In this article we show that the large time asymptotics for the
Grinevich-Zakharov rational solutions of the Novikov-Veselov equation at
positive energy (an analog of KdV in 2+1 dimensions) is given by a finite sum
of localized travel waves (solitons)

Kazeykina, Anna

Novikov, Roman

English

arXiv

AbstractIn this article we show that the large time asymptotics for the Grinevich–Zakharov rational solutions of the Novikov–Veselov equation at positive energy (an analog of KdV in 2+1 dimensions) is given by a finite sum of localized travel waves (solitons)

Kazeykina, A.V.

Novikov, R.G.

Elsevier - Publisher Connector 

Large time asymptotics for the Grinevich–Zakharov potentials 

International audienceIn this article we show that the large time asymptotics for the Grinevich-Zakharov rational solutions of the Novikov-Veselov equation at positive energy (an analog of KdV in 2+1 dimensions) is given by a finite sum of localized travel waves (solitons)

HAL-Polytechnique

Large time asymptotics for the Grinevich-ZakharovpotentialsAnna Kazeykina, Roman NovikovTo cite this version:Anna Kazeykina, Roman Novikov. Large time asymptotics for the Grinevich-Zakharovpotentials. Bulletin des Sciences Mathe´matiques, Elsevier, 2011, 135 (4), pp.374-382.<10.1016/j.bulsci.2011.02.003>. <hal-00536791v2>HAL Id: hal-00536791https://hal.archives-ouvertes.fr/hal-00536791v2Submitted on 3 Feb 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Large time asymptotics for the Grinevich-ZakharovpotentialsA. V. Kazeykina and R. G. NovikovAbstract. In this article we show that the large time asymptotics forthe Grinevich–Zakharov rational solutions of the Novikov–Veselov equationat positive energy (an analog of KdV in 2+1 dimensions) is given by a finitesum of localized travel waves (solitons).1 IntroductionWe consider the following 2 + 1–dimensional analog of the KdV equation(Novikov–Veselov equation):∂tv = 4Re(4∂3zv + ∂z(vw)− E∂zw),∂z¯w = −3∂zv, v = v¯, E ∈ R,v = v(x, t), w = w(x, t), x = (x1, x2) ∈ R2, t ∈ R,(1.1)where∂t =∂∂t, ∂z =12(∂∂x1− i∂∂x2), ∂z¯ =12(∂∂x1+ i∂∂x2). (1.2)We assume thatv is sufficiently regular and has sufficient decay as |x| → ∞,w is decaying as |x| → ∞.(1.3)Equation (1.1) is contained implicitly in the paper of S.V. Manakov [M]as an equation possessing the following representation:∂(L− E)∂t= [L−E,A] +B(L− E) (1.4)(Manakov L − A − B triple), where L = −∆ + v(x, t), ∆ = 4∂z∂z¯, A andB are suitable differential operators of the third and zero order respectively.Equation (1.1) was written in an explicit form by S.P. Novikov and A.P.Veselov in [NV1], [NV2], where higher analogs of (1.1) were also constructed.Note that both Kadomtsev–Petviashvili equations can be obtained from (1.1)by considering an appropriate limit E → ±∞ (see [ZS], [Gr2]).In the present article we are focused on a very interesting family of so-lutions for equation (1.1) for E = Efix > 0 constructed by P.G. Grinevich1and V.E. Zakharov, see [Gr1], [Gr2] (containing also a reference to privatecommunication from V.E. Zakharov). The solutions of this family are givenbyv(x, t) = −4∂z∂z¯ ln detA,w(x, t) = 12∂2z ln detA,(1.5)where A = (Alm) is 4N × 4N–matrix,All =iE1/22(z¯ −zλ2l)− 3iE3/2t(λ2l −1λ4l)− γl,Alm =1λl − λmfor l 6= m,(1.6)E1/2 > 0, z = x1 + ix2, z¯ = x1 − ix2, ∂z , ∂z¯ are defined in (1.2), andλ1, . . . , λ4N , γ1, . . . , γ4N are complex numbers such thatλj 6= 0, |λj| 6= 1, j = 1, . . . , 4N, λl 6= λm for l 6= m,λ2j = −λ2j−1, γ2j−1 − γ2j =1λ2j−1, j = 1, . . . , 2N,λ4j−1 =1λ4j−3, γ4j−1 = λ¯24j−3γ¯4j−3, j = 1, . . . , N.(1.7)The functions v, w of (1.5)–(1.7) satisfy the Novikov–Veselov equation(1.1) for positive E of (1.6) and have also, in particular, the following prop-erties (see [Gr1], [Gr2]):v = v¯, w ∈ C∞(R2 × R),v(x, t), w(x, t) are rational functions of x and t,v(x, t) = O(|x|−2), w(x, t) = O(|x|−2), |x| → ∞, for each t ∈ R;(1.8)the Schro¨dinger equation Lψ = Eψ, where L = −∆+ v(x, t),has zero scattering amplitude for fixed E > 0 and t ∈ R. (1.9)Because of property (1.9) the potentials v of (1.5)–(1.7) are called trans-parent potentials.Note that in many respects the solutions (1.5)–(1.7) of (1.1) are similarto related solutions of the KP1 equation, see [FA], [Gr1], [Gr2].Note that the potentials v of (1.5)–(1.7) play an important role in the di-rect and inverse scattering theory