The Benjamin-Ono Hierarchy with Asymptotically Reflectionless Initial Data in the Zero-Dispersion Limit

THE BENJAMIN-ONO HIERAR CHY WITH ASYMPTOTICALL Y REFLECTIONLESS INITIAL D A T A IN THE ZER O-DISPERSION LIMIT PETER D. MILLER AND ZHENGJIE XU Abstract. W e study the Benjamin-Ono hierarch y with positive initial data of a general type, in the limit when the dispersion parameter tends to zero. W e establish simple form ulae for the limits (in appropriate w eak or distributional senses) of an infinite family of simultaneously conserved densities in terms of alternating sums of branches of solutions of the inviscid Burgers hierarch y . 1. Introduction The Benjamin-Ono equation is a nonlinear evolution equation gov erning certain types of internal waves . In ternal w av es are disturbances — set into motion by gra vit y — of the in terface betw een t w o immiscible fluids of different densities. A num ber of assumptions are employ ed to deduce the Benjamin-Ono equation as a simplified mo del from the full equations of three-dimensional fluid mechanics: • One supp oses that b oth fluid lay ers consist of inviscid and incompressible fluids, and (for stability) the less-dense fluid rests on top of the denser fluid. • One supp oses that the w a v es are propagating in one direction only , whic h reduces the problem to that of t wo-dimensional fluid mec hanics (only the v ertical direction and the propagation direction surviv e). • One supp oses that the top lay er containing the less-dense fluid is thin compared to a typical w a ve- length of the interface. This allo ws the top lay er to b e treated by a depth-av eraging approach. The top of this lay er is idealized to a rigid horizontal lid. • One supposes that the bottom la y er con taining the denser fluid is infinitely thick. This simplifies the contribution from this la yer to the disp ersion relation of small-amplitude wa ves on the in terface in the linear approximation. • One supp oses that the amplitude of the wa v es is small compared to the thickness of the top la yer. This allows the nonlinear eff ects to be brough t in p erturbatively , and also means w e are assuming the interface do es not breac h the surface by meeting the rigid lid. In the linear and disp ersionless appro ximation, deformations of the interface satisfy the one-dimensional w av e equation. If one p erturbatively introduces the balanced effects of w eak nonlinearit y and disp ersion on a solution of the wa ve equation propagating to the right (say) at constant velocity , then the Benjamin-Ono equation arises as the first correction in the mo ving frame of reference of the wa v e, describing the slow v ariation of the dimensionless wa v e height u as a function of a spatial co ordinate x in the propagation direction and the time t : (1) u t + 2 uu x +  H [ u xx ] = 0 . Here, subscripts denote partial deriv atives, and H is the Hilb ert transform op erator defined on L 2 ( R ) by the singular integral (2) H [ f ]( x ) := 1 π − Z R f ( y ) dy y − x , and  > 0 is a dimensionless measure of the relativ e strength of disp ersiv e effects compared with nonlinear effects. If an initial condition u = u 0 ( x ) is given at t = 0 where u 0 is indep enden t of  , then of course the Date : Octob er 24, 2018. This work was supp orted by the National Science F oundation under grant DMS-0807653. 1 corresp onding s olution of (1) will dep end on  , but we will often not be explicit ab out this dependence in our notation. It is quite useful in applications to hav e accurate and easily analyzable mo dels for internal wa ves. Indeed, one application that is particularly timely is the mo deling of submerged “plumes” of oil as were rep orted follo wing the Deep water Horizon leak in the Gulf of Mexico in Ma y–August 2010. Recen t exp eriments [1] p erformed by Roberto Camassa and Richard McLaughlin at the Universit y of North Carolina, Chap el Hill ha ve demonstrated that if oil is emitted as a turbulen t jet from an ocean floor leak and a densit y stratification is present in the surrounding fluid, then most of the oil will b ecome trapp ed at the in terface b etw een the dense and less-dense fluid la y ers rather than floating to the surface, even though the oil is less dense stil l than the upp er layer . In these circumstances, mo deling the motion of the submerged oil plumes within the fluid column amounts to mo deling the motion of the density interface, that is, mo deling internal wa ves. The purp ose of this short pap er is to sho w how metho ds we hav e recently dev eloped [9] to study the asymptotic b eha vior of solutions of the Benjamin-Ono equation as  ↓ 0 in the weak topology extend b oth to the whole hierarc hy of “higher-order” Benjamin-Ono equations and also to the whole hierarch y of densities coming from conserv ation la ws. It is a pleasure to b e able to contribute to this sp ecial v olume of papers in honor of Da v e Levermore. It will be clear to all of those who hav e follo w ed his w ork that our small con tribution is directly inspired by his groundbreaking analysis with P eter Lax of the zero-disp ersion limit for the Kortew eg-de V ries equation [7], a pro ject that has had a tremendous impact on fields of study ranging from asymptotic analysis of nonlinear ev olution equations to the theory of orthogonal p olynomials and of random matrices. 