Gino Biondini's publications

This page lists articles in refereed journals, in reverse chronological order, as well as arXiv preprints, with links to the full text.
It does not include chapters in books, refereed/unrefereed/invited conference proceedings, theses, editorial material etc.
[Updated 8/28/2026]

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Preprints

  • “Wedge problems and dispersive shock waves in the two-dimensional Toda lattice”,
    M. Calabrese, G. Biondini, C. Chong and P. G. Kevrekidis,
    arXiv:2608.27415 [nlin.ps] (Abstract)
    We study the formation and interaction of dispersive shock waves (DSWs) in the two-dimensional Toda lattice subject to wedge-type initial conditions, and show that their interaction gives rise to a discrete analog of Mach reflection for dispersive shock waves in discrete systems. The initial jump across each leg of the wedge acts locally as a Riemann problem for the one-dimensional Toda lattice, producing two oblique DSWs whose leading-edge soliton amplitude is determined explicitly by the one-dimensional Whitham modulation theory. The two-dimensional nature of the problem manifests when these oblique DSWs meet along the symmetry axis. We show that, for compressive wedges (i.e., when the initial conditions are such that two oblique DSWs that are generated propagate toward each other), a critical slope qcr separates two regimes: in the subcritical regime ($q < q_{cr}$) the interaction is resonant and it produces an expanding DSW whose amplitude, length and velocity are explicitly computed by using exact soliton solutions of the two-dimensional Toda lattice; in the supercritical regime ($q > q_{cr}$) the interaction is ordinary and produces a localized peak whose amplitude is determined analytically. We also show qualitatively that a similar dichotomy between two regimes exists for expansive wedges (i.e., when the initial conditions are such that the two oblique DSWs propagate away from each other). We confirm all analytical predictions by comparing them with the results of direct numerical simulations. Finally, we show that the continuum limit of the result is consistent with the analogous theory for the Kadomtsev-Petviashvili equation, providing an independent validation of the analytical framework.
  • “Learning Lax pairs: Revisiting the classical paradigm”,
    J. Adriazola, G. Biondini, W. Zhu, and P. G. Kevrekidis,
    arXiv:2607.01493 [nlin.si] (Abstract)
    A Lax pair $(L,P)$ is sometimes thought of as a structural certificate, in that the spatial operator $L$ carries the spectral data of an integrable system, and its isospectral evolution under $\partial_t L = [L,P]$ encodes the nonlinear dynamics. Yet, experience shows that the correspondence between equations and Lax pairs is much more nuanced than this picture suggests. Equations can admit Lax pairs that fail to encode the expected integrable structure. This paper probes that anomalous corner of the Lax pair landscape through five case studies (the Euler top, the free Schrödinger equation, the inviscid Burgers equation, the shallow water system, and the Korteweg--de Vries equation), each illustrating a different way the link to integrability can be distorted. The approach combines analytical calculations with the Sparse Identification of Lax Operators (SILO) framework, which proved useful throughout, in some cases confirming the textbook pair and in others surfacing alternatives worth understanding on their own terms. The recurring lesson across the five cases is that compatibility underdetermines the Lax representation, so that anomalous pairs are regular features of the landscape rather than pathologies. Notably, we show that a spectrally degenerate Korteweg--de Vries Lax pair, classified as fake by standard criteria, still generates the full conservation hierarchy through its operator algebra, which shows that a blunt dichotomy between true and fake Lax pairs can be too reductive.
  • “Kinetic equations for a two-dimensional soliton gas”,
    G. Biondini, T. Bonnemain, B. Doyon, G. A. El and G. Roberti,
    arXiv:2606.30582 [nlin.ps] (Abstract)
    We formulate a general system of kinetic equations for a non-stationary two-dimensional gas of elastically interacting line solitons and apply it to the description of a soliton gas governed by the Kadomtsev-Petviashvili II (KPII) equation. We then verify the predictions of the kinetic theory in two analytically tractable problems: the oblique interaction of a KPII line soliton with a one-dimensional soliton condensate of the Korteweg-de Vries equation, and the interaction of a trial KPII soliton with a monochromatic KPII soliton gas. In both cases, we compare the analytical results with direct numerical simulations obtained by constructing two-dimensional soliton gases via exact KPII $N$-soliton solutions for large $N$, using appropriately chosen random distributions of soliton parameters. The comparison demonstrates excellent agreement, thereby providing strong validation of the proposed kinetic theory of 2D non-equilibrium soliton gases.
  • “Traveling waves and dispersive shock waves in two-dimensional lattices”,
    C. Chong, P. G. Kevrekidis, G. Biondini and W. Reichel,
    arXiv:2606.30353 [nlin.ps] (Abstract)
    In the present work we analyze traveling and dispersive shock waves of a two-dimensional Fermi-Pasta-Ulam-Tsingou lattice. In the first part of the paper, using variational techniques we prove the existence of both periodic and solitary traveling waves for convex potentials. In the case of unimodal profiles we are able to remove the assumption of convexity. The variational formulation also provides a natural algorithm for the numerical computation of traveling waves, which we use to explore both solitary and periodic traveling waves. The numerical computations are compared with analytical approximations based on the derivation of the KdV equation for quasi-one-dimensional propagation. In the second part of the paper, we focus on dispersive shock waves (DSWs), which are expanding modulated waves that connect states of different amplitude. In particular, we focus on line DSWs, which are constant along one direction and propagate in the direction orthogonal to which it is constant. Such solutions form when subject to quasi-one-dimensional jump initial data. We find that while the shape of the DSW depends on the direction of travel, properties such as the speed and amplitude do not. The systematic numerical study of the line\~DSWs is then compared to those predicted by the KdV equation along the line of propagation. Key characteristics of the DSWs, such as the speeds of the trailing and leading edges, are investigated for various jump heights, yielding good agreement between simulation and KdV approximation in the limit of vanishing jump height. Finally, we apply the DSW fitting method to study the trailing and leading edge characteristics of the DSW, finding even better agreement to the numerics when compared to the KdV prediction. The KdV prediction and DSW fitting predictions agree in the limit of small jump height.
  • “Modulation theory for lumps and interactions between lumps and mean flow in the Kadomtsev-Petviashvili equation”,
    G. Biondini S. Dyachenko, M. A. Hoefer and N. Ossi,
    arXiv:2606.14986 [nlin.si], to appear in Journal of Nonlinear Waves (Abstract)
    A (2+1)-dimensional hyperbolic system of four quasi-linear partial differential equations is derived that describes the modulations of lump solutions of the Kadomtsev-Petviashvili I (KPI) equation in the presence of a mean field. The system is then shown to satisfy the necessary conditions for integrability of hydrodynamic chains. Moreover, a suitable reduction of the resulting modulation system is applied to study the interactions between lumps and a rarefaction wave for the mean field. Precise conditions are derived that describe how the lump parameters change as a result of the interaction, and which in particular determine whether the lump is transmitted through or trapped inside the rarefaction wave. The theoretical predictions are compared to direct numerical simulations of the KPI equation, showing excellent agreement.
  • "Inverse spectral theory for self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the defocusing nonlinear Schrodinger equation with periodic boundary conditions"
    G. Biondini and Z. Zhang
    arXiv:2311.18127 [math.ap], to appear in Journal of Nonlinear Waves (Abstract)
    The inverse spectral theory for a self-adjoint one-dimensional Dirac operator associated periodic potentials is formulated via a Riemann-Hilbert problem approach. The resulting formalism is also used to solve the initial value problem for the nonlinear Schrodinger (NLS) equation. A uniqueness theorem for the solutions of the Riemann-Hilbert problem is established, which provides a new method for obtaining the potential from the spectral data. Two additional, scalar Riemann-Hilbert problems are also formulated that provide conditions for the periodicity in space and time of the solution generated by arbitrary sets of spectral data. The formalism applies for both finite-genus and infinite-genus potentials. The formalism also shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the defocusing NLS equation, in contrast with the representation of the solution of the NLS equation via the finite-genus formalism, in which two different sets of Dirichlet eigenvalues are used. 2026:

2026:

  • “Semiclassical dynamics and coherent soliton ensembles in the derivative nonlinear Schrödinger equation with periodic initial conditions”,
    Z. Wolski, Z. Zhang, G. Biondini and G. Kovacic
    Studies in Applied Mathematics 157 e70260 (2026) (Abstract)
    The semiclassical limit of the derivative nonlinear Schrödinger equation with periodic initial conditions is studied analytically and numerically. The spectrum of the associated scattering problem for a certain class of initial conditions, referred to as periodic single‐lobe potentials, is numerically computed, and it is shown that the spectrum becomes confined to the real and imaginary axes or the spectral parameter in the semiclassical limit. A formal Wentzel–Kramers–Brillouin expansion is computed for the scattering eigenfunctions, which allows one to obtain asymptotic expressions for the number, location, and size of the spectral bands and gaps. The results of these calculations suggest that, in the semiclassical limit, all excitations in the spectrum become effective solitons. Finally, the analytical predictions are compared with direct numerical simulations as well as with numerical calculations of the Lax spectrum, and the results are shown to be in excellent agreement.
  • "First-order continuum models for nonlinear dispersive waves in the granular crystal lattice”,
    G. Biondini, C. Chong, P. G. Kevrekidis and S. Yang
    Studies in Applied Mathematics 156 e70190 (2026) (Abstract)
    We derive and analyze, theoretically and numerically, two first‐order continuum models to approximate the nonlinear dynamics of granular crystal lattices, focusing specifically on solitary waves, periodic waves, and dispersive shock waves. The dispersive shock waves predicted by the two continuum models are studied using modulation theory, DSW fitting techniques, and direct numerical simulations. The PDE‐based predictions show good agreement with the DSWs generated by the discrete model simulation of the granular lattice itself, even in cases where no precompression is present and the lattice is purely nonlinear. Such an effective description could prove useful for future, more analytically amenable approximations of the original lattice system. 2025:

2025:

  • “Experimental observation of the spatio-temporal dynamics of breather gases in a recirculating fiber loop” F. Copie, G. Biondini, J. Oregero, G. A. El, P. Suret and S. Randoux
    Optics Letters 50, 7043 (2025) [Supplementary material] (Abstract)
    We report the experimental observation of breather gases (BGs) in optics, realized in a recirculating fiber loop that enables virtually lossless propagation over 1200 km. Initiated by a slowly modulated optical background perturbed by noise, the BGs form through a nonlinear fission process and demonstrate spatiotemporal dynamics that align closely with numerical simulations of the focusing one-dimensional nonlinear Schrödinger equation. The minimal dissipation in our setup enables a statistical characterization of the BGs and confirms the theoretically predicted doubling of kurtosis during the evolution of the BGs. To our knowledge, these results open new avenues for experimental studies of integrable turbulence involving solitons on a finite background.
  • “Spectral theory for non-self-adjoint Dirac operators with periodic potentials and inverse scattering transform for the focusing nonlinear Schrödinger equation with periodic boundary conditions”
    G. Biondini, G. Kovačič, A. Tovbis, Z. Wolski and Z. Zhang
    Physica D 148, 134970 (2025) (Abstract)
    We formulate the inverse spectral theory for a non-self-adjoint one-dimensional Dirac operator associated with periodic potentials via a Riemann–Hilbert problem approach. We use the resulting formalism to solve the initial value problem for the focusing nonlinear Schrödinger equation. We establish a uniqueness theorem for the solutions of the Riemann–Hilbert problem, which provides a new method for obtaining the potential from the spectral data. The formalism applies for both finite- and infinite-genus potentials. As in the defocusing case, the formalism shows that only a single set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the potential of the Dirac operator and the corresponding solution of the focusing NLS equation.
  • "Nonlinear stage of modulational instability in repulsive two-component Bose-Einstein condensates"
    S. Mossman, S.I. Mistakidis, G. C. Katsimiga, A. Romero-Ros, G. Biondini, P. Schmelcher, P. Engels, P. G. Kevrekidis
    Physical Review Letters 135, 113401 (2025) [Supplementary material] (Abstract)
    Modulational instability (MI) is a fundamental phenomenon in the study of nonlinear dynamics, spanning diverse areas such as shallow water waves, optics, and ultracold atomic gases. In particular, the nonlinear stage of MI has recently been a topic of intense exploration and has been shown to manifest, in many cases, in the generation of dispersive shock waves (DSWs). In this Letter, we experimentally probe the MI dynamics in an immiscible two-component ultracold atomic gas with exclusively repulsive interactions, catalyzed by a hard-wall-like boundary produced by a repulsive optical barrier. We analytically describe the expansion rate of the DSWs in this system, generalized to arbitrary inter- component interaction strengths and species ratios. We observe excellent agreement among the analytical results, an effective 1D numerical model, full 3D numerical simulations, and experimental data. Additionally, we extend this scenario to the interaction between two counterpropagating DSWs, which leads to the production of Peregrine soliton structures. These results further demonstrate the versatility of atomic platforms toward the controlled realization of DSWs and rogue waves.
  • “Mach reflection and expansion of two-dimensional dispersive shock waves”
    G. Biondini, A. Bivolcic and M. A. Hoefer
    Physical Review Letters 135, 067201 (2025)
    [Supplementary material] (Abstract)
    The oblique collisions and dynamical interference patterns of two-dimensional dispersive shock waves are studied numerically and analytically via the temporal dynamics induced by wedge-shaped initial conditions for the Kadomtsev-Petviashvili II equation. Various asymptotic wave patterns are identified, classified, and characterized in terms of the incidence angle and the amplitude of the initial step, which can give rise to either subcritical or supercritical configurations, including the generalization to dispersive shock waves of the Mach reflection and expansion of viscous shocks and line solitons. An eightfold amplification of the amplitude of an obliquely incident flow upon a wall at the critical angle is demonstrated. Applications of the results include bore interactions in geophysical fluid dynamics.
  • “Two-dimensional stationary soliton gas”
    T. Bonnemain, G. Biondini, B. Doyon, G. Roberti and G. A. El
    Physical Review Research 7, 013143 (2025) (Abstract)