for the aforementioned Schro¨dinger equa-tion Lψ = Eψ at fixed energy E > 0 and in the theory of integrable systems,see [Gr2], [JS], [VW], [WY], [FN], [N] and references therein. However, the2properties of these potentials were not yet studied sufficiently in the litera-ture.In order to formulate our results we use the following definition. We saythat a solution (v, w) of (1.1) is a travel wave iffv(x, t) = V (x− ct), w(x, t) = W (x− ct), x ∈ R2, t ∈ R, (1.10)for some functions V , W on R2 and some velocity c ∈ R2. In addition, weidentify c = (c1, c2) ∈ R2 with c = c1 + ic2 ∈ C.The main results of the present note consist of the following:(1) We show that (v, w) of the form (1.5)–(1.7) is a travel wave iff N = 1.See Lemma 2.1 of Section 2.(2) We show that there are no travel waves of the form (1.5)–(1.7), N = 1,for c ∈ UE , and that there is an unique (modulo translations) travelwave of the form (1.5)–(1.7), N = 1, for c ∈ C\UE, where c denotestravel wave velocity and UE is defined by formula (2.2). In additionwe show that there is one–to–one correspondence between permittedvelocities c ∈ C\UE and λ–sets {λ1, λ2, λ3, λ4} of (1.5)–(1.7), N = 1.See Lemma 2.2 of Section 2.(3) We show that the large time asymptotics for the Grinevich–Zakharovpotentials, that is for (v, w) defined by (1.5)–(1.7), is described by asum of N localized travel waves propagating with different velocities.See Theorem 2.1 of Section 2. For KP1 equation a prototype of thisresult was given for the first time in [MZBIM].2 Main resultsThe main results of this article consist of Lemmas 2.1, 2.2 and Theorem 2.1presented below.Lemma 2.1. Let (v, w) be defined by (1.5)–(1.7). Then (v, w) admits therepresentation (1.10) (and is a travel wave solution for (1.1)) if and only ifN = 1. In addition,c = 6E(λ¯2 +1λ2+λ2λ2)(2.1)where c is the travel wave velocity and λ is any of λj, j = 1, 2, 3, 4, which, invirtue of (1.7), determines uniquely the λ set {λ1, λ2, λ3, λ4} for E > 0.3Lemma 2.1 is proved in Section 3.LetU = {u ∈ C : u = reiϕ, r ≤ |6(2e−iϕ + e2iϕ)|, ϕ ∈ [0, 2pi]},UE = {u ∈ C : u/E ∈ U}.(2.2)One can see that U1 = U.Lemma 2.2. (a) Let c ∈ UE. Then there is no travel wave solution of(1.1) of the form (1.5)–(1.7) with N = 1 and the given travel wavevelocity c.(b) Let c ∈ C\UE. Then there exists unique (modulo translations) solutionof (1.1) of the form (1.5)–(1.7) with N = 1 and the given travel wavevelocity c.(c) There is a one–to–one correspondence between c ∈ C\UE and the sets{λ1, λ2, λ3, λ4} satisfying (1.7).The proof of this Lemma is given in Section 3 and is based principally onthe following auxiliary lemma.Lemma 2.3. (a) Let c ∈ UE. Then equation (2.1) has no solution λ sat-isfying |λ| 6= 1.(b) Let c ∈ C\UE, then equation (2.1) has exactly four solutions λ1, λ2,λ3, λ4 satisfying the conditions indicated in (1.7) for N = 1.This Lemma is a corollary of Lemma 3.1 from [KN].Theorem 2.1. Let (v, w) be a solution of (1.1) constructed via (1.5)–(1.7).Then the asymptotical behavior of (v, w) can be described as follows:v ∼N∑k=1νk(ξk), w ∼N∑k=1ωk(ξk) as t→∞, (2.3)where ξk = z − c4kt andcl = 6E(λ¯2l +1λ2l+λ2lλ2l). (2.4)The functions νk, ωk are defined by the formulasνk = −4∂z∂z¯ ln detA(k),ωk = 12∂2z ln detA(k),(2.5)4where matrix A(k) is a 4×4 submatrix of matrix A, defined by formulas (1.6),such thatA(k) = {Alm}4kl,m=4(k−1)+1. (2.6)Remark. The relation (2.3) is understood in the following sense:limt→∞v = limt→∞N∑k=1νk(ξk) for fixed ξ = z − ct, (2.7)wherelimt→∞νk(ξk) ={0, for fixed ξ = z − ct, c 6= c4k,νk(ξ), for fixed ξ = z − c4kt.(2.8)Theorem 2.1 is proved in Section 3. The scheme of the proof of thistheorem follows principally the scheme of the derivation of the large timeasymptotics for the multi–soliton solutions of the classic KdV equation (see,for example, [G]).3 Proofs of Lemmas 2.1, 2.2 and Theorem 2.1The text of the proofs presented below does not completely follow the orderof statements in Section 2 as it was constructed to form a whole logical unit.However, we specify in due course which statement is being proved.3.1 Proof of the sufficiency part of Lemma 2.1Let us first consider the Grinevich–Zakharov potentials defined by (1.5)–(1.7) with N = 1. Then A is a 4×4 matrix and, in virtue of (1.7), the choiceof any of λj , j = 1, 2, 3, 4, uniquely determines the set {λ1, λ2, λ3, λ4}. Letus find cj such that Ajj = Ajj(z − cjt). Such value cj is a solution of thefollowing equationi2E1/2(z¯ − c¯t−1λ2j(z − ct))=i2E1/2(z¯ −zλ2j)− 3iE3/2t(λ2j −1λ4j).