2. The Benjamin-Ono Hierar chy Let us take the phase space of fields u to be (3) P := { u ∈ L 2 ( R ) ∩ C ∞ ( R ) , u ( k ) ∈ L 2 ( R ) , ∀ k } . This is clearly a linear space ov er R , and it is also an algebra that is closed under differentiation and Hilb ert transforms. W e will ha ve use b elo w for the Cauch y op erators C ± densely defined on L 2 ( R ) by singular in tegrals as follows: (4) C ± [ f ]( x ) := lim δ ↓ 0 1 2 π i Z R f ( y ) dy y − x ∓ iδ . The Cauch y op erators are b ounded with resp ect to the L 2 ( R ) norm and hence extend to b ounded op erators with the same norms on all of L 2 ( R ). In fact, the op erators C + and −C − are just the self-adjoint orthogonal pro jections from L 2 ( R ) onto the Hardy subspaces of functions analytic in the upp er and low er half-planes, resp ectiv ely . They satisfy the identities C + ◦ ( −C − ) = ( −C − ) ◦ C + = 0 , C 2 + = C + , ( −C − ) 2 = −C − , C + − C − = 1 , C + + C − = − i H . Since C ± are self-adjoint, it follows that H is sk ew-adjoint, that is, (5) Z R f H [ g ] dx = − Z R g H [ f ] dx whenev er f and g area real-v alued functions in the phase space P . The operators C ± and H comm ute with differentiation in x . Also, for functionals I [ u ] defined on the phase space P , we define the v ariational deriv ative δ I /δ u by (6) d dt I [ u + tv ]     t =0 = Z R δ I δ u [ u ]( x ) v ( x ) dx. F or functionals I with I [0] = 0, assuming the existence of the v ariational deriv ative of I for each u ∈ P we can recov er the functional from its deriv ative b y the formula (7) I [ u ] = Z 1 0 Z R δ I δ u [ tu ]( x ) u ( x ) dx dt = Z R  Z 1 0 δ I δ u [ tu ]( x ) dt  u ( x ) dx. 2 2.1. Conserv ation Laws for the Benjamin-Ono Equation. The Benjamin-Ono equation conserves an infinite num b er of functionals of u . These ma y b e obtained b y sev eral different metho ds, sev eral of whic h w e review (in historical order of discov ery) for the reader’s conv enience. 2.1.1. The Nakamur a Scheme. A. Nak amura [10] (see also [8]) was the first to deduce an infinite num b er of conserv ed quantities for the Benjamin-Ono equation (1). His deriv ation is based on a B¨ ac klund transforma- tion for (1). The B¨ ac klund transformation is (8) − iµ C + [ n x ] + 1 − e − n = µu where µ is an arbitrary parameter. F rom this equation it can b e shown that regardless of the v alue of µ , (9) d dt Z R n dx = 0 when u satisfies (1). Therefore, by expanding n in a p ow er series in µ with co efficients dep ending on u , the co efficien ts will all b e densities of conserved functionals of u . W riting n = µn 1 + µ 2 n 2 + µ 3 n 3 + · · · one easily obtains a recurrence in whic h the conserved density n n is explicitly given in terms of n 1 , n 2 , . . . , n n − 1 and u . In fact, the recurrence can b e made more explicit by noting that the nonlinearity of the scheme is only quadratic in n x : differen tiating (8) with resp ect to x and then using (8) to eliminate e − n from the result one arrives at (10) n x = µ ( un x + u x + in x C + [ n x ] + i C + [ n xx ]) , and therefore n 1 ,x = u x and (11) n m,x = un m − 1 ,x + i C + [ n m − 1 ,xx ] + i m − 2 X j =1 n j,x C + [ n m − 1 − j,x ] , m = 2 , 3 , 4 , . . . . The first several densities are giv en b y: n 1 = u n 2 = 1 2 u 2 +   1 2 H [ u x ] + 1 2 iu x  n 3 = 1 3 u 3 +   1 2 u H [ u x ] + iuu x + 1 2 H [ uu x ]  +  2  1 2 i H [ u xx ] − 1 2 u xx  n 4 = 1 4 u 4 +   1 2 u 2 H [ u x ] + 1 2 u H [ uu x ]  −  2 1 2 uu xx +    1 2 iu 3 + 1 6 H [ u 3 ]  +  2  3 4 iu H [ u x ] − 3 4 uu x + 1 4 H [ u H [ u x ]]  x −  2 1 4 u x H [ u x ] +  2 1 8  H [ u x ] 2 − ( u x ) 2  ) +  2 3 4 i H  ( u x ) 2 + uu xx  . (12) In the expression for n 4 , only the terms on the first line contribute to the integral ov er R . Indeed, those on the second line are deriv ativ es of functions in P , and those on the third line hav e zero integral b ecause H is sk ew-adjoint and H 2 = − 1. T o deduce that the terms on the fourth line hav e zero integral, it is necessary to correctly interpret the integral as the in tegrand is not of class L 1 ( R ). The key iden tity here is the follo wing: if f ∈ P then (13) lim R ↑∞ Z R − R H [ f ]( x ) dx = 0 . 