Front Matter

Published:2022
Boris A. Malomed, "Front Matter", Multidimensional Solitons, Boris A. Malomed
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Written by the preeminent researcher in this field, Multidimensional Solitons provides a comprehensive survey of selftrapped modes in two and threedimensional nonlinear media. It reviews studies covering fundamentals and recently reported theoretical and experimental results in diverse areas, such as nonlinear optics and BoseEinstein condensates. This vital work explores fundamental solitons, topologically organized ones, such as solitary vortices and hopfions, and a broad spectrum of other soliton species, including those in lattice and dissipative media.
Multidimensional Solitons:
Offers a general introduction to the modern “soliton science”
Provides context on the fundamentals of solitons in nonlinear optics and BoseEinstein condensates
Instructs how the stabilization of two and threedimensional solitons is possible, which is a crucially important problem for multidimensional setups
Presents theoretical (analytical and numerical) and experimental results on optical and matterwave solitons in diverse settings
Researchers – both theorists and experimentalists – working in optics and photonics, lowtemperature and atomic physics, mathematical physics, and applied mathematics will all find this an instrumental reference. Graduate students in these disciplines will also find it as a valuable resource.
To my mother Bertha, my wife Marina, my son Ilya, my daughter Olga, and my granddaughters Eleanor, Natalie, Negev, and Nili
Acronyms
 1D
Onedimensional
 (1+1+1)D
Twodimensional, with one temporal and one transverse spatial coordinates, while the propagation coordinate plays the role of the evolutional variable
 2D
Twodimensional
 (2+1)D
Twodimensional, with two transverse coordinates, while the third (propagation) coordinate plays the role of the evolutional variable
 3D
Threedimensional
 (3+1)D
Threedimensional, with the propagation coordinate playing the role of the evolutional variable and the temporal variable playing the role of an additional coordinate
 AL
Ablowitz‒Ladik (integrable discretization of the NLS equation)
 a.r.
Aspect ratio
 b.c.
Boundary condition(s)
 BdG
Bogoliubov‒de Gennes (linearized equations for perturbations around stationary solutions of GP/NLS equations)
 BEC
Bose‒Einstein condensate
 BG
Bragg grating
 BW
Bloch wave
 BZ
Brillouin zone
 CGL
Complex Ginzburg‒Landau (equation)
 CQ
Cubic‒quintic (nonlinearity)
 CSV
Cratershaped vortex (the usual localized vortex mode)
 CW
Continuous wave
 DM
Dispersion management
 DS
Dissipative soliton
 FF
Fundamental frequency
 FK
Frenkel‒Kontorova (a dynamical lattice model)
 FP
Fixed point
 FR
Feshbach resonance
 FT
Flattop (profiles of solitons and quantum droplets)
 FWHM
Fullwidth at halfmaximum
 FWM
Fourwave mixing
 GP
Gross‒Pitaevskii (equation)
 GS
Gap soliton
 GVD
Groupvelocity dispersion
 HO
Harmonicoscillator (potential)
 HV
Hidden vorticity
 IS
Intersite
 IST
Inversescattering transform (the method for solving integrable nonlinear partial differential equations)
 JJ
Josephson junction
 KdV
Korteweg‒de Vries (equation)
 KP
Kadomtsev‒Petviashvili (equation)
 LB
“Light bullet” (a 3D spatiotemporal optical soliton)
 LFE
Localfield effect
 LHY
Lee‒Huang‒Yang (correction to the MF theory)
 LI
Lévy index
 MF
Meanfield (approximation)
 MM
Mixed mode
 MW
Microwave
 NLS
Nonlinear Schrödinger (equation)
 NM
Nonlinearity management
 OAM
Orbital angular momentum
 OL
Optical lattice
 OS
Onsite
 PhC
Photonic crystal
 PhR
Photorefractive (optical material)
 $PT$
Paritytime (symmetry)
 QD
Quantum droplet
 RL
Rayleigh length (alias diffraction length)
 SG
SineGordon (equation)
 SH
Second harmonic
 SM
Skyrme model
 SOC
Spin‒orbit coupling
 SPM
Selfphase modulation
 SSB
Spontaneous symmetry breaking
 STOV
Spatiotemporal vortex
 SV
Semivortex
 TB
Threebody (loss)
 TF
Thomas‒Fermi (approximation)
 TOF
Time of flight
 TPA
Threephoton absorption
 TS
Townes soliton
 VA
Variational approximation
 VAV
Vortex‒antivortex (a twocomponent bound state in SOC systems)
 VK
Vakhitov‒Kolokolov (stability criterion)
 VR
Vortex ring
 XPM
Crossphase modulation
 ZS
Zeeman splitting
Preface
The absolute majority of work that has been performed in a huge area of theoretical and experimental studies of solitons, i.e., selftrapped solitary waves found in a great variety of nonlinear systems, dealt with onedimensional (1D) settings. Extension of the soliton concepts to the multidimensional world is a very promising, but also really challenging, direction of the work for theorists and experimentalists. The expected and, to a certain extent, realized gain is the fascinating possibility to create completely new species of solitary states, as the two and threedimensional (2D and 3D) geometries make it possible to build localized modes with intrinsic topological arrangements. The most obvious possibility is to create 2D and 3D vortex solitons, which are described by a complex wave function. This means building solitons with an intrinsic angular momentum, which