(3.1)If |λj| 6= 1, then this equation is uniquely solvable and its solution is givenbycj = 6E(λ¯2j +1λ2j+λ2jλ¯2j). (3.2)It is easy to see that due to (1.7) c1 = c2 = c3 = c4. Thus A = A(z − ct),and the representation (1.10) with c defined by (2.1) holds. Thus sufficiencyin Lemma 2.1 is proved.53.2 Proof of Lemma 2.2If c ∈ UE, then, as follows from item (a) of Lemma 2.3, c 6= cj , defined by(3.2), j = 1, 2, 3, 4 for any λ1, λ2, λ3, λ4 satisfying the conditions indicatedin (1.7) for N = 1. This and the sufficiency part of Lemma 2.1 imply item(a) of Lemma 2.2.If c ∈ C\UE, then, as follows from item (b) of Lemma 2.3, it determinesvia (3.2) uniquely the set of λ1, λ2, λ3, λ4 satisfying the conditions indicatedin (1.7) for N = 1. Then the solution (v, w), constructed according to for-mulas (1.5)–(1.7) with N = 1, constitutes a travel wave solution of equation(1.1) with the given velocity c. In the construction procedure one of the pa-rameters γj can be chosen arbitrarily and it determines uniquely the wholeset {γ1, . . . , γ4}.One can see that the transformz → z + ζ,t→ t+ τturns the potential (v, w) into another Grinevich–Zakharov potential (v˜, w˜)with the parameters {λ1, . . . , λ4, γ˜1, . . . , γ˜4}, whereγj − γ˜j =iE2(ζ¯ −ζλ2j)− 3iE3/2τ(λ2j −1λ4j)(3.3)for j = 1, 2, 3, 4.On the other hand, if (v˜, w˜) is a Grinevich–Zakharov potential with theset of parameters {λ1, . . . , λ4, γ˜1, . . . , γ˜4}, then it can be obtained from (v, w)by a translation, i.e. v˜(z, t) = v(z + ζ, t + τ), w˜(z, t) = w(z + ζ, t + τ) forappropriate ζ ∈ C and t ∈ R such that (3.3) holds for some j (equations(3.3) are equivalent for j = 1, 2, 3, 4 in virtue of (1.7)). In addition, one canassume, for example, that τ = 0 in this translation.Thus we have proved that any c ∈ C\UE determines uniquely, modulotranslations, the solution of (1.1) of the form (1.5)–(1.7) with N = 1 and thegiven travel velocity c. This proves the point (b) of Lemma 2.2.Item (c) of Lemma 2.2 follows immediately from Lemma 2.3. Lemma 2.2is proved.3.3 Proof of Theorem 2.1Let us consider a more convenient representation of (v, w) defined by(1.5)–(1.7). For this purpose we first perform the differentiation with respectto z¯ in the right–hand side of formula for v in (1.5):v = −4∂z[(detA)−1∂z¯(detA)]= −4∂z[(detA)−14N∑i,j=1∂Aij∂z¯Aˆij], (3.4)6where Aˆij is the (i, j) cofactor of the matrix A. Similarly,w = 12∂z[(detA)−14N∑i,j=1∂Aij∂zAˆij]. (3.5)In matrix A only diagonal elements depend on z, z¯, thusv = −2iE1/2∂z[(detA)−14N∑j=1Aˆjj], w = −6iE1/2∂z[(detA)−14N∑j=11λ2jAˆjj].(3.6)Let us consider the following families V (j) and W (j) of systems of linearalgebraic equations for functions ψ(j)k , η(j)k , j, k = 1, . . . 4N :V (j) :4N∑k=1Amkψ(j)k = −2iE1/2δmj , m = 1, . . . , 4N, (3.7)W (j) :4N∑k=1Amkη(j)k = −6iE1/2 1λ2jδmj , m = 1, . . . , 4N. (3.8)Then the functions v, w can be represented in the following formv =4N∑j=1∂ψ(j)j∂z, w =4N∑j=1∂η(j)j∂z. (3.9)In order to write a system of linear algebraic equations for∂ψ(j)k∂z=(ψ(j)k)z,we differentiate (3.7) with respect to z:4N∑k=1(Amk)z ψ(j)k +4N∑k=1Amk(ψ(j)k)z= 0, m = 1, . . . , 4N,and thus obtain4N∑k=1Amk(ψ(j)k)z=iE1/22λ2mψ(j)m , m = 1, . . . , 4N. (3.10)Now let us note that Ajj can be represented in the formAjj =iE1/22[(z¯ − c¯jt)−1λ2j(z − cjt)]− γj,7where cj is given by formula (2.4). As follows from item (c) of Lemma 2.2cj = ck iff ⌊(j−1)/4⌋ = ⌊(k−1)/4⌋, where ⌊x⌋ denotes the integer part of x.Now let us fixξ = z − ctand find the limits ψ(j)k∣∣∣ t→∞ξ fixed,(ψ(j)k)z∣∣∣ t→∞ξ fixed. We note thatAjj =iE1/22[{ξ¯ + (c¯− c¯j)t} −1λ2j{ξ + (c− cj)t}].If c = cj , then Ajj =iE1/22[ξ¯ − 1λ2jξ]and is independent of t. Otherwise,|Ajj| → ∞ as t → ∞ at fixed ξ. We substitute this into (3.7) and considerthe leading term in the Cramer’s formula for ψ(j)k as t→∞. Thus we obtainψ(j)k∣∣∣ t→∞ξ fixed= ψˆ(j)k (ξ), k, j : ck = cj = c,ψ(j)k∣∣∣ t→∞ξ fixed= 0, k : ck 6= c or j : cj 6= c.Here ψˆ(j)k (ξ) denotes some function of ξ independent of t at fixed ξ.Similarly, from (3.10) we obtain that(ψ(j)k)z∣∣∣ t→∞ξ fixed= ψ¯(j)k (ξ), k, j : ck = cj = c,(ψ(j)k)z∣∣∣ t→∞ξ fixed= 0, k : ck 6= c or j : cj 6= c,and, as previously, ψ¯(j)k (ξ) denotes