3 2.1.2. The F okas-F uchssteiner Scheme. The follo wing metho d is due to F ok as and F uchssteiner [5]. It is based on Lie-theoretic analysis of one-parameter symmetry groups of the Benjamin-Ono equation (1). One b egins with (14) δ I 1 δ u [ u ] = 1 and δ I 2 δ u [ u ] = u and then recursively defines (15) δ I m δ u [ u ] = 1 m − 2 δ δ u Z R  2 xuu x + u 2 +   x H [ u xx ] + 3 2 H [ u x ]  δ I m − 1 δ u [ u ] dx, m = 3 , 4 , 5 , . . . . T o compute the v ariational deriv ativ e on the right-hand side one needs the identit y (16) H [ xf ] = x H [ f ] if Z R f ( x ) dx = 0 . Although the explicit function x app ears in the in tegrand on the righ t-hand side of (15), the recursion guaran tees that the v ariational deriv ativ es pro duced are all generated from u and its deriv atives and a finite n umber of applications of H . F or example, (17) δ I 3 δ u [ u ] = δ δ u Z R  2 xu 2 u x + u 3 +   xu H [ u xx ] + 3 2 u H [ u x ]  dx = u 2 +  H [ u x ] and (18) δ I 4 δ u [ u ] = u 3 +   3 2 u H [ u x ] + 3 2 H [ uu x ]  −  2 u xx . As x do es not app ear explicitly in the resulting expressions, it follo ws from the formula (7) that the functionals I m are in tegrals of densities that also do not inv olve x explicitly . These densities ma y b e easily obtained from the v ariational deriv atives simply b y first multiplying through term-b y-term b y u and then dividing eac h term by its homogeneous degree in u . F or example, from eac h δ I m /δ u we obtain a corresp onding conserved densit y f m as follows: f 1 = u f 2 = 1 2 u 2 f 3 = 1 3 u 3 +  1 2 u H [ u x ] f 4 = 1 4 u 4 +   1 2 u 2 H [ u x ] + 1 2 u H [ uu x ]  −  2 1 2 uu xx . (19) Unlik e the densities pro duced b y the Nak amura sc heme, these densities are all absolutely in tegrable if u ∈ P . 2.1.3. The Kaup-Matsuno Scheme. The following method was deriv ed from the inv erse-scattering transform for the Benjamin-Ono equation by Kaup and Matsuno [6]. Set k 1 := u and then define recursively (20) k m := u C + [ k m − 1 ] + i  k m − 1 u  x , m = 2 , 3 , . . . . 4 Then, the quantities k m are all densities of functionals conserved b y (1). The first several densities obtained b y the Kaup-Matsuno scheme are k 1 = u k 2 = 1 2 u 2 − 1 2 iu H [ u ] k 3 = 1 4 u 3 − 1 4 u H [ u H [ u ]] − 1 4 iu 2 H [ u ] − 1 4 iu H [ u 2 ] +   1 2 iuu x + 1 2 u H [ u x ]  k 4 = 1 8 u 4 − 1 8 u 2 H [ u H [ u ]] − 1 8 u H [ u 2 H [ u ]] − 1 8 u H [ u H [ u 2 ]] + 1 8 u H [ u H [ u H [ u ]]] − 1 8 iu 3 H [ u ] − 1 8 iu 2 H [ u 2 ] − 1 8 iu H [ u 3 ] +   1 2 u 2 H [ u x ] + 3 4 u H [ uu x ] + 1 4 uu x H [ u ] + 3 4 iuu x − 1 2 iu H [ u H [ u x ]] − 1 4 iu H [ u x H [ u ]]  +  2  − 1 2 uu xx + 1 2 iu H [ u xx ]  . (21) As is the case for the F ok as-F uchssteiner sc heme, and unlik e the Nak am ura scheme, the densities pro duced b y the Kaup-Matsuno scheme are all absolutely integrable for u ∈ P . 2.1.4. Comp arison of the Schemes. Although the three sc hemes clearly do not giv e rise to identical densities, they apparently pro duce exactly the same integrals. That is, for u ∈ P w e typically hav e n m , f m , and k m b eing different (unequal) expressions generated by deriv atives of u and applications of H . How ev er, (22) I m [ u ] := Z R n m dx = Z R f m dx = Z R k m dx, m = 1 , 2 , 3 , . . . . Here in the case of R R n m dx the integral must generally be interpreted in the “principal v alue at infinity” sense as in (13). F rom eac h of the three schemes it follows that if u is a smo oth solution of (1), then dI m /dt = 0 for all m . W e do not pro vide a direct pro of of the equiv alence of the integrals generated by each of the three schemes here, although in the case where  = 0 all three sc hemes pro duce the same result: if u ∈ P is indep enden t of  , then (23) lim  ↓ 0 I m ( t ) = 1 m Z R u ( x ) m dx. This is rather straightforw ard to sho w from the Nak am ura and F ok as-F uchssteiner sc hem es, while for the Kaup-Matsuno scheme it follows from a lemma prov ed in the appendix of [9]. In any case, this fact easily establishes the functional indep endence of the integrals I m [ u ], m = 1 , 2 , 3 , . . . . The v ariational deriv ativ es δ I m /δ u in fact play a dual role, as is pro ved in both [5] and [8] (in the resp ective con text of tw o different schemes): they also serve as densities of integrals: (24) Z R δ I m δ u dx = ( m − 1) I m − 1 [ u ] , m ≥ 2 , or I m [ u ] = Z R 1 m δ I m +1 δ u dx, m ≥ 1 . The main adv antage of the Nak amura scheme is that it can b e used to place all of the equations of the Benjamin-Ono hierarch y (see b elo w) in bilinear form after which Hirota’s metho d can b e applied to deduce the form of the simultaneous N -soliton solution of the entire hierarch y . This is done in [8], and we will