may be considered as a classical counterpart of spin of quantum particles. Solitons can carry an orbital angular momentum in a different form if they are rotating modes with a nonaxiallysymmetric shape. Multicomponent solitons can be used to build more sophisticated topological structures, such as famous skyrmions, hopfions (alias twisted vortex tori in the 3D space), monopoles coupled to nonAbelian gauge fields, knots, and others, which have no counterparts in 1D realizations. As concerns dynamics of multidimensional solitons, they may demonstrate various scenarios of interactions and formation of bound states (often called “soliton molecules”), as well as manysoliton structures (such as soliton lattices). These and other possibilities are considered in detail in the corresponding chapters of this book.
On the other hand, the work with solitons in the 2D and 3D geometry encounters fundamental difficulties. In terms of the underlying mathematical theory, the fact that the most fundamental models that give rise to 1D solitons, such as the Korteweg–de Vries (KdV), sineGordon (SG), and nonlinear Schrödinger (NLS) equations, are integrable is limited to one dimension. Therefore, the unprecedentedly powerful methods [the inversescattering transform (IST), the Darboux transform, the Hirota method, etc.] that produce highly nontrivial exact solutions, such as those for collisions between solitons, formation of bound states in the form of breathers, prediction of the soliton content of a given input, etc., are not available in the studies of 2D and 3D models. In 2D, there are some exceptional integrable equations [most notably, the celebrated Kadomtsev–Petviashvili (KP) equations], but most important solitongenerating models, such as 2D and 3D NLS equations, lose the integrability that they had in 1D and admit no exact solutions. In 3D settings, no integrable equations, which would be capable to produce physically relevant solutions for solitons, are known.
The lack of the integrability of basic 2D and 3D equations underlying the soliton theory may be considered a technical difficulty because relevant solutions, once they are not available in an exact form, can often be constructed by means of approximate analytical or semianalytical methods, such as the ubiquitous variational approximation (VA). In any case, the availability of powerful computational facilities suggests a possibility to produce numerical solutions of nearly all theoretical problems, even if simulations of 2D and, especially, 3D nonlinear equations are more difficult than their 1D counterparts.
However, the exit from 1D models to the real 3D world (or intermediate 2D settings) leads to principal problems, related to stability of the expected soliton states. First, a wellknown result is that stationary solutions of the 3D nonlinear Klein–Gordon equations are always subject to instability, as they cannot represent an energy minimum (Derrick, 1964). Next, the instability problem is very clearly exhibited by the most important model based on the NLS equation with the selfattractive cubic nonlinearity (a commonly known example is the Kerr term in various models originating from optics): while stationary soliton solutions of the 1D NLS equations are completely stable, the 2D and 3D versions of the same equation produce soliton families (which can be found in an approximate form by means of the VA, or easily constructed, with an extremely high accuracy, as numerical stationary solutions) that are completely unstable, due to the fact that precisely the same 2D and 3D NLS equations give rise to the collapse, alias blowup (catastrophic selfcompression leading to the formation of a true singularity after a finite evolution time). The collapse is critical in 2D, and supercritical in 3D, which means that the 2D collapse sets in if the norm of the input (total power, in terms of optics) exceeds a certain finite critical (threshold) value, while in 3D, the threshold is zero; i.e., an arbitrarily weak input may initiate the supercritical collapse. In 2D, the input whose norm falls below the threshold value does not blow up but rather decays into “radiation” (smallamplitude waves). Small perturbations added to any soliton of the 3D NLS equation trigger its blowup, while in 2D, the addition of small perturbations initiates either the blowup or fast decay. In this connection, it is relevant to mention that the first species of solitons, which was ever considered in optics, is the family of the socalled Townes solitons [TSs, predicted by Chiao et al. (1964)]. These are stationary solutions of the 2D NLS equation, which predict selftrapped shapes of laser beams propagating in the bulk Kerr medium under the condition of paraxial diffraction. Actually, many other species of solitons, which were predicted later, had been created in the experiment [see the books of Kivshar and Agrawal (2003) and Dauxois and Peyrard (2006)], but the TSs in their pure form have never been created, as they are unstable states, which represent a separatrix between collapsing and decaying solutions of the 2D NLS equation.