some function of ξ independent of t atfixed ξ.In addition, one can see that if there exists k such that c = c4(k−1)+1 =. . . = c4k, thenv|t→∞ξ fixed=4k∑j=4(k−1)+1ψ¯(j)j (ξ) = νk(ξ), (3.11)where νk is defined by formulaνk = −4∂z∂z¯ ln detA(k), (3.12)8matrix A(k) is a 4 × 4 submatrix of matrix A from (1.6), such that A(k) ={Alm}4kl,m=4(k−1)+1.Similarly, for the case of function w we haveη(j)k∣∣∣ t→∞ξ fixed= ηˆ(j)k (ξ),(η(j)k)z∣∣∣ t→∞ξ fixed= η¯(j)k (ξ), k, j : ck = cj = c,η(j)k∣∣∣ t→∞ξ fixed= 0,(η(j)k)z∣∣∣ t→∞ξ fixed= 0, k : ck 6= c or j : cj 6= c,where ηˆjk(ξ), η¯jk(ξ) are some functions of ξ independent of t at fixed ξ. Ifthere exists k such that c = c4(k−1)+1 = . . . = c4k, thenw|t→∞ξ fixed=4k∑j=4(k−1)+1η¯(j)j (ξ) = ωk(ξ), (3.13)where ωk is defined by formulaωk = 12∂2z ln detA(k) (3.14)and matrix A(k) is the same as in (3.12).From (3.9), (3.11)–(3.14) it follows thatv ∼N∑k=1νk(ξk), w ∼N∑k=1ωk(ξk), t→∞. (3.15)Here ξk = z − c4kt, νk, ωk are defined by (2.5)–(2.6) and the meaning of therelation (3.15) is specified by (2.7)–(2.8). Theorem 2.1 is proved.3.4 Proof of the necessity part of Lemma 2.1From (3.15), taking into account (2.7)–(2.8), one can see that (v, w) canbe a travel wave only if N = 1. This completes the proof of Lemma 2.1.References[FA] Fokas A.S., Ablowitz M.J.: On the inverse scattering of thetime–dependent Schro¨dinger equation and the associated Kadomtsev–Petviashvili (I) equation. Studies in Appl. Math. 69, 211–228 (1983)[FN] Francoise J.–P., Novikov R.G.: Rational solutions of KdV–type equa-tions in dimension 2 + 1 and m–body problems on the line. C.R. Acad.Sci. Paris Se´r. I Math. 314(2), 109–113 (1992) (in French)9[G] Gabov S.A.: Introduction to the theory of non–linear waves. MoscowState University Press (1988) (in Russian)[Gr1] Grinevich P.G.: Rational solitons of the Veselov–Novikov equation arereflectionless potentials at fixed energy. TMF. 69(2), 307-310 (1986),translation in Theor. Math. Phys. 69, 1170-1172 (1986)[Gr2] Grinevich P.G.: The scattering transform for the two–dimensionalSchro¨dinger operator with a potential that decreases at infinity at fixednon–zero energy. Uspekhi Mat. Nauk. 55(6(336)), 3-70 (2000), transla-tion in Russian Math. Surveys. 55(6), 1015-1083 (2000)[JS] Joshi M.S., Sa´ Barreto A.: Recovering asymptotics of short range po-tentials. Comm. Math. Phys. 193(1), 197–208 (1998)[KN] Kazeykina A. V., Novikov R. G.: A large time asymptotics for trans-parent potentials for the Novikov–Veselov equation at positive energy.arXiv: 1010.2897 (2010)[M] Manakov S.V.: The inverse scattering method and two-dimensional evo-lution equations. Uspekhi Mat. Nauk. 31(5), 245-246 (1976) (in Russian)[MZBIM] Manakov S.V., Zakharov V.E., Bordag L.A., Its A.R., MatveevV.B.: Two–dimensional solitons of the Kadomtsev–Petviashvili equationand their interaction. Physics Letters A. 63(3), 205–206 (1977)[N] Novikov R.G.: Absence of exponentially localized solitons for theNovikov–Veselov equation at positive energy. Physics Letters A, to ap-pear; arXiv:1010.0770[NV1] Novikov S.P., Veselov A.P.: Finite-zone, two-dimensional, potentialSchro¨dinger operators. Explicit formula and evolutions equations. Dokl.Akad. Nauk SSSR. 279, 20-24 (1984), translation in Sov. Math. Dokl.30, 588-591 (1984)[NV2] Novikov, S.P., Veselov A.P.: Finite-zone, two-dimensional Schro¨dingeroperators. Potential operators. Dokl. Akad. Nauk SSSR. 279, 784-788(1984), translation in Sov. Math. Dokl. 30, 705-708 (1984)[VW] Vasy A., Wang X.–P.: Inverse scattering with fixed energy for dilation–analytic potentials. Inverse Problems. 20(4), 1349–1354 (2004)[WY] Weder R., Yafaev D.: On inverse scattering at a fixed energy for po-tentials with a regular behaviour at infinity. Inverse Problems. 21(6),1937–1952 (2005)10[ZS] Zakharov V.E., Shulman E.I.: Integrability of nonlinear systems andperturbation theory // What is integrability? Berlin: Springer–Verlag.185–250 (1991)(A.V. Kazeykina) Centre de Mathe´matiques Applique´es, Ecole Polytech-nique, Palaiseau, 91128, FranceLomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991,Russiae–mail: kazeykina@cmap.polytechnique.fr(R.G. Novikov) Centre de Mathe´matiques Applique´es, Ecole Polytechnique,Palaiseau, 91128, FranceIIEPT RAS–MITPAN, Profsoyuznaya str., 84/32, Moscow, 117997, Russiae–mail: novikov@cmap.polytechnique.fr11

