Self-accelerating solitons Boris A. Malomed12 1Department of Physical Electronics School of Electrical Engineering Faculty of Engineering

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Self-accelerating solitons

Boris A. Malomed1,2

1Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering,

and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel

2Instituto de Alta Investigaci´on, Universidad de Tarapac´a, Casilla 7D, Arica, Chile

Basic models which give rise to one- and two-dimensional (1D and 2D) solitons, such as the

Gross-Pitaevskii (GP) equations for BEC, feature the Galilean invariance, which makes it possible

to generate families of moving solitons from quiescent ones. A challenging problem is to ﬁnd models

admitting stable self-accelerating (SA) motion of solitons. SA modes are known in linear systems

in the form of Airy waves, but they are poorly localized states. This brief review presents two-

component BEC models which make it possible to predict SA solitons. In one system, a pair of

interacting 1D solitons with opposite signs of the eﬀective mass is created in a binary BEC trapped

in an optical-lattice potential. In that case, opposite interaction forces, acting on the solitons with

positive and negative masses, produce equal accelerations, while the total momentum is conserved.

The second model is based on a system of GP equations for two atomic components, which are

resonantly coupled by a microwave ﬁeld. The latter model produces an exact transformation to an

accelerating references frame, thus predicting 1D and 2D stable SA solitons, including vortex rings.

Introduction. – A basic property of one- and two-dimensional (1D and 2D) equations which produce solitons is the

Galilean invariance, which generates solitons moving with an arbitrary velocity from a quiescent one. A challenging

issue is to construct localized states moving at a constant acceleration, rather than constant velocity [1, 2]. A well-

known fact is that the linear Schr¨odinger equation, iuz+ (1/2)uxx = 0 (it is written in the form of the paraxial

propagation equation in optics, with propagation distance zand transverse coordinate x) admits self-accelerating

(SA) solutions in the form of Airy waves [3]. Later, it was predicted [4] and experimentally demonstrated, originally

in optics [5], and then in electron beams [6], plasmonics [7], Bose-Einstein condensates (BECs) [8], acoustics [9], gas

discharge [10], and water waves [11] that truncated Airy waves (TAWs) can be created in these media, the respective

exact solution of the linear Schr¨odinger equation being

uTAW (x, z) = u0Ai αx −α4

4z2+iℵα2zexp (ℵαx)

×exp −iα6

12 z3+iα3

2xz −ℵα4

2z2+iℵ2α2

2z,(1)

where Ai is the Airy function, and constants u0,α, and ℵ>0 deﬁne, respectively, the amplitude, internal scale, and

truncation of the Airy wave. Accordingly, the input which generates solution (1) is u(x;z= 0) = u0Ai (αx) exp (ℵαx).

The truncation factor is necessary to make the total norm (power, in terms of optics) of the input ﬁnite:

NTAW ≡Z+∞

−∞ |u(x)|2dx =u2

0√8πℵα−1exp 2ℵ3/3,(2)

while the TAW’s momentum,

P=iZ+∞

−∞

u∗

xudx, (3)

is zero, in spite of the self-acceleration featured by this wave (both Nand Pare dynamical invariants of the Schr¨odinger

equation).

In fact, the truncation leads to gradual degradation of the TAW, as shown, in particular, by factor exp −ℵα4z2/2

in solution (1). Another source of degradation is the action of nonlinearity, as TAW is the eigenmode of the linear

medium. Eﬀects of nonlinearity on the Airy waves were considered in many works [12]-[18], chieﬂy demonstrating

decay into solitons.

A possibility to create well-localized (unlike the Airy waves) SA two-component pulses in optics was elaborated

in terms of a system of coupled nonlinear Schr¨odinger (NLS) equations with opposite signs of the group-velocity

dispersion (GVD) in them [19], aiming to create two pulse components with opposite signs of their eﬀective masses.

In this case, the opposite interaction forces with which the coupled components act on each other give rise to identical

signs of the acceleration. A solution for such an SA bound state was constructed approximately, taking an unperturbed

NLS soliton in the anomalous-GVD component, and applying the Thomas-Fermi approximation to the normal-GVD

arXiv:2210.01507v1 [nlin.PS] 4 Oct 2022

one. The so predicted optical SA pulses were demonstrated experimentally in a roughly similar temporal-domain

form, using a pair of ﬁber loops with diﬀerent lengths, coupled to each other at one point [20].