As concerns 2D and 3D solitons with embedded vorticity (alias vortex rings, VRs), they are subject to still stronger instability, which develops faster than the collapse, viz., spontaneous splitting of the VR into two or several fragments, which are close to the corresponding fundamental (zerovorticity) solitons. At a later stage of the evolution, the secondary solitons are destroyed by the intrinsic collapse.
The NLS equation with the selfattractive nonlinearity may be a physically correct model for many physical realizations in optics, dynamics of Bose–Einstein condensates (BECs) in ultracold atomic gases, physics of plasma (Langmuir waves), etc., but the occurrence of the collapse implies that these physical settings cannot be used for the creation of multidimensional solitons, once the solitons are the objective of the work. Therefore, a cardinal problem is search for physically realistic multidimensional systems, which include additional ingredients that make it possible to suppress the collapse and help to predict and create stable 2D and 3D solitons. This is possible in various physical setups. For instance (as is discussed in detail in this book), stable 2D and 3D optical solitons can be predicted and, eventually, experimentally created if the optical medium features, in addition to the cubic selffocusing, higherorder quintic selfdefocusing, which arrests the blowup and provides the stabilization of 2D and 3D optical solitons (2D solitons stabilized by the quintic selfdefocusing have been reported in experimental works, while the creation of 3D solitons remains a challenging problem). Another recently discovered and extremely interesting option is to consider a binary BEC with the collapse driven by the attractive cubic interaction between its two intrinsically selfrepulsive components. In this system, the collapse is actually arrested by a higherorder quartic selfrepulsive term, which is induced in each component by the socalled Lee–Huang–Yang (LHY) effect, i.e., a correction to the usual cubic meanfield (MF) interaction induced by quantum fluctuations around the MF state (Lee et al., 1957). As a result, the binary BEC creates completely stable 3D and quasi2D selftrapped “quantum droplets” (QDs), which seem as multidimensional solitons (even if they are not usually called “solitons,” as the name of QDs is preferred in the literature). The prediction of QDs, made by Petrov (2015), was quickly realized experimentally by several experimental groups (Cabrera et al., 2018; and Semeghini et al., 2018). These theoretical and experimental findings (as well as many others) are considered in detail in this book.
The collapsesuppressing mechanisms mentioned above, i.e., the cubic–quintic (CQ) focusing– defocusing nonlinearity in optical media, and the attractive cubic MF interaction in the binary BEC with the LHY quartic selfrepulsive correction may stabilize not only 3D and 2D fundamental (i.e., structureless) solitonlike states, but also ones with embedded vorticity (3D and 2D VRs), although these results remain, as yet, a theoretical prediction [optical solitonlike modes with embedded vorticity were observed by Eilenberger et al. (2013) and Reyna et al. (2016), but only as transient states in different experimental settings]. The stabilization of vortex solitons is another topic considered in detail in this book.
The broad topic of multidimensional solitons was a subject of several review articles (Malomed et al., 2005; 2016; Malomed, 2016; 2018; 2019; Mihalache, 2017; and Kartashov et al., 2019). These reviews were focused on particular aspects of the theme; see a detailed discussion in Chap. 1. However, the topic as a whole was not previously summarized in a relatively full form. This is the objective of the present book. Because the stabilization is the critically important issue in the theoretical and experimental work with multidimensional solitons, it is natural to organize a survey of this broad subject according to particular stabilization mechanisms. This principle determines the structure of the book, as can be seen in the Table of Contents. However, it is relevant to stress that, although the intention is to include many essential aspects of the subject, the book is not designed as a comprehensive presentation of all aspects relevant to studies of multidimensional solitons. Some topics that are not included are briefly mentioned in Chap. 15 (Conclusion). One such topic, which has become a subject of many recent works, is spatiotemporal propagation of light in nonlinear multimode fibers, and another recently active one, which also belongs to the realm of photonics, is nonlinear selftrapping in exciton–polariton condensates.
