equation. I, II, III.
Comm. Pure Appl. Math. ,36(3, 5, 6):253–290, 571–593, 809–829, 1983.[19] Y. Martel and F. Merle. Review on blow up and asymptotic dy-namics for critical and subcritical gKdV equations. In
Noncompactproblems at the intersection of geometry, analysis, and topology ,volume 350 of
Contemp. Math. , pages 157–177. Amer. Math. Soc.,Providence, RI, 2004.[20] A. Pazy.
Semigroups of linear operators and applications to par-tial differential equations , volume 44 of
Applied Mathematical Sci-ences . Springer-Verlag, New York, 1983.[21] T. Tao.
Nonlinear dispersive equations , volume 106 of
CBMSRegional Conference Series in Mathematics . Published for theConference Board of the Mathematical Sciences, Washington, DC,2006. Local and global analysis.[22] S. Venakides. The zero dispersion limit of the Korteweg-de Vriesequation for initial potentials with nontrivial reflection coefficient.
Comm. Pure Appl. Math. , 38(2):125–155, 1985., 38(2):125–155, 1985.