It is relevant to mention that a soliton moving with a constant acceleration (although not of the SA type) can be

readily predicted taking the single NLS equation with the self-attractive cubic term and an attractive or repulsive

local defect which moves with acceleration a:

iuz+ (1/2)uxx +|u|2u=−εδ x−az2/2u, (4)

where δis the delta-function, ε > 0 or ε < 0 corresponding to the attractive or attractive defect, respectively. In terms

of the spatial-domain NLS equation (4) in optics, the “accelerating” defect represents a narrow parabolic (x=az2/2)

stripe in the (x, z) plane, with the locally increased (ε > 0) or decreased (ε < 0) value of the refractive index. It is

convenient to rewrite Eq. (4) in the co-moving reference frame, applying the corresponding boost transformation [1]:

ξ≡x−az2/2, u(x, z)≡v(ξ, z) exp iaξz +ia2z3/6,(5)

ivz+ (1/2)vξξ +|v|2v=aξv −εδ(ξ)v. (6)

In the moving frame, the soliton may be approximated by the simple stationary solution to Eq. (6),

vsol(ξ, z)=(N/2)sech ((N/2) (ξ−ξ0)) exp iN2z/8,(7)

where Nis the soliton’s norm (see Eq. (2)), and ξ0is a shift of the soliton from the position of the defect. Treating

the terms on the right-hand side of Eq. (6) by perturbations [21], the soliton is considered as a quasi-particle under

the action of an eﬀective potential,

U(ξ0) = Naξ0−εN2/4sech2(Nξ0/2) .(8)

This potential has an equilibrium position with sign sgn (ξ0) = −sgn (aε), which exists, for given strength εof the

defect, if the acceleration does not exceed a critical value,

|amax|=|ε|N2/6√3.(9)

The corresponding largest equilibrium value of the shift is (|ξ0|)max =N−1ln √3+12/2≈1.32/N . Note that

the largest acceleration with which the soliton can be dragged by the moving defect, as given by Eq. (9), does not

depend on the sign of ε.

A diﬀerent result is produced for dragging solitons by the local defect moving with constant acceleration in the

framework of the NLS equation with the quintic, rather than cubic, nonlinearity (which may also be realized in optical

media [22]),

iuz+ (1/2)uxx +|u|4u=−εδ x−az2/2u. (10)

In the uniform space (ε= 0), Eq. (10) gives rise to commonly known 1D Townes solitons [23], u=

(3k)1/4rsech 2√2kxexp (ikz), with arbitrary propagation constant k > 0. This soliton family is degenerate,

as its norm takes a single value, which does not depend on k,N=p3/2π/2, and the family is completely unstable

against the onset of the critical collapse [24]. However, all the solitons are stabilized by the interaction with the

quiescent attractive defect (a= 0, ε > 0) [25]. Then, the above consideration can be developed for the solitons of Eq.

(10) pulled by the defect with constant acceleration. In particular, the largest acceleration which can be supported

by the defect with given ε > 0 is |amax|= (4/π)εk, cf. Eq. (9).

The objective of this perspective is to produce a brief summary of results which predict possibilities of true SA

motion of 1D and 2D solitons in speciﬁc BEC models, one based on the spatially-periodic optical-lattice (OL) potential,

and another one making use of a binary BEC whose components are resonantly coupled by a microwave (MW) ﬁeld.

Co-accelerating bound states of solitons with positive and negative masses. – A two-component model

which allows one to predict stable SA bound states of solitons with positive and negative eﬀective masses is represented

by a system of Gross-Pitaevskii (GP) equations for wave functions φand ψof the binary BEC, including the spatially-

periodic potential of the OL type in each equation, with strengths U1and U2[26]:

iφt=−(1/2)φxx −g1|φ|2+γ|ψ|2+U1cos (2πx) + fxφ,

iψt=−(1/2)ψxx −γ|φ|2−g2|ψ|2+U2cos (2πx) + fxψ. (11)

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摘要：

Self-acceleratingsolitonsBorisA.Malomed1;21DepartmentofPhysicalElectronics,SchoolofElectricalEngineering,FacultyofEngineering,andCenterforLight-MatterInteraction,TelAvivUniversity,TelAviv69978,Israel2InstitutodeAltaInvestigacion,UniversidaddeTarapaca,Casilla7D,Arica,ChileBasicmodelswhichgiveriseto...

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