PARTIAL DEGENERATION OF FINITE GAP SOLUTIONS TO THE KORTEWEG-DE VRIES EQUATION SOLITON GAS AND SCATTERING ON ELLIPTIC BACKGROUNDS

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PARTIAL DEGENERATION OF FINITE GAP SOLUTIONS TO THE

KORTEWEG-DE VRIES EQUATION: SOLITON GAS AND SCATTERING ON

ELLIPTIC BACKGROUNDS

M. BERTOLA♠,♣,♥,♦AND R. JENKINS?AND A. TOVBIS?

Abstract. We obtain Fredholm type formulas for partial degenerations of Theta functions on (irreducible)

nodal curves of arbitrary genus, with emphasis on nodal curves of genus one. An application is the study

of “many-soliton” solutions on an elliptic (cnoidal) background standing wave for the Korteweg-de Vries

(KdV) equation starting from a formula that is reminiscent of the classical Kay-Moses formula for N-

solitons. In particular, we represent such a solution as a sum of the following two terms: a “shifted” elliptic

(cnoidal) background wave and a Kay-Moses type determinant containing Jacobi theta functions for the

solitonic content, which can be viewed as a collection of solitary disturbances on the cnoidal background.

The expressions for the travelling (group) speed of these solitary disturbances, as well as for the interaction

kernel describing the scattering of pairs of such solitary disturbances, are obtained explicitly in terms of

Jacobi theta functions. We also show that genus N+ 1 ﬁnite gap solutions with random initial phases

converge in probability to the deterministic cnoidal wave solution as Nbands degenerate to a nodal curve

of genus one. Finally, we derive the nonlinear dispersion relations and the equation of states for the KdV

soliton gas on the residual elliptic background.

Contents

1. Introduction and results 1

2. Generalities about nodal hyperelliptic and elliptic curves 7

2.1. Terminology. 8

2.2. General properties of nodal curves and their period matrix 9

2.3. KdV tau function 14

3. Velocity of a single soliton on elliptic background: Bright-and-forward versus dim-and-retrograde 17

4. Scattering of soliton pairs over cnoidal background 18

5. Elliptic gas of solitons for KdV 24

Appendix A. Arbitrary genus 26

A.1. Notations and conventions. 26

Appendix B. Average and convergence in probability 28

References 30

1. Introduction and results

The Korteweg-de Vries (KdV) equation

ut+uxxx + 6uux= 0, u =u(x, t)(1.1)

♠SISSA, International School for Advanced Studies, via Bonomea 265, Trieste, Italy

♣Istituto Nazionale di Fisica Nucleare (INFN)

♥Centre de recherches math´

ematiques, Universit´

e de Montr´

eal, C. P. 6128, succ. centre ville, Montr´

eal,

Qu´

ebec, Canada H3C 3J7

♦Department of Mathematics and Statistics, Concordia University,, 1455 de Maisonnueve W., Montr´

eal,

Qu´

ebec, Canada H3G 1M8

?Department of Mathematics, University of Central Florida, Orlando, FL, USA

E-mail addresses:marco.bertola@concordia.ca, robert.jenkins@ucf.edu, alexander.tovbis@ucf.edu, .

2000 Mathematics Subject Classiﬁcation. 35Q53, 35C08, 14H70.

Key words and phrases. solitons, soliton gas, integrable PDE, Korteweg-deVries equation.

arXiv:2210.01350v3 [math-ph] 17 May 2023

2 PARTIAL DEGENERATION OF FINITE GAP SOLUTIONS OF KDV

is historically the ﬁrst equation shown [30] to admit solitary waves. The simplest such solution is the single

“soliton” (solitary wave) solution

u(x, t) = |b|/2

cosh2√|b|x−|b|3

2t+φ

2

(1.2)

which is a simple traveling wave moving to the right with speed v=|b|, where φ∈Ris an arbitrary

constant. The parameter b < 0 corresponds to the unique eigenvalue of the Sturm–Liouville operator

(stationary Schr¨odinger equation with potential −u(x, 0))

−f00(x)−u(x, 0)f(x) = b

4f(x), f ∈L2(R).(1.3)

If fis the Jost solution, then the associated norming constant γand shift φare given by γ=kfk−1

2and

φ= 2 log γ2

√|b|.