It is relevant to stress that settings which give rise to multidimensional solitons and solitonlike states may be essentially conservative (if losses are negligible on temporal and spatial scales relevant to the experiment) or dissipative (if losses are essential, and the system must include gain or external pump, necessary for compensation of the losses). In this book, dissipative multidimensional solitons are considered, in some detail, in Chap. 14, while Chaps. 2–13 address lossless systems or ones including weak losses.
As concerns physical realizations in which multidimensional solitons can be (and have been) created, they are included in the chapters presenting the respective results. In fact, all the realizations considered in the book in detail belong to two broad areas: first, optics and photonics, and second, various realizations of BEC in atomic gases. The Table of Contents presents the material in a sufficiently detailed form to help the reader find particular physical realizations and particular results predicted and/or experimentally reported in the corresponding settings. Another vast realm in which 3D solitons (such as famous skyrmions) are a subject of great significance is the classical field theory, with applications to particles and nuclei, ferromagnetic media, semiconductors, etc. This topic is discussed in the Introduction (Chap. 1) but is not addressed in other chapters of the book.
Of course, the selection of the material for the presentation in the book is biased by personal work and interests of the author. Actually, different topics included in the book are considered with different degrees of detail. Some settings, which play a paradigmatic role, are presented in a sufficiently full form (dropping minor technicalities, in many cases), while some others, which seem as less significant additions to the basic points, are considered schematically or, sometimes, are only briefly mentioned.
Different chapters of the book are linked by relevant crossreferences. Nevertheless, some basic principles are briefly repeated in particular chapters to make them relatively independent from each other. In particular, the fact that the zerovorticity (fundamental) Townes solitons (TSs), i.e., solutions of the 2D NLS equation with the cubic selfattraction, are subject to the instability driven by the critical collapse, and similar solitons with embedded vorticity (vortex rings, VRs) are prone to the still stronger splitting instability, is mentioned in several different chapters, as these facts are basically important for the work with 2D solitons in various settings.
The book is drafted as a monograph, which may be appropriate for reading and using by active researchers, both theorists and experimentalists, working in the realm of solitons and nonlinear waves. It may also be useful for members of research communities in broad areas of nonlinear optics and photonics, matter waves (in BEC), the general theory of nonlinear waves and nonlinear dynamics, and some others. The book may also be used by graduate students who start or continue their work in these areas.
I am deeply thankful to colleagues with whom I had a privilege to collaborate on and/or discuss various topics comprised in this book and on related topics:
Fatkhulla Abdullaev, Sadhan Adhikari, Najdan Aleksić, Egor Alfimov, Anderson Amaral, Dan Anderson, Ady Arie, Javid Atai, Grigori Astrakharchik, Mark Azbel (deceased), Alon Bahabad, Bakhtiyor Baizakov, Yehuda Band, Igor Barashenkov, Milivoj Belić, Eshel BenJacob (deceased), Anders Berntson, Ishfaq Ahmad Bhat, Alan Bishop, Olga Borovkova, George Boudebs, Marijana Brtka, Gennadiy Burlak, Alexander Buryak, Thomas Busch, JeanGui Caputo, Wesley Cardoso, Ricardo CarreteroGonzález, Márcio Carvalho, Alan Champneys, Stathis Charalampidis, Zhigang Chen, Kwok Chow, Demetri Christodoulides, Pak Chu (deceased), Lucian Crasovan, Jesús Cuevas, Cid de Araújo, Dongmei Deng, Martijn de Sterke, Anton Desyatnikov, Fotis Diakonos, Paolo Di Trappani, Guangjiong Dong, Peter Drummond, Vanja Dunjko, Omjyoti Dutta, Paul Dyke, Nikos Efremidis, Christoph Etrich, Henrique Fabrelli, Edilson Falcão Filho, BaoFeng Feng, Giovanni Filatrella, William Firth, Mario Floría, Nikos Flytzanis, Dimitri Frantzeskakis, Shenhe Fu, Jorge Fujioka, Arnaldo Gammal, Goran Glicorić, Jesús GómezGardeñez, Arjunan Govindarajan, Roger Grimshaw, Damia Gomila, Er'el Granot, Evgeny Gromov, Ljupco Hadzievski, Hao He, Jingsong He, Shangling He, Yingji He, Kyriakos Hizanidis, ChunQing Huang, Randy Hulet, Erik Infeld (deceased), Soumendu Jana, Irina Kabakova, Yaroslav Kartashov, Kenichi Kasamatsu, Dave Kaup, Emmanuel Kengne, Panos Kevrekidis, Avinash Khare, Maxim Khlopov, Yuri Kivshar, Patrick Köberle, Natasha Komarova, Roberto Kraenkel, Vladimir Konotop, Valentin Krinsky, Gershon Kurizki, Taras Lakoba, Muthusami Lakshmanan, Oleg Lavrentovich, Hervé Leblond, Falk Lederer, RayKuang Lee, Maciej Lewenstein, Ben Li, Lu Li, Pengfei Li, Qian Li, Zhaoxin Liang, Mietek Lisak (deceased), WuMing Liu, Valery Lobanov, De Luo, Zhihuan Luo, Xuekai Ma, Andrei Maimistov, Aleksandra Maluckov, Shukhrat Mardonov, Michal Matuszewski, Gérard Maugin (deceased), Dumitru Mazilu, Torsten Meier, Curtis Menyuk, Dumitru Mihalache, Thudiyangal Mithun, Michele Modugno, Jerry Moloney, Alex Nepomnyashchy, Alan Newell, Jason Nguyen, Markus Oberthaler, Patrik Öhberg, Maxim Olshanii, Richard Osgood, Jieli Qin, Wei Pang, Nicolae Panoiu, Bob Parmentier (deceased), Pavel Paulau, Dmitry Pelinovsky, GangDing Peng, Thomas Peschel, Ulf Peschel, Miguel Porras, K. Porsezian (deceased), Han Pu, Radha Ramaswamy, Albert Reyna, Nikolay Rosanov, Carlos RuizJiménez, Hidetsugu Sakaguchi, Luca Salasnich, Mario Salerno, Avadh Saxena, Stefan Schumacher, Evgeny Sherman, Yasha Shnir, Vladimir Skarka, Iain Skinner, Dmitry Skryabin, Ivan Smalyukh, Yuri Stepanyants, Leticia Tarruell, Flavio Toigo, Lluis Torner, Juan Torres, Mikhail Tribelsky, Marek Trippenbach, Alexey Ustinov, Ivan Uzunov, Amichai Vardi, Victor Vysloukh, Herbert Winful, Frank Wise, Logan Wright, Günter Wunner, Alexander Yakimenko, Zhenya Yan, Jianke Yang, Fangwei Ye, Vladimir Yurovsky, Damir Zajec, Liangwei Zeng, Dmitry Zezyulin, YiCai Zhang, WeiPing Zhong, and ZhengWei Zhou.
I also thank younger colleagues who worked with me on these topics, originally as my research students or postdocs:
Zeev Birnbaum, Miriam Blaauboer, Zhaopin Chen, Radik Driben, Nir Dror, Zhiwei Fan, Arik Gubeskis, Nir Hacker, Yongyao Li, Vitaly Lutsky, Eitam Luz, William Mak, Oleksander Marchukov, Thawatchai Mayteevarunyoo, Elad Shamriz, Alex Shnirman, Rich Tasgal, Igor Tikhonenkov, Isaac Towers, and Jianhua Zeng.
I keep in great esteem the memory of the work with my Ph.D. adviser, the great physicist Yakov Borisovich Zeldovich (1914–1987).
I would like to thank representatives of the publishing house of the American Institute of Physics for their invitation to write this book and the great help provided by them in the course of the work.
The cover image is reprinted with permission from Y.C. Zhang, Z.W. Zhou, B. A. Malomed, and H. Pu, “Stable solitons in three dimensional free space without the ground state: Selftrapped BoseEinstein condensates with spinorbit coupling,” Phys. Rev. Lett. 115, 253902 (2015). Copyright 2015 American Physical Society.