Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation Bo Yang1 Jianke Yang2

2025-05-02 0 0 935.25KB 11 页 10玖币

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Partial-rogue waves that come from nowhere but leave with a trace in the

Sasa-Satsuma equation

Bo Yang1, Jianke Yang2

1School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China

2Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401, U.S.A

Partial-rogue waves, i.e., waves that “come from nowhere but leave with a trace”, are analytically

predicted and numerically conﬁrmed in the Sasa-Satsuma equation. We show that, among a class of

rational solutions in this equation that can be expressed through determinants of 3-reduced Schur

polynomials, partial-rogue waves would arise if these rational solutions are of certain orders, where

the associated generalized Okamoto polynomials have real but not imaginary roots, or imaginary

but not real roots. We further show that, at large negative time, these partial-rogue waves approach

the constant-amplitude background, but at large positive time, they split into several fundamental

rational solitons, whose numbers are determined by the number of real or imaginary roots in the

underlying generalized Okamoto polynomial. Our asymptotic predictions are compared to true

solutions, and excellent agreement is observed.

I. INTRODUCTION

Rogue waves has been a subject of intensive theoreti-

cal and experimental studies in mathematical and phys-

ical communities in the past decade. Hundreds of pa-

pers and several books have been published on it, and

more are still coming. Rogue waves are often deﬁned

as “waves that come from nowhere and leave without

a trace” [1]. For example, they can be localized wave

excitations that arise from the constant-amplitude back-

ground, reach higher amplitude, and then retreat back

to the same background, as time progresses. Almost all

rogue waves that have been theoretically derived or ex-

perimentally observed belong to this category (see [2–6],

among many others).

However, there exists another type of waves that “come

from nowhere but leave with a trace”. Speciﬁcally,

these waves also arise from the constant-amplitude back-

ground (thus “come from nowhere”), stay localized, and

reach higher amplitude. Afterwards, instead of retreat-

ing back to the same constant background with no trace,

they evolve into localized waves on the constant back-

ground that persist at large time, thus leaving a trace.

The ﬁrst report of such peculiar waves seems to be in

[7] for the Davey-Stewartson-II equation, where a two-

dimensional localized wave arose from the constant back-

ground and then split into two localized lumps at large

time (see Fig. 4 of that paper). Later, a similar but

one-dimensional solution was reported in [8] for the Sasa-

Satsuma equation. These peculiar waves resemble rogue

waves in the ﬁrst half of evolution, but contrast them in

the second half of evolution. Due to these peculiar be-

haviors, let us call them partial-rogue waves. Note that

although two examples of partial-rogue waves can be seen

in [7, 8], there was no explanation for their appearance, as

if they were pure accidents. It was also unclear whether

additional types of partial-rogue waves could be found in

those two systems.

In this paper, we predict partial-rogue waves in the

Sasa-Satsuma equation through large-time asymptotic

analysis on its rational solutions. We show that, among

a class of rational solutions in this equation that can be

expressed through determinants of 3-reduced Schur poly-

nomials, partial-rogue waves arise if and only if these

rational solutions are of certain orders, where the associ-

ated generalized Okamoto polynomials have real but not

imaginary roots, or imaginary but not real roots. We fur-

ther show that, at large negative time, these partial-rogue

waves approach the constant-amplitude background, but

at large positive time, they split into several fundamen-

tal rational solitons, whose numbers are determined by

the number of real or imaginary roots in the underlying

generalized Okamoto polynomial. Our asymptotic pre-

dictions are compared to true solutions, and excellent

agreement is observed.

II. PRELIMINARIES

The Sasa-Satsuma equation was proposed as a higher-

order nonlinear Sch¨odinger equation for optical pulses

that includes some additional physical eﬀects such

as third-order dispersion and self-steepening [9, 10].

Through a variable transformation, this equation can be

written as

ut=uxxx + 6|u|2ux+ 3u(|u|2)x.(1)

Sasa and Satsuma [9] showed that this equation is inte-

grable.

A. A class of rational solutions

Soliton solutions on the zero background in this equa-

tion were derived by Sasa and Satsuma in their original

paper [9]. Later, rational solutions on a constant back-

ground, including rogue waves, were also derived [8, 11–

19]. The solutions that will be the starting point of this

paper are a certain class of rational solutions which, in

the language of Darboux transformation, are associated

arXiv:2210.03414v1 [nlin.SI] 7 Oct 2022

with a scattering matrix admitting a triple eigenvalue.