Large time asymptotics for the Grinevich-Zakharov potentials

Bull. Sci. math. 135 (2011) 374–382
www.elsevier.com/locate/bulsci
Large time asymptotics for the Grinevich–Zakharov
potentials
A.V. Kazeykina a,b,∗, R.G. Novikov c,d
a Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 91128, France
b Lomonosov Moscow State University, GSP-1, Leninskie Gory, Moscow, 119991, Russia
c CNRS: UMR 7641, Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 91128, France
d IIEPT RAS–MITPAN, Profsoyuznaya str., 84/32, Moscow, 117997, Russia
Received 7 February 2011
Available online 19 February 2011
Abstract
In this article we show that the large time asymptotics for the Grinevich–Zakharov rational solutions of
the Novikov–Veselov equation at positive energy (an analog of KdV in 2 + 1 dimensions) is given by a
finite sum of localized travel waves (solitons).
© 2011 Elsevier Masson SAS. All rights reserved.
1. Introduction
We consider the following 2 + 1-dimensional analog of the KdV equation (Novikov–Veselov
equation):
∂tv = 4 Re
(
4∂3z v + ∂z(vw) − E∂zw
)
,
∂z¯w = −3∂zv, v = v¯, E ∈ R,
v = v(x, t), w = w(x, t), x = (x1, x2) ∈ R2, t ∈ R, (1.1)
where
∂t = ∂
∂t
, ∂z = 12
(
∂
∂x1
− i ∂
∂x2
)
, ∂z¯ = 12
(
∂
∂x1
+ i ∂
∂x2
)
. (1.2)
* Corresponding author at: Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, 91128, France.
E-mail addresses: kazeykina@cmap.polytechnique.fr (A.V. Kazeykina), novikov@cmap.polytechnique.fr
(R.G. Novikov).0007-4497/$ – see front matter © 2011 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.bulsci.2011.02.003
A.V. Kazeykina, R.G. Novikov / Bull. Sci. math. 135 (2011) 374–382 375We assume that
v is sufficiently regular and has sufficient decay as |x| → ∞,
w is decaying as |x| → ∞. (1.3)
Eq. (1.1) is contained implicitly in the paper of S.V. Manakov [8] as an equation possessing
the following representation:
∂(L − E)
∂t
= [L − E,A] + B(L − E) (1.4)
(Manakov L–A–B triple), where L = −+v(x, t),  = 4∂z∂z¯, A and B are suitable differential
operators of the third and zero order respectively. Eq. (1.1) was written in an explicit form by
S.P. Novikov and A.P. Veselov in [11,12], where higher analogs of (1.1) were also constructed.
Note that both Kadomtsev–Petviashvili equations can be obtained from (1.1) by considering an
appropriate limit E → ±∞ (see [15,5]).
In the present article we are focused on a very interesting family of solutions for Eq. (1.1)
for E = Efix > 0 constructed by P.G. Grinevich and V.E. Zakharov, see [4,5] (containing also a
reference to private communication from V.E. Zakharov). The solutions of this family are given
by
v(x, t) = −4∂z∂z¯ ln detA,
w(x, t) = 12∂2z ln detA, (1.5)
where A = (Alm) is 4N × 4N -matrix,
All = iE
1/2
2
(
z¯ − z
λ2l
)
− 3iE3/2t
(
λ2l −
1
λ4l
)
− γl,
Alm = 1
λl − λm for l = m, (1.6)
E1/2 > 0, z = x1 + ix2, z¯ = x1 − ix2, ∂z, ∂z¯ are defined in (1.2), and λ1, . . . , λ4N , γ1, . . . , γ4N
are complex numbers such that
λj = 0, |λj | = 1, j = 1, . . . ,4N, λl = λm for l = m,
λ2j = −λ2j−1, γ2j−1 − γ2j = 1
λ2j−1
, j = 1, . . . ,2N,
λ4j−1 = 1
λ4j−3
, γ4j−1 = λ¯24j−3γ¯4j−3, j = 1, . . . ,N. (1.7)
The functions v, w of (1.5)–(1.7) satisfy the Novikov–Veselov equation (1.1) for positive E
of (1.6) and have also, in particular, the following properties (see [4,5]):
v = v¯, v,w ∈ C∞(R2 × R),
v(x, t),w(x, t) are rational functions of x and t,
v(x, t) = O(|x|−2), w(x, t) = O(|x|−2), |x| → ∞, for each t ∈ R; (1.8)
the Schrödinger equation Lψ = Eψ , where L = − + v(x, t),
has zero scattering amplitude for fixed E > 0 and t ∈ R. (1.9)
376 A.V. Kazeykina, R.G. Novikov / Bull. Sci. math. 135 (2011) 374–382Because of property (1.9) the potentials v of (1.5)–(1.7) are called transparent potentials.
Note that in many respects the solutions (1.5)–(1.7) of (1.1) are similar to related solutions of