This particular solution can be written suggestively as

u(x, t)=2∂2

xln 

1 + e√|b|x−|b|3

2t+φ

2p|b|

.(1.4)

Despite the equation being nonlinear, KdV admits N-soliton solutions that describe a superposition of

the simple solitons introduced above. The N-soliton solution can be concisely described by the Kay-Moses

[21] formula of Fredholm type:

u(x, t) =2∂2

xln det [1N+G(x, t)] ,where

G`m(x, t) =√C`Cmeϑ`+ϑm

p|b`|+p|bm|, ϑ`:= p|b`|x− |b`|3

2t, ` = 1, . . . , N.(1.5)

Here: 1Ndenotes the identity matrix of size N;Gis the N×Nmatrix indicated above; the parameters b`

are arbitrary negative numbers, and; C`, ` = 1, . . . , N, are arbitrary positive numbers. It is well known that

for an N-soliton solution (1.5) the spectrum of (1.3) is {b1, . . . , bN}; the constant C`is called a norming

constant associated with b`,`= 1, . . . , N.

A diﬀerent family is the so-called ﬁnite-gap family of solutions [18,8], which can be written as

u(x, t)=2∂2

xln τ(x, t),(1.6)

where the tau–function τ(x, t) for the ﬁnite-gap solutions is expressed in terms of the Riemann theta function

associated to underlying hyperelliptic Riemann surface. (For the N–soliton solutions, the tau–function is

given by the Kay-Moses determinant (1.5).)

The starting point of this work is the obsrevation that N-soliton solutions can be obtained from the

ﬁnite-gap solutions by degenerating the hyperelliptic surface to a nodal curve of genus zero (ﬁrst observed

by Its and Matveev in [18]); the computation is contained essentially in the last chapter of Mumford’s book

[24], where a determinantal formula of diﬀerent type was derived (see also [19], [23] and [20], where the

degeneration procedure was used and a determinant formula for the N-soliton solutions of the focusing NLS

equation was obtained). In the recent work [15] the Kay-Moses formula is recovered from the degeneration

via the equivalence with the Wronskian formula of Matveev [23].

In this paper wepresent a generalization of the Kay-Moses formula to the case where the hyperelliptic

curve is partially degenerated to a nodal curve of genus 1. The core of the computation is, in fact, more

general; it is based on a formula (presented in Appendix A) for the limit of the Riemann Theta function

(with appropriate characteristics) when the curve degenerates to a nodal curve whose resolution has an

arbitrary genus g.

A formula for the partial degeneration can also be found in the monograph [2] (Chapter 4, p. 138) in

the context of the Nonlinear Schr¨odinger equation). This formula provides a diﬀerent, not determinantal

description of the degeneration (see Theorem 2.2 below), which also could be used to study the scattering

of the NLS solitons on the cnoidal background. We now describe the setting of the problem.

PARTIAL DEGENERATION OF FINITE GAP SOLUTIONS OF KDV 3

Consider the real elliptic curve

w2= 4z3−g2z−g3= 4(z−e1)(z−e2)(z−e3), e3< e2< e1, e1+e2+e3= 0(1.7)

with half-periods

$1:= Ze2

2p(z−e3)(z−e2)(z−e1)+∈R+

(1.8)

$3:= Ze2

2p(z−e3)(z−e2)(z−e1)∈iR−,(1.9)

where the radical is chosen with branchcuts [e3, e2]∪[e1,∞) and with the determination such that it is

in iR−in the gap [e2, e1]. Then the stationary cnoidal wave solution (genus one ﬁnite-gap or one-phase

nonlinear wave solution) is given by

u(x, t) = 2∂2

xln τ(x, t) = 2∂2

xln e−ζ($3)

8$3x2θ3x

4i$3

, τ,(1.10)

where τ:= $1/$3∈iR+, and θ1,2,3,4(β, τ) denote the standard Jacobi elliptic theta functions.