Such solutions have been studied in [8, 14] by Darboux

transformation. However, their solutions are not general

nor explicit for our purpose. For this reason, we will ﬁrst

present general and explicit expressions for this class of

rational solutions through Schur polynomials.

Before presenting these solutions, we need to specify

the nonzero background. Through variable scalings, we

can normalize the background amplitude to be unity.

Then, this background can be written as

ubg(x, t) = ei[α(x+6t)−α3t],(2)

where αis a free wavenumber parameter, which cannot

be removed since the Sasa-Satsuma equation (1) is not

Galilean-invariant. But αcan be restricted to be posi-

tive, since the Sasa-Satsuma equation is invariant under

the axes reﬂection of (x, t)→(−x, −t), and negative-

αsolution can be related to positive-αsolution through

this axes reﬂection.

To present these explicit rational solutions, we also

need to introduce elementary Schur polynomials. These

polynomials Sj(x) with x= (x1, x2, . . .) are deﬁned by

the generating function

∞

j=0

Sj(x)j= exp 



∞

j=1

xjj

.(3)

In addition, we deﬁne Sj(x) = 0 when j < 0.

Our expressions for general rational solutions corre-

sponding to a triple eigenvalue in the scattering matrix

of Darboux transformation are given by the following the-

orem.

Theorem 1 When α= 1/2, the

Sasa-Satsuma equation (1) admits bounded

(N1, N2)-th order rational solutions

uN1,N2(x, t) = gN1,N2

fN1,N2

ei[α(x+6t)−α3t],(4)

where N1and N2are arbitrary non-negative

integers,

fN1,N2=σ0,0, gN1,N2=σ1,0,(5)

σk,l = det σ[1,1]

k,l σ[1,2]

k,l

σ[2,1]

k,l σ[2,2]

k,l !,(6)

σ[I,J]

k,l =φ(k,l I,J)

3i−I, 3j−J1≤i≤NI,1≤j≤NJ

,(7)

matrix elements in σ[I,J]

k,l are deﬁned by

φ(k,l,I,J)

i,j =

min(i,j)

ν=0 p2

4p2

0ν

Si−ν(x+

I(k, l) + νs)Sj−ν(x−

J(k, l) + νs),(8)

vectors x±

I(k, l)=(x±

1,I , x±

2,I ,···) are given

r,I (k, l) = pr(x+ 6t) + βrt+kθr+lθ∗

r+ar,I ,(9)

x−

r,J (k, l) = pr(x+ 6t) + βrt−kθ∗

r−lθr+ar,J ,(10)

βrand θrare coeﬃcients from expansions

p3(κ) =

∞

r=0

βrκr,ln p(κ)+iα

p0+ iα=

∞

r=1

θrκr,(11)

the function p(κ) with expansion p(κ) =

P∞

r=0 prκrand real expansion coeﬃcients pr

is deﬁned by the equation

Q1[p(κ)] = Q1(p0)

3"eκ+ 2e−κ/2cos √3

2κ!#,(12)

with

Q1(p)≡1

p−iα+1

p+ iα+p, (13)

p0=±√3/2, the real vector s= (s1, s2,···)

is deﬁned by the expansion

ln 2p0

p1κp(κ)−p0

p(κ) + p0=

∞

r=1

srκr,(14)

the asterisk ‘*’ represents complex conjuga-

tion, and

(a1,1,··· , a3N1−1,1),(a1,2,··· , a3N2−2,2) (15)

are free real constants.

In these solutions, subscripts of matrix elements in

Eq. (7) jump by 3. Thus, the corresponding determi-

nant in Eq. (6) is called a determinant of 3-reduced Schur

polynomials [20].

Note 1 When we choose p0=√3/2, the ﬁrst few

coeﬃcients of pr,βr,θr, and srare

p1=121/6

2, p2=12−1/6

2, p3=1

4√3,(16)

β1=9

8121/6, β2=9

8·35/6

21/3, β3=19√3

16 ,(17)

θ1=121/6

√3+i, θ2=i

121/6√3+i2, θ3= 0,(18)

s1= 0, s2= 0, s3=−1

40.(19)

If we choose p0=−√3/2, then prand βrwould switch

sign, θrchange to θ∗

r, and srremain the same.