the KP1 equation, see [1,4,5].
Note that the potentials v of (1.5)–(1.7) play an important role in the direct and inverse scat-
tering theory for the aforementioned Schrödinger equation Lψ = Eψ at fixed energy E > 0 and
in the theory of integrable systems, see [5,6,13,14,2,10] and references therein. However, the
properties of these potentials were not yet studied sufficiently in the literature.
In order to formulate our results we use the following definition. We say that a solution (v,w)
of (1.1) is a travel wave iff
v(x, t) = V (x − ct), w(x, t) = W(x − ct), x ∈ R2, t ∈ R, (1.10)
for some functions V , W on R2 and some velocity c ∈ R2. In addition, we identify c =
(c1, c2) ∈ R2 with c = c1 + ic2 ∈ C.
The main results of the present note consist of the following:
(1) We show that (v,w) of the form (1.5)–(1.7) is a travel wave iff N = 1. See Lemma 2.1 of
Section 2.
(2) We show that there are no travel waves of the form (1.5)–(1.7), N = 1, for c ∈ UE , and
that there is an unique (modulo translations) travel wave of the form (1.5)–(1.7), N = 1,
for c ∈ C \ UE , where c denotes travel wave velocity and UE is defined by formula (2.2).
In addition we show that there is one-to-one correspondence between permitted velocities
c ∈ C \ UE and λ-sets {λ1, λ2, λ3, λ4} of (1.5)–(1.7), N = 1. See Lemma 2.2 of Section 2.
(3) We show that the large time asymptotics for the Grinevich–Zakharov potentials, that is for
(v,w) defined by (1.5)–(1.7), is described by a sum of N localized travel waves propagating
with different velocities. See Theorem 2.1 of Section 2. For KP1 equation a prototype of this
result was given for the first time in [9].
2. Main results
The main results of this article consist of Lemmas 2.1, 2.2 and Theorem 2.1 presented below.
Lemma 2.1. Let (v,w) be defined by (1.5)–(1.7). Then (v,w) admits the representation (1.10)
(and is a travel wave solution for (1.1)) if and only if N = 1. In addition,
c = 6E
(
λ¯2 + 1
λ2
+ λ
2
λ2
)
(2.1)
where c is the travel wave velocity and λ is any of λj , j = 1,2,3,4, which, in virtue of (1.7),
determines uniquely the λ set {λ1, λ2, λ3, λ4} for E > 0.
Lemma 2.1 is proved in Section 3.
Let
U = {u ∈ C: u = reiϕ, r  ∣∣6(2e−iϕ + e2iϕ)∣∣, ϕ ∈ [0,2π]},
UE = {u ∈ C: u/E ∈ U}. (2.2)
One can see that U1 = U.
A.V. Kazeykina, R.G. Novikov / Bull. Sci. math. 135 (2011) 374–382 377Lemma 2.2.
(a) Let c ∈ UE . Then there is no travel wave solution of (1.1) of the form (1.5)–(1.7) with N = 1
and the given travel wave velocity c.
(b) Let c ∈ C \ UE . Then there exists unique (modulo translations) solution of (1.1) of the form
(1.5)–(1.7) with N = 1 and the given travel wave velocity c.
(c) There is a one-to-one correspondence between c ∈ C \ UE and the sets {λ1, λ2, λ3, λ4} sat-
isfying (1.7).
The proof of this lemma is given in Section 3 and is based principally on the following auxil-
iary lemma.
Lemma 2.3.
(a) Let c ∈ UE . Then Eq. (2.1) has no solution λ satisfying |λ| = 1.
(b) Let c ∈ C \ UE , then Eq. (2.1) has exactly four solutions λ1, λ2, λ3, λ4 satisfying the condi-
tions indicated in (1.7) for N = 1.
This lemma is a corollary of Lemma 3.1 from [7] and is based on an analysis of the following
algebraic equation
λ6 − c¯
6E
λ4 + c
6E
λ2 − 1 = 0
depending on the complex parameter u = c/E.
Theorem 2.1. Let (v,w) be a solution of (1.1) constructed via (1.5)–(1.7). Then the asymptotical
behavior of (v,w) can be described as follows:
v ∼
N∑
k=1
νk(ξk), w ∼
N∑
k=1
ωk(ξk) as t → ∞, (2.3)
where ξk = z − c4kt and
cl = 6E
(
λ¯2l +
1
λ2l
+ λ
2
l
λ2l
)
. (2.4)
The functions νk , ωk are defined by the formulas
νk = −4∂z∂z¯ ln detA(k),
ωk = 12∂2z ln detA(k), (2.5)
where matrix A(k) is a 4 × 4-submatrix of matrix A, defined by formulas (1.6), such that
A(k) = {Alm}4kl,m=4(k−1)+1. (2.6)
Remark. The relation (2.3) is understood in the following sense:
lim
t→∞v = limt→∞
N∑
νk(ξk) for fixed ξ = z − ct, (2.7)k=1
378 A.V. Kazeykina, R.G. Novikov / Bull. Sci. math. 135 (2011) 374–382where
lim
t→∞νk(ξk) =
{
0, for fixed ξ = z − ct, c = c4k,
νk(ξ), for fixed ξ = z − c4kt. (2.8)
One can see that the asymptotics (2.3) are the same for ∞ specified as +∞ or as −∞. Thus
the solitons of (1.5)–(1.7) for the Novikov–Veselov equation (1.1) do not interact at all in a
similar way with the result of [9] for a family of rational soliton solutions for the KP1 equation.
Theorem 2.1 is proved in Section 3. The scheme of the proof of this theorem follows princi-
pally the scheme of the derivation of the large time asymptotics for the multi-soliton solutions of
the classic KdV equation (see, for example, [3]).
3. Proofs of Lemmas 2.1, 2.2 and Theorem 2.1
The text of the proofs presented below does not completely follow the order of statements in
Section 2 as it was constructed to form a whole logical unit. However, we specify in due course
which statement is being proved.
3.1. Proof of the sufficiency part of Lemma 2.1
Let us first consider the Grinevich–Zakharov potentials defined by (1.5)–(1.7) with N = 1.
Then A is a 4 × 4 matrix and, in virtue of (1.7), the choice of any of λj , j = 1,2,3,4, uniquely
determines the set {λ1, λ2, λ3, λ4}. Let us find cj such that Ajj = Ajj (z − cj t). Such value cj is
a solution of the following equation
i
2
E1/2
(
z¯ − c¯t − 1
λ2j
(z − ct)
)
= i
2
E1/2
(
z¯ − z
λ2j
)
− 3iE3/2t
(
λ2j −
1
λ4j
)
. (3.1)
If |λj | = 1, then this equation is uniquely solvable and its solution is given by
cj = 6E
(
λ¯2j +
1
λ2j
+ λ
2
j
λ¯2j
)
. (3.2)
It is easy to see that due to (1.7) c1 = c2 = c3 = c4. Thus A = A(z − ct), and the representa-
tion (1.10) with c defined by (2.1) holds. Thus sufficiency in Lemma 2.1 is proved.
3.2. Proof of Lemma 2.2
If c ∈ UE , then, as follows from item (a) of Lemma 2.3, c = cj , defined by (3.2), j = 1,2,3,4
for any λ1, λ2, λ3, λ4 satisfying the conditions indicated in (1.7) for N = 1. This and the suffi-
ciency part of Lemma 2.1 imply item (a) of Lemma 2.2.
If c ∈ C \ UE , then, as follows from item (b) of Lemma 2.3, it determines via (3.2) uniquely
the set of λ1, λ2, λ3, λ4 satisfying the conditions indicated in (1.7) for N = 1. Then the solution
(v,w), constructed according to formulas (1.5)–(1.7) with N = 1, constitutes a travel wave so-
lution of Eq. (1.1) with the given velocity c. In the construction procedure one of the parameters
γj can be chosen arbitrarily and it determines uniquely the whole set {γ1, . . . , γ4}.
One can see that the transform
z → z + ζ,
t → t + τ
A.V. Kazeykina, R.G. Novikov / Bull. Sci. math. 135 (2011) 374–382 379turns the potential (v,w) into another Grinevich–Zakharov potential (v˜, w˜) with the parameters
{λ1, . . . , λ4, γ˜1, . . . , γ˜4}, where
γj − γ˜j = iE2
(
ζ¯ − ζ
λ2j
)
− 3iE3/2τ
(
λ2j −
1
λ4j
)
(3.3)
for j = 1,2,3,4.
On the other hand, if (v˜, w˜) is a Grinevich–Zakharov potential with the set of parameters
{λ1, . . . , λ4, γ˜1, . . . , γ˜4}, then it can be obtained from (v,w) by a translation, i.e. v˜(z, t) = v(z +
ζ, t +τ), w˜(z, t) = w(z+ζ, t +τ) for appropriate ζ ∈ C and t ∈ R such that (3.3) holds for some
j (Eqs. (3.3) are equivalent for j = 1,2,3,4 in virtue of (1.7)). In addition, one can assume, for
example, that τ = 0 in this translation.
Thus we have proved that any c ∈ C \UE determines uniquely, modulo translations, the solu-
tion of (1.1) of the form (1.5)–(1.7) with N = 1 and the given travel velocity c. This proves the
point (b) of Lemma 2.2.
Item (c) of Lemma 2.2 follows immediately from Lemma 2.3. Lemma 2.2 is proved.
3.3. Proof of Theorem 2.1
Let us consider a more convenient representation of (v,w) defined by (1.5)–(1.7). For this