Note that the choice e1+e2+e3= 0 in (1.7) is made without loss of generality. If ˜u(x, t) is a solution of

(1.1) corresponding to the elliptic curve with ee1+ee2+ee3=v, then using the Galilean symmetry of (1.1)

˜u(x, t) = u(x−2vt, t) + v

3, where uis a solution of (1.1) associated with an elliptic curve with branchpoints

e1, e2, e3satisfying (1.7). In this way we can obtain any cnoidal travelling wave solution from the family of

stationary cnoidal solutions.

Choose Lpoints bj∈(−∞, e3), j = 1,...L and N−Lpoints bj∈(e2, e1), j =L+ 1,...N. These

points represent the centers of O(ε), ε→0+, fast shrinking bands of some genus N+1 hyperelliptic Riemann

surface RN(ε). If Nis ﬁxed, the rate of decay of the shrinking bands is not essential. (But in the case of

soliton gases, where N→ ∞ (see below), εis linked with Nand the rate of decay of the bands is important.)

Let Θ be the Riemann Theta function [13] on RN(ε):

Θ (X;Ω) := X

ν∈ZN+1

eiπν|Ων+2iπν|X,X∈CN+1,(1.11)

where Ω=Ω(ε) is the Riemann period matrix (our choice of Ajand Bjcycles is shown on Figure 5). Let

us set X= [ψ1, . . . , ψN, β]|∈CN+1. Our main observation, see Theorem 2.2, states that in the limit ε→0

the Riemann Theta function

lim

ε→0ΘX−1

2Ω(ε)u;Ω(ε)= det [1N+G]θ3(β− A),(1.12)

where the shift Adepends only on b1, . . . , bN, and the N×Nmatrix Gdepends on b1, . . . , bNand also on

X. The limiting elliptic curve (1.7) with Npairs of identiﬁed points b1, . . . , bN(on both sheets) will be

denoted by RN(0).

The factorization (1.12) represents the limiting Riemann Theta function as a product of Kay-Moses

type determinant and the (shifted) Riemann Theta function θ3on the residual elliptic background RN(0).

It is a direct generalization of Mumford’s approach to the case when all but one bands of the Riemann

surface are shrinking as ε→0 and, thus, in the limit, one obtains an N-soliton solution to the KdV on

the residual (elliptic) background. In Appendix A this result is generalized to an arbitrary hyperelliptic

residual background.

Our next main result, Theorem 2.5, states that

u(x, t)=2∂2

xln τ(x, t) with τ(x, t) := e−ζ($3)

8$3x2det [1N+G(x, t)] θ3x

4i$3− A

(1.13)

is a solution to the KdV (1.1), where ζdenotes the Weierstrass zeta function on the elliptic curve (1.7) and

the x, t dependence of the vector X, which is part of G, is deﬁned in terms of the limiting quasi-momentum

and quasi-energy meromorphic diﬀerentials on RN(ε) as ε→0, see Theorem 2.5 for details. Thus,

u(x, t)=2∂2

xln det [1N+G]+2∂2

xln θ3x

4i$3− A−ζ($3)

2$3

,(1.14)

4 PARTIAL DEGENERATION OF FINITE GAP SOLUTIONS OF KDV

b1b2b3b4b5b6b7b8b9b10 b11

Figure 1. The position of hot (red), j= 1, . . . , L, and cool (blue), j=L+ 1, . . . , N,

points bjof discrete spectra, relative to the spectral bands [e3, e2]∪[e1,∞) of the elliptic

background. These can be viewed as the degeneration of small bands—centered at each

discrete spectral point—on a higher genus surface.

i.e., up to a constant, solutions of KdV describing the Nsolitons on the elliptic background can be repre-

sented as a sum of the Kay-Moses type determinant “tweaked by the elliptic background” and the elliptic

background solution 2∂2

xln θ3(y) “shifted by the Nsolitons”.

