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APPEARED IN BULLETIN OF THE

arXiv:math/9210215v1 [math.AP] 1 Oct 1992APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 27, Number 2, October 1992, Pages 266-272NEW TYPES OF SOLITON SOLUTIONSF. Gesztesy, W. Karwowski, and Z. ZhaoAbstract.

We announce a detailed investigation of limits of N-soliton solutionsof the Korteweg-deVries (KdV) equation as N tends to infinity. Our main resultsprovide new classes of KdV-solutions including in particular new types of soliton-like(reflectionless) solutions.

As a byproduct we solve an inverse spectral problem for one-dimensional Schr¨odinger operators and explicitly construct smooth and real-valuedpotentials that yield a purely absolutely continuous spectrum on the nonnegative realaxis and give rise to an eigenvalue spectrum that includes any prescribed countableand bounded subset of the negative real axis.IntroductionIn this note we announce the construction of new types of soliton-like solutionsof the Korteweg-de Vries (KdV)-equation. More precisely, we offer a solution tothe following problem:Construct new classes of KdV-solutions by taking limits of N-soliton solutionsas N →∞.As it turns out, our solution to this problem is intimately connected with a solu-tion to the following inverse spectral problem in connection with one-dimensionalSchr¨odinger operators H = −d2/dx2 + V in L2(R):Given any bounded and countable subset {−κ2j}j∈N of (−∞, 0), construct a (smoothand real-valued) potential V such that H = −d2/dx2 + V has a purely absolutelycontinuous spectrum equal to [0, ∞) and the set of eigenvalues of H includes theprescribed set {−κ2j}j∈N.In addition, we also construct a new class of reflectionless KdV-solutions inwhich the underlying Schr¨odinger operator has infinitely many negative eigenvaluesaccumulating at zero.Although we present our results exclusively in the KdV-context, it will becomeclear later on that our methods are not confined to KdV-type equations but arewidely applicable in the field of integrable systems.Received by the editors November 19, 19911980 Mathematics Subject Classification (1985 Revision).

Primary 35Q51, 35Q53; Secondary58F07The first two authors were partially supported by the BiBoS-Research Center at the Facultyof Physics, University of Bielefeld and the Department of Mathematics, University of Bochum,FRG. The first author was also partially supported by the Norwegian Research Council for Scienceand the Humanities during a stay at the Division of Mathematical Sciences of the Universityof Trondheim, Norway.The second author was also partially supported by CNRS-Marseille,the University of Provence I, Marseille, France, and the German-Polish scientific collaborationprogramc⃝1992 American Mathematical Society0273-0979/92 $1.00 + $.25 per page1

2F. GESZTESY, W. KARWOWSKI, AND Z. ZHAOBefore formulating our results in detail we briefly review some background ma-terial.

The celebrated N-soliton solutions VN(t, x) of the KdV-equation(1)KdV(V ) = Vt −6V Vx + Vxxx = 0described, e.g., in [1, 2, 4](2)VN(t, x) = −2∂2x ln det[1N + CN(t, x)],(t, x) ∈R2,CN(t, x) =cjcℓκj + κℓe4(κ3j+κ3ℓ)t−(κj+κℓ)xNj,ℓ=1,cj > 0, κj > 0, 1 ≤j ≤N, N ∈Nare well known to be isospectral and reflectionless potentials VN(t, x) in connectionwith the one-dimensional Schr¨odinger operator(3)HN(t) = −d2dx2 + VN(t, . )in L2(R).

In particular, the spectrum σ(HN(t)) of HN(t) is independent of t andis given by(4)σ(HN(t)) = {−κ2j}Nj=1 ∪[0, ∞)with purely absolutely continuous spectrum [0, ∞). Hence HN(t) are isospectral de-formations of HN(0), which is clear from the Lax formalism connecting (1) and (3).The reflectionless property of VN manifests itself in the (t-independent) scatteringmatrix SN(k) in C2 associated with the pair (HN(t), H0)SN(k) =TN(k)00TN(k),TN(k) =NYj=1k + iκjk −iκj, k ∈C\{iκj}Nj=1,where H0 = −d2/dx2 and z = k2 is the spectral parameter corresponding to H0.Here TN(k) denotes the transmission coefficient and the vanishing of the off-diagonalterms in SN(k) exhibits reflection coefficients identical to zero at all energies.

Inmore intuitive terms this remarkable and highly exceptional behavior can be de-scribed as follows: If one views V as representing an “obstacle” for an incoming“signal” (wave, etc.) then the outgoing signal generically consists of two parts, atransmitted and a reflected one.