use the resulting formulae b elo w. Ho wev er, due to the presence of terms of the form H [ f ] for f ∈ P that are not themselves of the form f = g x for g ∈ P , the densities generated by the Nak am ura scheme are generally not absolutely integrable and require a more careful interpretation of the integral. The adv antages of the F ok as-F uchssteiner scheme are (i) that it provides a direct one-term recurrence for the v ariational deriv atives of the conserv ed quantities (whic h unlik e densities are uniquely determined b y the functional), (ii) that the v ariational deriv ativ es it generates are manifestly real and the corresponding densities f m are simpler than in either of the other tw o schemes, and (iii) that it allo ws a direct pro of of the fact that the functionals I m are all in inv olution with resp ect to the Poisson brac ket: (25) { I , J } := Z R δ I δ u ( x ) ∂ ∂ x δ J δ u ( x ) dx, 5 in other words, { I j , I k } = 0 for all j, k . In the Kaup-Matsuno scheme, the quantities N m := k m /u are the co efficients in the Laurent expansion ab out λ = ∞ of the eigenfunction N ( λ ; x, t ) that is a simultaneous solution of the tw o linear equations making up the Lax pair for the Benjamin-Ono equation. Here λ ∈ C is the sp ectral parameter. Because the in tegrals I m [ u ] := R R k n dx are obtained by expanding a scattering eigenfunction, they equiv alen tly enco de the scattering data in an explicit wa y , and in [6] one can find explicit formulae for I m in terms of the discrete sp ectrum and the reflection co efficient of the direct scattering problem. Thus, the Kaup-Matsuno sc heme leads not just to an infinite collection of integrals of motion of the Benjamin-Ono equation, but also a hierarch y of trace form ulae equiv alently expressing each I m b oth as a functional of u and also as functional of the scattering data. These formulae can b e used to deduce from initial data the asymptotic distribution of eigenv alues λ j of the scattering problem in the zero-disp ersion limit (see [8] and section 3.2 of [9]). 2.2. Construction of the Hierarch y. The Benjamin-Ono equation (1) can be written in Hamiltonian form as (26) u t = − ∂ ∂ x δ I 3 δ u . The Noetherian symmetry of (1) asso ciated with the conserv ed quan tit y I m is the Hamiltonian flo w with Hamiltonian I m : (27) u t k = − ∂ ∂ x δ I k +2 δ u , k = 1 , 2 , 3 , . . . . Here t k is the parameter of the symmetry group generated b y I k +2 . The fact that the in tegrals I k are all in in volution [5] implies that these flows are all compatible (that is, the symmetry group is ab elian), so given a smo oth function u 0 ∈ P and a p ositive in teger K , there will exist a function u ( x, t 1 , t 2 , . . . , t K ) satisfying u ( x, 0 , 0 , . . . , 0) = u 0 ( x ) and equations (27) for k = 1 , 2 , . . . , K . Equations (27) constitute the Benjamin-Ono hierarc hy . 3. Setting up the Zero-Dispersion Limit 3.1. F orm ulation of the Problem. The problem w e wish to consider is the follo wing. Let u 0 ∈ P b e giv en, an initial condition indep enden t of  . F or each  > 0 w e may construct the sim ultaneous solution u ( x, t 1 , t 2 , . . . , t K ) of the Benjamin-Ono hierarc h y (27) of commuting flows satisfying the initial condition u ( x, 0 , 0 , . . . , 0) = u 0 ( x ). The question of interest is the asymptotic b ehavior of u ( x, t 1 , t 2 , . . . , t K ) in the zero-disp ersion limit  ↓ 0. As a first step, we will address this problem by establishing the existence of the disp ersionless limits (in appropriately weak top ologies) of all of the conserved densities (see (24)) (28) D m := 1 m δ I m +1 δ u , m = 1 , 2 , 3 , . . . . Note that D 1 = u . In general, D m differs from all three of n m , f m , and k m b y a “trivial” density that in tegrates to zero for all u ∈ P . Ho w ever, the densities D m are those that most directly yield a dispersionless represen tation. 3.2. Admissible Initial Conditions. W e will further assume that u 0 ∈ P satisfies the following conditions adapted from [9]: • u 0 ( x ) > 0 for all x ∈ R . • There is a unique critical point x 0 ∈ R for whic h u 0 0 ( x 0 ) = 0, and u 00 0 ( x 0 ) < 0 making x 0 the global, nondegenerate maximizer of u 0 . • u 0 exhibits p ow er-law decay in its tails: lim x →±∞ u 0 ( x ) = 0 and (29) lim x →±∞ | x | q +1 u 0 0 ( x ) = C ± for some q > 1, where C + < 0 and C − > 0 are constants. • F or each k = 1 , 2 , . . . , K let f ( x ) = u 0 ( x ) k . Then in each b ounded interv al there exist at most finitely many p oints x = ξ at which f 00 ( ξ ) = 0, and each is a simple inflection p oint: f 000 ( ξ ) 6 = 0. Suc h u 0 ∈ P will b e called admissible initial conditions. An example is sho wn in Figure 1. 