purpose we first perform the differentiation with respect to z¯ in the right-hand side of formula
for v in (1.5):
v = −4∂z
[
(detA)−1∂z¯(detA)
]= −4∂z
[
(detA)−1
4N∑
i,j=1
∂Aij
∂z¯
Aˆij
]
, (3.4)
where Aˆij is the (i, j) cofactor of the matrix A. Similarly,
w = 12∂z
[
(detA)−1
4N∑
i,j=1
∂Aij
∂z
Aˆij
]
. (3.5)
In matrix A only diagonal elements depend on z, z¯, thus
v = −2iE1/2∂z
[
(detA)−1
4N∑
j=1
Aˆjj
]
, w = −6iE1/2∂z
[
(detA)−1
4N∑
j=1
1
λ2j
Aˆjj
]
. (3.6)
Let us consider the following families V (j) and W(j) of systems of linear algebraic equations
for functions ψ(j)k , η
(j)
k , j, k = 1, . . . ,4N :
V (j):
4N∑
k=1
Amkψ
(j)
k = −2iE1/2δmj , m = 1, . . . ,4N, (3.7)
W(j):
4N∑
k=1
Amkη
(j)
k = −6iE1/2
1
λ2j
δmj , m = 1, . . . ,4N. (3.8)
Then the functions v, w can be represented in the following form
v =
4N∑ ∂ψ(j)j
∂z
, w =
4N∑ ∂η(j)j
∂z
. (3.9)
j=1 j=1
380 A.V. Kazeykina, R.G. Novikov / Bull. Sci. math. 135 (2011) 374–382In order to write a system of linear algebraic equations for ∂ψ
(j)
k
∂z
= (ψ(j)k )z, we differentiate
(3.7) with respect to z:
4N∑
k=1
(Amk)zψ
(j)
k +
4N∑
k=1
Amk
(
ψ
(j)
k
)
z
= 0, m = 1, . . . ,4N,
and thus obtain
4N∑
k=1
Amk
(
ψ
(j)
k
)
z
= iE
1/2
2λ2m
ψ
(j)
m , m = 1, . . . ,4N. (3.10)
Now let us note that Ajj can be represented in the form
Ajj = iE
1/2
2
[
(z¯ − c¯j t) − 1
λ2j
(z − cj t)
]
− γj ,
where cj is given by formula (2.4). As follows from item (c) of Lemma 2.2 cj = ck iff (j −
1)/4	 = (k − 1)/4	, where x	 denotes the integer part of x.
Now let us fix
ξ = z − ct
and find the limits ψ(j)k
∣∣ t→∞
ξ fixed
, (ψ
(j)
k )z
∣∣ t→∞
ξ fixed
. We note that
Ajj = iE
1/2
2
[{
ξ¯ + (c¯ − c¯j )t
}− 1
λ2j
{
ξ + (c − cj )t
}]
.
If c = cj , then Ajj = iE1/22 [ξ¯ − 1λ2j ξ ] and is independent of t . Otherwise, |Ajj | → ∞ as t → ∞
at fixed ξ . We substitute this into (3.7) and consider the leading term in the Cramer’s formula for
ψ
(j)
k as t → ∞. Thus we obtain
ψ
(j)
k
∣∣ t→∞
ξ fixed
= ψˆ(j)k (ξ), k, j : ck = cj = c,
ψ
(j)
k
∣∣ t→∞
ξ fixed
= 0, k: ck = c or j : cj = c.
Here ψˆ(j)k (ξ) denotes some function of ξ independent of t at fixed ξ .
Similarly, from (3.10) we obtain that(
ψ
(j)
k
)
z
∣∣ t→∞
ξ fixed
= ψ¯(j)k (ξ), k, j : ck = cj = c,(
ψ
(j)
k
)
z
∣∣ t→∞
ξ fixed
= 0, k: ck = c or j : cj = c,
and, as previously, ψ¯(j)k (ξ) denotes some function of ξ independent of t at fixed ξ .
In addition, one can see that if there exists k such that c = c4(k−1)+1 = · · · = c4k , then
v| t→∞
ξ fixed
=
4k∑
j=4(k−1)+1
ψ¯
(j)
j (ξ) = νk(ξ), (3.11)
where νk is defined by formula
A.V. Kazeykina, R.G. Novikov / Bull. Sci. math. 135 (2011) 374–382 381νk = −4∂z∂z¯ ln detA(k), (3.12)
matrix A(k) is a 4 × 4-submatrix of matrix A from (1.6), such that A(k) = {Alm}4kl,m=4(k−1)+1.
Similarly, for the case of function w we have
η
(j)
k
∣∣ t→∞
ξ fixed
= ηˆ(j)k (ξ),
(
η
(j)
k
)
z
∣∣ t→∞
ξ fixed
= η¯(j)k (ξ), k, j : ck = cj = c,
η
(j)
k
∣∣ t→∞
ξ fixed
= 0, (η(j)k )z∣∣ t→∞ξ fixed = 0, k: ck = c or j : cj = c,
where ηˆjk (ξ), η¯
j
k (ξ) are some functions of ξ independent of t at fixed ξ . If there exists k such that
c = c4(k−1)+1 = · · · = c4k , then
w| t→∞
ξ fixed
=
4k∑
j=4(k−1)+1
η¯
(j)
j (ξ) = ωk(ξ), (3.13)
where ωk is defined by formula
ωk = 12∂2z ln detA(k) (3.14)
and matrix A(k) is the same as in (3.12).
From (3.9), (3.11)–(3.14) it follows that
v ∼
N∑
k=1
νk(ξk), w ∼
N∑
k=1
ωk(ξk), t → ∞. (3.15)
Here ξk = z − c4kt , νk , ωk are defined by (2.5)–(2.6) and the meaning of the relation (3.15) is
specified by (2.7)–(2.8). Theorem 2.1 is proved.
3.4. Proof of the necessity part of Lemma 2.1
From (3.15), taking into account (2.7)–(2.8), one can see that (v,w) can be a travel wave only
if N = 1. This completes the proof of Lemma 2.1.
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Large time asymptotics for the Grinevich-Zakharov potentials

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