It is in the exceptional case of N-soliton poten-tials VN such as (2) that the reflected part of the outgoing signal is entirely missingand hence the obstacle appears to be completely transparent independently of thewavelength of the incoming signal.Incidentally, (4) offers a solution to the following inverse spectral problem: Giventhe finite set {−κ2j}Nj=1 ⊂(−∞, 0), construct (smooth and real-valued) potentialsVN such that HN = −d2/dx2 + VN has a purely absolutely continuous spectrumequal to [0, ∞) and precisely the eigenvalues {−κ2j}Nj=1.A natural generalization of this fact would be to ask whether one can choose asequence {cj > 0}j∈N such that for an arbitrarily prescribed bounded and countableset {−κ2j}j∈N ⊂(−∞, 0), VN(t, x) converge to a smooth KdV-solution V∞(t, x) asN →∞with the associated Schr¨odinger operator H∞(t) = −d2/dx2 + V∞(t, . )having the purely absolutely continuous spectrum [0, ∞) and containing the set{−κ2j}j∈N in its point spectrum.The main goal of this note is to present an affirmative answer to this question.

NEW TYPES OF SOLITON SOLUTIONS3Main resultsTheorem 1. Assume {κj > 0}j∈N ∈ℓ∞(N), κj ̸= κℓfor j ̸= ℓ, and choose{cj > 0}j∈N such that {c2j/κj}j∈N ∈ℓ1(N).

Then VN converges pointwise to someV∞∈C∞(R2) ∩L∞(R2) as N →∞and(i) limx→+∞V∞(t, x) = 0 and(5)limN→∞sup(t,x)∈K|∂mt ∂nxVN(t, x) −∂mt ∂nxV∞(t, x)| = 0,m, n ∈N0,for any compact subset K ⊂R2. Moreover,(6)KdV(V∞) = 0.

(ii) Denoting H∞(t) = −d2/dx2 + V∞(t, .) we haveσess(H∞(t)) = {−κ2j}′j∈N ∪[0, ∞),(7)σac(H∞(t)) = [0, ∞),(8)[σp(H∞(t)) ∪σsc(H∞(t))] ∩(0, ∞) = ∅,(9){−κ2j}j∈N ⊆σp(H∞(t)) ⊆{−κ2j}j∈N.

(10)The spectral multiplicity of H∞(t) on (0, ∞) equals two while σp(H∞(t)) issimple. In addition, if {κj}j∈N is a discrete subset of (0, ∞) (i.e., if 0 is its onlylimit point) thenσsc(H∞(t)) = ∅,(11)σ(H∞(t)) ∩(−∞, 0) = σd(H∞(t)) = {−κ2j}j∈N.

(12)More generally, if {−κ2j}′j∈N is countable then (11) holds.Here A denotes the closure of A ⊂R, A′ is the derived set of A (i.e., the setof accumulation points of A), N0 = N ∪{0}, and σess(. ), σac(.

), σsc(. ), σd(.

), andσp(.) denote the essential, absolutely continuous, singularly continuous, discrete,and point spectrum (the set of eigenvalues) respectively.Under stronger assumptions on {κj}j∈N we obtainTheorem 2.

Assume {κj > 0}j∈N ∈ℓ1(N), κj ̸= κℓfor j ̸= ℓ, and choose {cj >0}j∈N such that {c2j/κj}j∈N ∈ℓ1(N). Then in addition to (5) and (6) we have(i)(13)limN→∞||∂mt ∂nxVN(t, .) −∂mt ∂nxV∞(t, .

)||p = 0,m, n ∈N0, 1 ≤p ≤∞. (ii)σess(H∞(t)) = σac(H∞(t)) = [0, ∞),σp(H∞(t)) ∩(0, ∞) = σsc(H∞(t)) = ∅,σd(H∞(t)) = {−κ2j}j∈N .

4F. GESZTESY, W. KARWOWSKI, AND Z. ZHAO(iii) The (t-independent) scattering matrix S∞(k) in C2 associated with the pair(H∞(t), H0) is reflectionless and given byS∞(k) =T∞(k)00T∞(k),T∞(k) =∞Yj=1k + iκjk −iκj, k ∈C\{{iκj}j∈N ∪{0}}.While Theorem 1 solves the two problems stated in the introduction, Theo-rem 2 constructs a new class of reflectionless potentials in connection with one-dimensional Schr¨odinger operators involving an infinite negative point spectrumaccumulating at zero.Moreover, suppose V ∈C∞(R2) to be real-valued with∂mx V (t, .) ∈L1(R), m ∈N, and either V (t, .) ∈L1(R; (1+|x|ε) dx) for some ε > 0 orthat T (k), k ∈R\{0}, the transmission coefficient associated with the pair (H(t) =−d2/dx2 +V (t, .