Note 2 If we choose p0=−√3/2 and keep all inter-

nal parameters (ar,1, ar,2) unchanged, then the resulting

solution ˜u(x, t) would be related to the solution u(x, t)

with p0=√3/2 as ˜u(x, t) = u∗(−x, −t).

Note 3 Internal parameters a3n,1and a3n,2(n=

1,2,···) do not aﬀect solutions in Theorem 1, for rea-

sons which can be found in [21]. Thus, we will set them

as zero in later text.

The simplest solution of this class — the fundamental

rational soliton, is obtained when we set N1= 0 and

N2= 1 in Theorem 1. In this case, the solution has a

single real parameter a1,2, which can be normalized to

zero through a shift of the xaxis. The resulting solution,

for both p0=±√3/2, is

u1(x, t) = ˆu1(x, t)ei[ 1

2(x+6t)−1

8t],(20)

where

ˆu1(x, t) = 3ˆx2+ 3iˆx−2

3ˆx2+ 1 ,(21)

and

ˆx≡x+33

4t(22)

is a moving coordinate. The graph of this solution is plot-

ted in Fig. 1. This solution is a rational soliton moving

on the constant-amplitude background (2) with velocity

−33/4. Its 3D graph shows a W-shape along the ˆxdirec-

tion and has sometimes been called a W-shaped rational

soliton in the literature [8, 18]. Its height, i.e., max(|u1|),

is 2.

FIG. 1: Graph of the fundamental rational soliton |u1(x, t)|in

Eq. (20). Left: 3D plot. Right: density plot. The horizontal

axes are ˆx=x+ (33/4)t.

B. Generalized Okamoto polynomials

We will show in later text that rational solutions in

Theorem 1 contain partial-rogue waves but are not all

partial-rogue waves. The question of what solutions in

Theorem 1 are partial-rogue waves turns out to be closely

related to root properties of generalized Okamoto poly-

nomials. So, we will introduce these polynomials and

examine their root structures next.

Original Okamoto polynomials arose in Okamoto’s

study of rational solutions to the Painlev´e IV equation

[22]. He showed that a class of such rational solutions can

be expressed as the logarithmic derivative of certain spe-

cial polynomials, which are now called Okamoto polyno-

mials. These original polynomials were later generalized,

and the generalized Okamoto polynomials provide a more

complete set of rational solutions to the Painlev´e IV equa-

tion [20, 23–25]. In addition, determinant expressions for

the original and generalized Okamoto polynomials were

discovered [20, 23–25].

Let pj(z) be Schur polynomials deﬁned by

∞

j=0

pj(z)j= exp z +2,(23)

with pj(z)≡0 for j < 0. Then, generalized Okamoto

polynomials QN1, N2(z), with N1, N2being nonnegative

integers, are deﬁned as

QN1, N2(z) = Wron[p2, p5,··· , p3N1−1, p1, p4,··· , p3N2−2],

(24)

or equivalently,

QN1, N2(z) =



p2p1··· p3−N1−N2

p3N1−1p3N1−2··· p2N1−N2

p1p0··· p2−N1−N2

p3N2−2p3N2−3··· p2N2−N1−1



,(25)

since p0

j+1(z) = pj(z) from the deﬁnition of pj(z) in

Eq. (23), where the prime represents diﬀerentiation. The

ﬁrst few QN1, N2(z) polynomials are

Q1,0(z) = 1

2(z2+ 2),

Q2,0(z) = 1

80(z6+ 10z4+ 20z2+ 40),

Q0,1(z) = z,

Q1,1(z) = 1

2(−z2+ 2)

Q2,1(z) = 1

20z(z4−20),

Q0,2(z) = 1

8(z4+ 4z2−4),

Q1,2(z) = 1

8(−z4+ 4z2+ 4),

Q2,2(z) = 1

80(−z6+ 10z4−20z2+ 40).

Note that our deﬁnition of generalized Okamoto polyno-

mials is diﬀerent from that by Clarkson in Refs. [24, 25].

Denoting the Qm,n(z) polynomial introduced in [24, 25]

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Partial-roguewavesthatcomefromnowherebutleavewithatraceintheSasa-SatsumaequationBoYang1,JiankeYang21SchoolofMathematicsandStatistics,NingboUniversity,Ningbo315211,China2DepartmentofMathematicsandStatistics,UniversityofVermont,Burlington,VT05401,U.S.APartial-roguewaves,i.e.,wavesthat\comefromnowhereb...

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Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation Bo Yang1 Jianke Yang2

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