6 - 10 - 5 5 10 0.5 1.0 1.5 2.0 - 10 - 5 5 10 - 1.5 - 1.0 - 0.5 0.5 1.0 1.5 - 10 - 5 5 10 - 6 - 4 - 2 2 Figure 1. An admissible initial condition for which u 0 0 ( x ) = − 5 x (1 + cos( π x ) / 2) / (1 + x 2 ) 2 . Left: u 0 ( x ). Cen ter: u 0 0 ( x ). Righ t: u 00 0 ( x ). Note that this particular admissible initial condition has an infinite num ber of inflection p oin ts asymptotically near integer v alues of x for x large. 3.3. Reflectionless Mo dification of the Initial Data. Our metho d will b e to study the Benjamin-Ono hierarc hy with admissible initial condition u 0 using the inv erse-scattering transform for the x -part of the Lax pair (see [4, 6]). The first step in the pro cess is to asso ciate to u 0 its scattering data, consisting of a complex-v alued reflection co efficient β ( λ ), λ > 0, as w ell as discrete eigenv alues { λ j < 0 } and corresponding phase constan ts { γ j ∈ R } . Matsuno was the first to observe that the conserv ation laws can b e used to deduce information ab out the scattering data (see [8], section 3.3) when the parameter  > 0 (which app ears parametrically in the scattering problem although the p otential u 0 is indep enden t of  ) is small. His analysis b ecomes more rigorous with the use of the trace formulae arising from the Kaup-Matsuno scheme. F or admissible initial conditions, Matsuno’s main results are: • The reflection co efficient β ( λ ) is small when   1. • The n um b er N of eigenv alues is large when   1, but (30) lim  ↓ 0 N = M := Z R u 0 ( x ) dx. The num b er N [ a, b ] of eigenv alues in the interv al [ a, b ], − L ≤ a ≤ b ≤ 0, L := max x ∈ R u 0 ( x ), satisfies (31) lim  ↓ 0 N [ a, b ] = Z b a F ( λ ) dλ, F ( λ ) := 1 2 π ( x + ( λ ) − x − ( λ )) . Here x − ( λ ) < x + ( λ ) are defined for − L < λ < 0 as the tw o solutions of the equation u 0 ( x ) = − λ . They play the role of turning p oints in this theory . (Note that the “mass” M defined b y (30) is finite for admissible u 0 although it is not so for general elements of P .) The solution of the Benjamin-Ono hierarc hy for an admissible initial condition is therefore in particular appro ximately reflectionless in the zero-dispersion limit. In the absence of reflection the exact solution of the hierarch y is a multi-soliton solution that takes the form [8]: (32) u ( x, t 1 , t 2 , . . . , t K ) = 2  ∂ ∂ x ={ log( τ ( x, t 1 , t 2 , . . . , t k )) } where the “tau function” is (33) τ ( x, t 1 , t 2 , . . . , t K ) := det( I + i − 1 A ) and A is an N × N Hermitian matrix with constan t off-diagonal elements (34) A nm := 2 i √ λ n λ m λ n − λ m , n 6 = m and diagonal elements dep ending explicitly on x, t 1 , t 2 , . . . , t K : (35) A nn := − 2 λ n ( X ( λ n ; x, t 1 , t 2 , . . . , t K ) + γ n ) , where for future conv enience we define a polynomial in λ b y (36) X ( λ ; x, t 1 , t 2 , . . . , t K ) := x − K X k =1 ( k + 1)( − λ ) k t k . 7 T o sp ecify an appropriate family of exact solutions of the Benjamin-Ono hierarch y , first define the exact n umber of approximate eigenv alues by (37) N (  ) :=  M   . Then, define approximations { ˜ λ n } N (  ) n =1 with − L < ˜ λ 1 < ˜ λ 2 < · · · < ˜ λ N (  ) < 0 by quantizing the Matsuno eigen v alue density: (38) Z ˜ λ n − L F ( λ ) dλ =   n − 1 2  , n = 1 , 2 , . . . , N (  ) . Next, define approximations for the phase constants { γ n } by setting [9]: (39) ˜ γ n := γ ( ˜ λ n ) , γ ( λ ) := − 1 2 ( x + ( λ ) + x − ( λ )) , − L ≤ λ < 0 . No w, for each  > 0, let ˜ u = ˜ u ( x, t 1 , t 2 , . . . , t K ) denote the exact solution of the Benjamin-Ono hierarch y giv en by the reflectionless solution formula (32) with determinantal tau function ˜ τ inv olving the N (  ) × N (  ) Hermitian matrix ˜ A whose elements are giv en by (34)–(35) with λ n and γ n replaced by ˜ λ n and ˜ γ n resp ectiv ely , for 1 ≤ n ≤ N (  ). In [9] it is prov ed that for admissible u 0 , (40) lim  ↓ 0 Z R | u 0 ( x ) − ˜ u ( x, 0 , 0 , . . . , 0) | 2 dx = 0 , so that the replacement of the scattering data we hav e just made amounts to a modification of the initial condition that is negligible in the L 2 ( R ) sense in the zero-disp ersion limit. 