Proofs of Theorems 2.2 and 2.5, including the appropriate notations from algebraic geometry, are the

subject of Section 2. Important steps in these proofs are the calculation of the Riemann period matrix Ω,

the normalized holomorphic diﬀerentials and the quasi-momentum and quasi-energy diﬀerentials in the limit

ε→0. Using results of Section 2, we then calculate the velocity of a single soliton on the elliptic background

(Section 3) and the phase shift of two interacting solitons on the elliptic background (Section 4). Here we

want to mention that the solitons corresponding to bj< e3, see Figure 1, appear to have larger than the

cnoidal wave amplitude and positive speed (bright-and-forward or simply hot), see Figure 3. On the other

hand, the solitons corresponding to bj∈(e2, e3) appear to have smaller than the cnoidal wave amplitude

and negative speed (dim-and-retrograde or simply cool). The smaller than the cnoidal wave amplitude of

the soliton can be identiﬁed with the dip in the cnoidal wave oscillations clearly visible on Figure 2; for

some early works about solitons on the elliptic background see, for example, [29], [22].

Dim (and retrograde) solitons, also mentioned as solitary disturbances, have negative group velocity, i.e.,

they are moving from the right to the left, which runs contrary to the common understanding that the

solitons for the KdV equation (1.1) are moving left to right. However, the motion in the opposite direction

is the result of the interaction between the soliton and the background. A similar phenomenon happens in

KdV soliton gases on zero background, where the faster (and taller) solitons are constantly pushing back

the slower (and smaller ) solitons due to the phase shift of their pair-wise interaction, so that the eﬀective

velocity of suﬃciently small solitons is negative, see [5]. Numerical observation of KdV solitons moving in

the negative direction was ﬁrst reported in [25].

In Section 5we use the results described above to derive the main equations for a KdV soliton gas on

the elliptic background, such as the nonlinear dispersion relations (NDR) and the equation of state. The

concept of a soliton gas, which can be traced back to some ideas of V. Zakharov [31] and S. Venakides [28],

was formulated by G. El in [9]. A soliton gas can be considered as a large N→ ∞ limit of an ensemble

of Nsolitons, viewed as particles with 2-particle interactions. Alternatively, it can be viewed as a speciﬁc

(thermodynamic) limit of genus Nﬁnite-gap solutions when N→ ∞ and simultaneously the size of all

(or all but ﬁnitely many) bands go to zero exponentially fast in N. Thus, we are interested in the large

Nlimit of various quantities associated with a hyperelliptic Riemann surface RN(e−νN ), where ν > 0.

In this setting the NDR become simply the thermodynamic limit of the Riemann bilinear identities on

RN(e−νN ), which involve the quasi-momentum and quasi-energy meromorphic diﬀerentials on one side and

the normalized holomorphic diﬀerentials on the other. Many details about soliton gases for the NLS and

KdV equation can be found in [11,9], see also [10], [27]. In particular, the thermodynamic limits of the

NDR were derived in [11] for the focusing NLS soliton (all bands are shrinking) and breather (all but one

band are shrinking) gases. In both cases, the genus of the residual (degenerate) surface was zero. In Section

5we make the next step in this direction by deriving the NDR and the equation of state for a KdV gas on

the genus one background. In light of Appendix A, it is quite clear that one can use the same technique

to derive the NDR for a KdV gas on backgrounds of any ﬁnite genus. We want to mention that the error

estimate of the limiting NDR is outside the scope of this paper, although some partial results related to

this issue in the context of the NLS soliton gas can be found in [27].

In the Appendices Aand Bwe respectively state and sketch the proof of Theorem 2.2 for any genus

g≥1 background (residual) Riemann surface, as well as prove that any solution described by Theorem 2.5

PARTIAL DEGENERATION OF FINITE GAP SOLUTIONS OF KDV 5

converges to the backgound elliptic solution with respect to some natural probability measure uniformly on

compact subsets of (x, t).

The following theorem summarizes the main results of Sections 2-4. Its proof follows from Theorems

2.2,2.5,4.2. Proof of (1.16) requires repeated use of Theorem 4.2.