), H0), has an analytic continuation into {C+\{iκj}−κ2j∈σd(H(0))}∪{k ∈C\{0}|k| < η, Im k ≤0} for some η > 0 (C+ = {k ∈C| Im k > 0}). Intro-ducing the KdV-invariants χn ∈C∞(R2) byχ1 = V,χ2 = −Vx,χn+1 = −∂x χn −n−1Xm=1χn−mχm,n ≥2,an extension of the results in [5, 13] yields the conservation laws (trace relations)(14)−ZRdxχ2n+1(t, x) = 22(n+1)2n + 1X−κ2j∈σd(H)κ2n+1j+ (−1)n22(n+1) 1πZ ∞0dkk2n ln |T (k)|,n ∈N0assumingX−κ2j∈σd(H) κj < ∞(see [6] for details).

Since |T (k)| ≤1 for k > 0 bythe unitarity of the associated scattering matrix, this yields the bounds−ZRdxχ4m+1(t, x) ≤22(2m+1)4m + 1X−κ2j∈σd(H)κ4m+1j,m ∈N0,(15)−ZRdxχ4m+3(t, x) ≥22(2m+2)4m + 3X−κ2j∈σd(H)κ4m+3j,m ∈N0. (16)For m = 0 the bound (15) can be found in [12].

In the case where ∂mx V (t, .) ∈L1(R; (1+|x|) dx), m ∈N0, and hence σd(H(t)) is finite, (15) and (16) are discussed,e.g., in [8, 10].

By (14), the bounds (15) and (16) saturate iff|T (k)| = 1 for k > 0,i.e., iffV is reflectionless. Consequently, the bounds (15) and (16) saturate if Vequals the N-soliton solutions VN in (2) and, in particular, if V is an element ofour new class of reflectionless KdV-solutions V∞described in Theorem 2.

NEW TYPES OF SOLITON SOLUTIONS5Sketch of proofsThe hypotheses in Theorem 1 guarantee that CN(t, x) (viewed as an operator inℓ2(N)) converges for any fixed (t, x) ∈R2 in trace norm to some trace class operatorC∞(t, x) ∈B1(ℓ2(N)) as N →∞and hencelimN→∞VN(t, x) = V∞(t, x) = −2∂2x ln det1[1 + C∞(t, x)],where det1(.) denotes the corresponding Fredholm determinant.

A crucial identityin proving (5) and (6) is(17)V∞(t, x) = −4∞Xj=1κjψ∞,j(t, x)2,where {ψ∞,j(t, x)}j∈N turn out to be the eigenfunctions of H∞(t) corresponding tothe eigenvalues {−κ2j}j∈N, determined by(18)Ψ∞(t, x) = [1 + C∞(t, x)]−1Ψ0∞(t, x),Ψ0∞(t, x) = {cje−κjx}Tj∈N,Ψ∞(t, x) = {ψ∞,j(t, x)}Tj∈Nin ℓ2(N). Equations (17) and (18) are well known in the context of VN and canbe obtained by pointwise limits as N →∞.Equation (10) then follows fromstrong resolvent convergence of HN(t) to H∞(t) and σess(H∞(t)) ⊇[0, ∞) is aconsequence of V∞(t, x) →x→+∞0.Next one constructs the Weyl m-functionsm±∞(t, z) associated with H±∞,D(t), the restriction of H∞(t) to the interval (0, ±∞)with a Dirichlet boundary condition at 0.

One obtainsm±∞(t, z) = ± i√z ∓[1 ∓i∞Xj=1(√z ± iκj)−1cjψ∞,j(t, 0)]−1× i∞Xj=1(√z ± iκj)−1cj[∂xψ∞,j(t, 0) −κjψ∞,j(t, 0)],z ∈C\R,defining the branch of √z by limε↓0p|λ| ± iε = ±|λ|1/2, limε↓0p−|λ| ± iε =i|λ|1/2. Since m±∞(t, .) are bounded on any region of the type Jδ,R1,R2 = {z =λ + iνR1 < λ < R2, 0 < ν < δ}, δ, R1, R2 >0 andlimε↓0 Im[m±∞(t, λ + iε)]is bounded away from zero for λ ∈(R1, R2), the spectrum of H∞(t) in (0, ∞) ispurely absolutely continuous by Theorem 3.1 of [11] and hence (9) and σac(H∞(t)) ⊃(0, ∞) follow.