4. Distributional Limits of Conser ved Densities The conserved densities ˜ D m , m ≥ 1 (we are using tildes to remind the reader that these are expressions in ˜ u , its deriv ativ es in x , and Hilb ert transforms thereof ), all ha ve representations in terms of the tau function asso ciated with ˜ u . Indeed, since ˜ u satisfies (27) it follows from using the formula (32) for ˜ u that (41) 2  ∂ 2 ∂ x∂ t k ={ log( ˜ τ ) } = − ( k + 1) ∂ ∂ x ˜ D k +1 . In tegration in x using decay at x = ±∞ to fix the integration constant yields the form ulae (42) ˜ D m = − 2  m ∂ ∂ t m − 1 ={ log( ˜ τ ) } , m = 2 , 3 , . . . , K + 1 . Of course since ˜ D 1 = ˜ u a slightly different formula holds for ˜ D 1 according to (32). In principle, this gives a wa y of ev aluating ˜ D m for arbitrary m , although one must include dependence on a sufficien t num b er of times t k b y c ho osing K large enough. Let α 1 ≤ α 2 ≤ · · · ≤ α N (  ) denote the (real) eigenv alues of ˜ A . Then, w e ma y write the densities in the form (43) ˜ u = ˜ D 1 = ∂ ˜ U ∂ x , ˜ D m = − 1 m ∂ ˜ U ∂ t m − 1 , m = 2 , 3 , . . . , K + 1 , where (44) ˜ U ( x, t 1 , t 2 , . . . , t K ) :=  N (  ) X n =1 2 arctan(  − 1 α n ) . Remark ably , the function ˜ U has a completely explicit zero-disp ersion limit: Prop osition 1. Uniformly on c omp act subsets of R K +1 , (45) lim  ↓ 0 ˜ U ( x, t 1 , t 2 , . . . , t K ) = V ( x, t 1 , t 2 , . . . , t K ) := Z R π sgn( α ) G ( α ; x, t 1 , t 2 , . . . , t K ) dα 8 wher e (46) G ( α ; x, t 1 , t 2 , . . . , t K ) := − 1 4 π Z 0 − L χ I ( α ) dλ λ and wher e χ I ( α ) denotes the indic ator function of the interval (47) I := [ − 2 λ ( X ( λ ; x, t 1 , t 2 , . . . , t K ) − x + ( λ )) , − 2 λ ( X ( λ ; x, t 1 , t 2 , . . . , t K ) − x − ( λ ))] . Pr o of. This is a simple generalization of Prop osition 4.2 from [9] and it is prov ed in exactly the same wa y (see in particular sections 4.1–4.3 of that reference). F or the reader’s con venience we will simply describ e the idea of the pro of. The key observ ation is that the eigen v alues of the matrix ˜ A ha ve a limiting density G ( α ; x, t 1 , t 2 , . . . , t K ); that is, the normalized (to total mass M ) counting measures of eigenv alues of ˜ A conv erge in the w eak- ∗ sense to G dα as  ↓ 0. This fact is prov ed using Wigner’s metho d of momen ts. One studies the asymptotic b eha vior of traces of arbitrary p ow ers of the N (  ) × N (  ) matrix ˜ A in the limit N (  ) → ∞ , and with the use of some combinatorial arguments and approximation in terms of diagonal and T o eplitz matrices one obtains leading-order asymptotic formulae for these, which in turn are prop ortional to moments of the eigen v alue counting measures. Then one solv es the momen t problem for the limiting momen ts to obtain G dα . Finally , b y estimating the extreme eigen v alues of ˜ A one is able to conv ert con v ergence of moments to w eak- ∗ con v ergence. Next, one observes that the exact formula (44) can b e written as the integral of the function 2 arctan(  − 1 α ) against the normalized counting measure of eigenv alues of ˜ A . Poin t wise, the integrand con v erges to π sgn( α ) as  ↓ 0, and by a careful dominated conv ergence argument one then establishes the desired lo cally uniform con vergence of ˜ U to V .  Before giving our next result, w e recall the inviscid Burgers hierarc hy . Consider the equation (48) u B = u 0 x − K X k =1 ( k + 1) u k B t k ! . F or t 1 , . . . , t K all sufficien tly small (giv en x ∈ R ) it follows from the implicit function theorem that there exists a unique solution u B ( x, t 1 , t 2 , . . . , t K ) ≈ u 0 ( x ). As t k increases from zero, there will b e bifurcation p oin ts at which the num b er of solutions of (48) increases by a finite even integer (this is due to the condition on inflection points of f ( x ) = u 0 ( x ) k satisfied b y admissible u 0 ). Therefore, near a giv en x and at giv en v alues of t 1 , t 2 , . . . , t K , there will generically b e an o dd finite num ber 2 P ( x, t 1 , t 2 , . . . , t K ) + 1 of distinct solutions u B , 0 < u B , 1 < · · · < u B , 2 P to (48), and each is differen tiable with resp ect to all of the indep endent v ariables x and t 1 , t 2 , . . . , t K . By differentiation of (48) one observes that each of the solution branc hes is a function u B ( x, t 1 , t 2 , . . . , t K ) that simultaneously satisfies the equations (49) ∂ u B ∂ t k + ( k + 1) u k B ∂ u B ∂ x = 0 or ∂ u B ∂ t + ∂ ∂ x u k +1 B = 0 , k = 1 , 2 , . . . , K. These are the partial differential equations of the inviscid Burgers hierarch y . The simultaneous solution of these equations with initial condition u B ( x, 0 , 0 , . . . , 0) = u 0 ( x ) is accomplished b y the metho d of c haracter- istics and produces the solution in implicit form (48). W e note that whereas t ypically when Burgers-t yp e equations app ear in the theory of partial differential equations one is interested in single-v alued weak solu- tions representing sho ck wa v es, our interest here is in the multiv alued solution pro duced by finding all real solutions of the implicit equation (48). See Figures 1 and 2. Prop osition 2. The function V is of class C 1 ( R K +1 ) , and (50) ∂ V ∂ x = 2 P X n =0 ( − 1) n u B ,n ( x, t 1 , t 2 , . . . , t K ) while (51) − 1 m ∂ V ∂ t m − 1 = 2 P X n =0 ( − 1) n 1 m u B ,n ( x, t 1 , t 2 , . . . , t K ) m , m = 2 , 3 , . . . , K + 1 , 9 - 10 - 5 5 10 0.5 1.0 1.5 2.0 - 10 - 5 5 10 0.5 1.0 1.5 2.0 5.0 5.1 5.2 5.3 5.4 5.