Fix b1< b2<··· < L ∈(−∞, e3) and bL+1 <··· < bN∈(e2, e3). Let ℘(s) denote the Weierstass

function on the elliptic curve (1.7). Deﬁne implicitly βksatisfying bk=℘(2$3βk), k = 1, . . . , N, and

Re(βk)∈[0,1),Im(βk)∈ {0,Im τ/2}. Denote by β?

k:= 1 −βk+χτ the second pre-image of each bkin the

fundamental rectangle, where χ= 1 if Im β= Im τ /2 and χ= 0 if Im β= 0.

Theorem 1.1. [1] The solution of the KdV equation (1.1) with Nsolitons on a cnoidal background satisfying

(1.7) is given by

u(x, t)=2∂2

xln det [1N+G]+2∂2

xln θ3x−x0

4i$3− A−ζ($3)

2$3

where the background shift A:= 1

j=1

(βj−β?

j),

G`,m :=

θ3β`−β?

m+x−x0

4i$3− A

θ1(β`−β?

m)θ3x−x0

4i$3− ApC`Cmeiπ(ψ`+ψm), `, m = 1, . . . , N,

ψj(x, t) := (x−x(0)

j)Pj

2π+tEj

2π, Ej:= −1

2℘0(2$3βj), Pj:= 1

2$3

θ0

1(β;τ)

θ1(β;τ)

β=βj

β=β?

(1.15)

and the norming constants Cjare positive numbers given explicitly in Theorem 2.2 and x(0)

j, x0∈Rare

arbitrary shifts.

[2] The points βj∈(0,1

2), j = 1, . . . , L correspond to right-propagating solitons (solitary disturbances)

whereas the points βj∈(0,1

2) + τ

2, j ≥L+ 1 correspond to left propagating solitons (solitary disturbances).

[3] The proﬁle of such solutions for t→ ±∞ consists of a cnoidal stationary background (with period X=

4i$3∈R+) modulated by solitons (solitary disturbances) that are localized around the lines xj=Vjt+ Φ(±)

Each Vj=−Ej

Pjgives the modiﬁed velocity of the solitons on the elliptic background; the order of velocities

is preserved, i.e., V1> V2> . . . . The phase shifts Φ(±)

jdepend on the norming constants, but their averaged

diﬀerence (the averaged total shift of the jth soliton) does not:

DΦ(+)

jE−DΦ(−)

jE=2

|Pj|X

k>j

ln 

θ1(βj−β?

θ1(βj−βk)−2

|Pj|X

k<j

ln 

θ1(βj−β?

θ1(βj−βk)

(1.16)

where the average is over the period of the cnoidal wave (see description in Section 4and Figures 2,3,4).

Remark 1.2. Part [3] in Theorem 1.1 shows that solitons on the elliptic background aﬀect a phase shift

in the cnoidal background. A consequence of this fact is that the solitary disturbance is not localized.

For N= 1 this can be directly veriﬁed by computing the asymptotic behavior of the solitonic disturbance

2∂2

xln(1 + G) as x→ ±∞. One ﬁnds that

2∂2

xln(1 + G)∼(0x→+∞

2∂2

xln θ3β1−β?

1+x−x0

4i$3− A−2∂2

xln θ3x−x0

4i$3− Ax→ −∞

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PARTIALDEGENERATIONOFFINITEGAPSOLUTIONSTOTHEKORTEWEG-DEVRIESEQUATION:SOLITONGASANDSCATTERINGONELLIPTICBACKGROUNDSM.BERTOLA;|;~;}ANDR.JENKINS?ANDA.TOVBIS?Abstract.WeobtainFredholmtypeformulasforpartialdegenerationsofThetafunctionson(irreducible)nodalcurvesofarbitrarygenus,withemphasisonnodalcurvesof...

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PARTIAL DEGENERATION OF FINITE GAP SOLUTIONS TO THE KORTEWEG-DE VRIES EQUATION SOLITON GAS AND SCATTERING ON ELLIPTIC BACKGROUNDS

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