Next one proves the following lemma on the basis of Hp-theory,0 < p < 1 (see, e.g., [3]).

6F. GESZTESY, W. KARWOWSKI, AND Z. ZHAOLemma 3.3.

Let {aj}j∈N, {bj}j∈N ⊂R, {aj}j∈N ∈ℓ1(N). Then there exists areal-valued function f on [0, ∞) withm({x ∈[0, ∞)f(x) = c}) = 0for each c ∈Rsuch thatlimε↓0∞Xj=1aj√x −iε −bj= f(x)for m-a.e.

x ≥0. (Here m denotes the Lebesgue measure on R.) Identifying x = −λ, λ < 0, bj =∓κj, aj = cjψ∞,j(t, 0) resp.

aj = cj[∂xψ∞,j(t, 0) −κjψ∞,j(0, t)], Lemma 3 appliedto m±∞(t, .) yields the existence of real-valued and finite limits of m±∞(t, λ + iε) fora.e.

λ < 0 as ε ↓0. Thusµ±∞,ac((−∞, 0)) = 0,where µ±∞,ac is the absolutely continuous part (with respect to m) of the Stieltjesmeasure generated by the spectral function of H±∞,D(t).

Consequently σac(H±∞,D(t)),being the topological support of µ±∞,ac, is contained in [0, ∞). This together withσac(H∞(t)) = σac(H+∞,D(t)) ∪σac(H−∞,D(t)) yieldsσac(H∞(t)) ∩(−∞, 0) = ∅and hence (8).

The rest of Theorem 1 is plain.Finally we turn to Theorem 2. Due to (17), the fact that ∂2x ψ∞,j = (V∞+κ2j)ψ∞,j and the KdV-equation (6) for V∞one can show it suffices to prove (13) for0 ≤m ≤2 and n = 0.

This is accomplished in a series of steps. First one provesthe crucial identityZRdxV∞(t, x) = −4∞Xj=1κj,which follows fromLemma 4.

Assume the hypotheses in Theorem 1. Thendet1[1 + C∞(0, x)] = 1 +XI∈Pa∞,Ie−2 Pj∈I κjx,where P is the family of all finite, nonempty subsets of N and a∞,I > 0 are positivenumbers (whose precise value turns out to be immaterial for the proof of Theorem 2)and a detailed study of the asymptotic behavior of det1[1 + C∞(0, x)] as |x| →∞.In the sequel one repeatedly invokes the identity (17) and Vitali’s theorem ([9,p.

203]). The rest of Theorem 2 follows from Theorem 1(ii) and scattering theoryfor L1(R)-potentials.Detailed proofs can be found in [7].We feel that the simplicity of constructing KdV-solutions producing these re-markable spectral (resp.

scattering) properties represents a significant result thatdeserves further investigations.In particular, generalizations, replacing the N-soliton KdV-solutions VN by N-gap quasi-periodic KdV-solutions and studying the

NEW TYPES OF SOLITON SOLUTIONS7limit N →∞involving accumulations of spectral gaps and bands, appear to offera variety of interesting and challenging problems.Due to the close resemblance of the determinant structure of the N-soliton solu-tions of (hierarchies of) integrable systems such as the AKNS-class (particularly thenonlinear Schr¨odinger and Sine-Gordon equations), the Toda lattice, and especiallythe Kadomtsev-Petviashvili equation, the methods in this paper are by no meansconfined to KdV-type equations but are widely applicable in this field.AcknowledgmentsWe are indebted to N. Kalton and W. Kirsch for several stimulating discussions.In particular, we would like to thank N. Kalton for invaluable help in connectionwith Lemma 3.References1. P. A. Deift, Applications of a commutation formula, Duke Math.

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5 (1971), 280–287.Department of Mathematics, University of Missouri, Columbia, Missouri 65211E-mail address: mathfg@umcvmb.bitnetInstitute of Theoretical Physics, University of Wroclaw, 50-205 Wroclaw, PolandE-mail address: iftuwr@plwrtu11Department of Mathematics, University of Missouri, Columbia, Missouri 65211E-mail address: mathzz@umcvmb.bitnet

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