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 2. Left and center: the solution u B of (48) with initial data as sp ecified in Figure 1 with t k = 0 for k > 3 and with t 1 = t 2 = t 3 = 0 . 1 and t 1 = t 2 = t 3 = 3 resp ectiv ely . Right: a close-up of the case when t 1 = t 2 = t 3 = 3 displaying interv als of x with P = 0 , 1 , 2 (one, three, and five branches, resp ectiv ely). wher e P = P ( x, t 1 , t 2 , . . . , t K ) . Both of these formulae assume that ( x, t 1 , t 2 , . . . , t K ) ∈ R K +1 is a p oint at which the inte ger P is wel l-define d; however, sinc e new solution br anches bifur c ate in p airs fr om the same value of u B it is cle ar that the formulae extend by c ontinuity to al l of R K +1 . Pr o of. The pro of is virtually identical to that of Lemma 4.12 from [9]. The idea is as follows. With the use of the explicit form ula for G ( α ; x, t 1 , t 2 , . . . , t K ) one can exchange the order of integration in the formula (45) to obtain (52) V ( x, t 1 , t 2 , . . . , t K ) = Z 0 − L J ( λ ; x, t 1 , t 2 , . . . , t K ) dλ where (53) J ( λ ; x, t 1 , t 2 , . . . , t K ) :=      − π F ( λ ) , X ( λ ; x, t 1 , t 2 , . . . , t K ) < x − ( λ ) X ( λ ; x, t 1 , t 2 , . . . , t K ) + γ ( λ ) , x − ( λ ) ≤ X ( λ ; x, t 1 , t 2 , . . . , t K ) ≤ x + ( λ ) π F ( λ ) , X ( λ ; x, t 1 , t 2 , . . . , t K ) > x + ( λ ) . The integrand therefore has a different form as a function of λ in three different t ypes of subin terv als of [ − L, 0], with b oundary p oin ts giv en by the solutions λ of the equations (54) X ( λ ; x, t 1 , t 2 , . . . , t K ) = x ± ( λ ) . Recalling that x ± ( λ ) are t w o branches of the inv erse function of u 0 in the sense that for − L < λ < 0, u 0 ( x ± ( λ )) = − λ , b oth of these equations can b e combined in the form (55) − λ = u 0 ( X ( λ ; x, t 1 , t 2 , . . . , t K )) , whic h one immediately notices is the same implicit equation (48) providing the multiv alued solution of the Burgers hierarch y , under the substitution u B = − λ . Differen tiation of (52) using Leibniz’ rule to take in to accoun t the moving b oundaries then yields the desired formulae.  No w w e may formulate our main result. Theorem 1. L et v ∈ L 2 ( R ) . Then (56) lim  ↓ 0 Z R ˜ D 1 ( x, t 1 , t 2 , . . . , t K ) v ( x ) dx = Z R 2 P X n =0 ( − 1) n u B ,n ( x, t 1 , t 2 , . . . , t K ) ! v ( x ) dx holds uniformly for ( t 1 , . . . , t K ) in c omp act subsets of R K . R e c al ling that ˜ D 1 = ˜ u , this me ans that ˜ u c onver ges to the alternating sum of br anches of the multivalue d solution of the Bur gers hier ar chy with initial c ondition u 0 in the we ak L 2 ( R x ) sense. Mor e over, the c onver genc e is in the str ong L 2 ( R x ) sense if t 1 , . . . , t K ar e al l sufficiently smal l that u B is single-value d as a function of x . Now let φ ∈ D ( R ) b e a test function. Then for m = 2 , 3 , . . . , K + 1 , (57) lim  ↓ 0 Z R ˜ D m ( x, t 1 , t 2 , . . . , t K ) φ ( t m − 1 ) dt m − 1 = Z R 2 P X n =0 ( − 1) n 1 m u B ,n ( x, t 1 , t 2 , . . . , t K ) m ! φ ( t m − 1 ) dt m − 1 10 holds uniformly for ( x, t 1 , . . . , t m − 2 , t m , . . . , t K ) in c omp act subsets of R K . Ther efor e ˜ D m c onver ges to the alternating sum of m -th p owers, weighte d by 1 /m , of the br anches of the multivalue d solution of the Bur gers hier ar chy with initial c ondition u 0 in the top olo gy of D 0 ( R t m − 1 ) , that is, distributional c onver genc e with r esp e ct to t m − 1 . Pr o of. The distributional conv ergence of ˜ D m for m ≥ 2 clearly follows from our abov e results. It is also easy to conclude that (56) holds if v is sp ecialized to a test function φ ∈ D ( R ). T o strengthen this to w eak L 2 ( R x ) con v ergence and strong L 2 ( R x ) con v ergence pre-breaking, one follo ws nearly verbatim the argumen ts on pages 254–256 of [9].  W e exp ect that with some additional effort, the nature of the conv ergence of ˜ D m for m ≥ 2 can b e strengthened to exactly the same type as is av ailable for ˜ D 1 = ˜ u , a type of conv ergence that is more suitable for ev aluation at a p oint in the phase space P of fields. This exp ectation is based on the reasonable h yp othesis that the weak (or distributional) nature of the conv ergence stems from the presence of wild oscillations that can b e mo deled by mo dulated P -phase wa ve exact solutions of the Benjamin-Ono hierarch y as hav e b een describ ed by Matsuno [8] using the bilinear metho d of Hirota. These P -phase wa ves ha v e also b een obtained directly from the Lax pair for (1) (the k = 1 case of the hierarch y only) by Dobrokhoto v and Kric hever [3], who further provided a formal Whitham-type mo dulation theory for these wa ves, noting that the mo dulation equations simply tak e the form of 2 P + 1 copies of the inviscid Burgers equation (equation (49) for k = 1). In ligh t of our results, it appears that these 2 P + 1 copies should b e globally view ed as sheets of the same multiv alued solution. In an y case, if one in terprets the distributional limits in D 0 ( R t m − 1 ) as lo cal av erages of ˜ D m o ver v anishingly small in terv als of t m − 1 , then assuming only that the w av enum b ers and frequencies are not rationally dep endent, these av erages could just as well b e calculated o ver small interv als in x , holding t 1 , t 2 , . . . , t K fixed. In other words, if the weak limits are necessary due to the presence of mo dulated multiphase w a v es of w av elengths and p eriods prop ortional to  , then there should at generic points b e no difference betw een conv ergence in D 0 ( R t m − 1 ) and con vergence in D 0 ( R x ). A proof of such a result probably requires resolution of the microstructure as could b e obtainable from an approach to the zero-disp ersion limit that starts with the nonlo cal Riemann-Hilb ert problem of inv erse scattering for Benjamin-Ono, and that inv olves the dev elopment of some new analogue of the Deift-Zhou asymptotic metho d as has b een applied [2] to strengthen the zero-disp ersion limit of Kortew eg-de V ries equation. W e hop e to b e able to announce progress in this direction in the near future. It seems to us that while the Benjamin-Ono equation lo oks at first glance to b e a more complicated mo del for wa ve propagation than the more famous Korteweg-de V ries equation due to the presence of the Hilb ert transform and its concomitant nonlo cality (and perhaps ev en at “second glance”, since the treatmen t of the Benjamin-Ono equation by the inv erse-scattering transform metho d is far less well-understoo d than in the case of the Korteweg-de V ries equation), in fact it is far simpler in the zero-dispersion limit. Indeed, the asymptotic formulae that are the analogues in the Korteweg-de V ries case of our limiting formulae for ˜ D m require the solution of a v ariational problem for a quadratic functional with constrain ts as was found b y Da ve Levermore and Peter Lax in their pioneering work [7], while for Benjamin-Ono it suffices to b e able to solv e the implicit algebraic equation (48) for u B , or alternatively to solv e the system of partial differential equations (49) numerically b y the metho d of characteristics. These are far more elemen tary tasks. W e w ant to stress this p oin t to hop efully encourage the use in the practical mo deling of internal w av es of the simple approximate form ulae av ailable for the Benjamin-Ono equation and its hierarc h y when the disp ersion parameter  can b e reasonably assumed to b e small. References [1] Rob erto Camassa and Ric hard McLaughlin, “How do underw ater oil plumes form?”, online video from http://www.youtube.com/watch?v=6Cp6fHINQ94 . [2] P . Deift, S. V enakides, and X. Zhou, “New results in small disp ersion KdV by an extension of the steep est descent metho d for Riemann-Hilbert problems”, Internat. Math. Res. Notic es , 1997 , 286–299, 1997. [3] S. Y u. Dobrokhotov and I. M. Krichev er, “Multi-phase solutions of the Benjamin-Ono equation and their av eraging”, Mat. Zametki , 49 , 42–58, 1991. English translation in Math. Notes , 49 , 583–594, 1991. [4] A. S. F ok as and M. J. Ablowitz, “The inv erse scattering transform for the Benjamin-Ono equation — a pivot to multidi- mensional problems”, Stud. Appl. Math. , 68 , 1–10, 1983. [5] A. S. F ok as and B. F uchssteiner, “The hierarch y of the Benjamin-Ono equation”, Phys. L ett. , 86A , 341–345, 1981. 11 [6] D. J. Kaup and Y. Matsuno, “The inv erse scattering transform for the Benjamin-Ono equation”, Stud. Appl. Math. , 101 , 73–98, 1998. [7] P . D. Lax and C. D. Lev ermore, “The small disp ersion limit of the Kortew eg-de V ries equation”, Comm. Pur e Appl. Math. , 36 , 253–290 (Part I), 571–593 (Part I I), 809–929 (Part I II), 1983. [8] Y. Matsuno, Biline ar T r ansformation Method , Mathematics in Science and Engineering, 174 , Academic Press, Orlando, FL, 1984. [9] Peter D. Miller and Zheng jie Xu, “On the zero-disp ersion limit of the Benjamin-Ono Cauch y problem for p ositiv e initial data”, Comm. Pure Appl. Math. , 64 , 205–270, 2011. [10] A. Nak amura, “B¨ acklund transform and conserv ation laws of the Benjamin-Ono equation”, J. Phys. So c. Jap an , 47 , 1335–1340, 1979. Dep ar tment of Ma thema tics, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI 48109 E-mail address , P . D. Miller: millerpd@umich.edu URL , P . D. Miller: http://www.math.lsa.umich.edu/~millerpd E-mail address , Z. Xu: zhengjxu@umich.edu Curr ent addr ess , Z. Xu: Blo omberg L. P . 12