    We study two-dimensional stationary soliton gas in the framework of the time-independent reduction of the Kadomtsev-Petviashvili (KPII) equation, which coincides with the integrable two-way “good” Boussinesq equation in the x y plane. This ( 2 \+ 0 ) D reduction enables the construction of the spatial analog of the kinetic equation for the stationary gas of KP solitons by invoking recent results on ( 1 \+ 1 ) D bidirectional soliton gases and generalized hydrodynamics of the Boussinesq equation. We then use the spectral kinetic theory to analytically describe two basic types of 2D soliton gas interactions: (i) refraction of a line soliton by a stationary soliton gas, and (ii) oblique interference of two soliton gases. We verify the analytical predictions by numerically implementing the corresponding KPII soliton gases via exact N \-soliton solutions with N \-large and appropriately chosen random distributions for the soliton parameters. We also explicitly evaluate the long-distance correlations for the two-component interference configurations. The results can be applied to a variety of physical systems, from shallow water waves to Bose-Einstein condensates.
  • “Breather gas fission from elliptic potentials in self-focusing media”
    G. Biondini, G. A. El, X.-D. Luo, J. Oregero and A. Tovbis
    Physical Review E 111, 014204 (2025) (Abstract)
    We present an analytical model of integrable turbulence in the focusing nonlinear Schrödinger (fNLS) equation, generated by a one-parameter family of finite-band elliptic potentials in the semiclassical limit. We show that the spectrum of these potentials exhibits a thermodynamic band/gap scaling compatible with that of soliton and breather gases depending on the value of the elliptic parameter m of the potential. We then demonstrate that, upon augmenting the potential by a small random noise (which is inevitably present in real physical systems), the solution of the fNLS equation evolves into a fully randomized, spatially homogeneous breather gas, a phenomenon we call breather gas fission. We show that the statistical properties of the breather gas at large times are determined by the spectral density of states generated by the unperturbed initial potential. We analytically compute the kurtosis of the breather gas as a function of the elliptic parameter m, and we show that it is greater than 2 for all nonzero m, implying non-Gaussian statistics. Finally, we verify the theoretical predictions by comparison with direct numerical simulations of the fNLS equation. These results establish a link between semiclassical limits of integrable systems and the statistical characterization of their soliton and breather gases.
  • "A regularized continuum model for the traveling waves and dispersive shock waves of the granular chain”,
    G. Biondini, C. Chong, P. G. Kevrekidis and S. Yang
    Journal of Nonlinear Waves 1, e2 (2025) (Abstract)
    In this paper, we focus on a discrete physical model describing granular crystals, whose equations of motion can be described by a system of differential difference equations. After revisiting earlier continuum approximations, we propose a regularized continuum model variant to approximate the discrete granular crystal model through a suitable partial differential equation. We then compute, both analytically and numerically, its travelling wave and periodic travelling wave solutions, in addition to its conservation laws. Next, using the periodic solutions, we describe quantitatively various features of the dispersive shock wave (DSW) by applying Whitham modulation theory and the DSW fitting method. Finally, we perform several sets of systematic numerical simulations to compare the corresponding DSW results with the theoretical predictions and illustrate that the continuum model provides a good approximation of the underlying discrete one.
  • “On the coupled Maxwell-Bloch system of equations with non-decaying fields at infinity”
    S. Li, G. Biondini and G. Kovačič,
    Studies in Applied Mathematics 154, e70055 (2025) (Abstract)
    We study an initial‐boundary‐value problem (IBVP) for a system of coupled Maxwell–Bloch equations (CMBE) that model two colors or polarizations of light resonantly interacting with a degenerate, two‐level, active optical medium with an excited state and a pair of degenerate ground states. We assume that the electromagnetic field approaches nonvanishing plane waves in the far past and future. This type of interaction has been found to underlie nonlinear optical phenomena including electromagnetically induced transparency, slow light, stopped light, and quantum memory. Under the assumptions of unidirectional, lossless propagation of slowly modulated plane waves, the resulting CMBE become completely integrable in the sense of possessing a Lax pair. In this paper, we formulate an inverse scattering transform (IST) corresponding to these CMBE and their Lax pair, allowing for the spectral line of the atomic transitions in the active medium to have a finite width. The scattering problem for this Lax pair is the same as for the Manakov system. The main advancement in this IST for CMBE is calculating the nontrivial spatial propagation of the spectral data and determining the state of the optical medium in the distant future from that in the distant past, which is needed for the complete formulation of the IBVP. The Riemann–Hilbert problem is used to extract the spatio‐temporal dependence of the solution from the evolving spectral data. We further derive and analyze several types of solitons and determine their velocity and stability, as well as find dark states of the medium, which fail to interact with a given pulse. 2024:

2024:

  • “Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening”
    G. Biondini, B. Prinari and Z. Zhang
    Adv. Nonlinear Anal. 13, 20240054 (2024) (Abstract)
    The Maxwell-Bloch system of equations with inhomogeneous broadening is studied, and the local and global well-posedness of the corresponding initial-boundary value problem is established by taking advantage of the integrability of the system and making use of the corresponding inverse scattering transform (IST). A key ingredient in the analysis is the L 2 {L}^{2} \-Sobolev bijectivity of the direct and IST established by Xin Zhou for the focusing Zakharov-Shabat problem.
  • "On the Whitham modulation equations for the Toda lattice and the quantitative description of its dispersive shocks"
    G. Biondini, C. Chong and P. G. Kevrekidis
    Physica D 469, 134315 (2024) (Abstract)
    The aim of this work is multifold. Firstly, it intends to present a complete, quantitative and self-contained description of the periodic traveling wave solutions and Whitham modulation equations for the Toda lattice, combining results from different previous works in the literature. Specifically, we connect the Whitham modulation equations and a detailed expression for the periodic traveling wave solutions of the Toda lattice. Along the way, some key details are filled in, such as the explicit expression of the characteristic speeds of the genus-one Toda–Whitham system. Secondly, we use these tools to obtain a detailed quantitative characterization of the dispersive shocks of the Toda system. Lastly, we validate the relevant analysis by performing a detailed comparison with direct numerical simulations.
  • "Integrable approximations of dispersive shock waves of the granular chain"
    A. Giesler, C. Chong P. Kevrekidis and G. Biondini
    Wave Motion 130, 103352 (2024) (Abstract)
    In the present work we revisit the shock wave dynamics in a granular chain with precompression. By approximating the model by an 𝛼-Fermi–Pasta–Ulam–Tsingou chain, we leverage the connection of the latter in the strain variable formulation to two separate integrable models, one continuum, namely the KdV equation, and one discrete, namely the Toda lattice. We bring to bear the Whitham modulation theory analysis of such integrable systems and the analytical approximation of their dispersive shock waves in order to provide, through the lens of the reductive connection to the granular crystal, an approximation to the shock wave of the granular problem. A detailed numerical comparison of the original granular chain and its approximate integrable-system-based dispersive shocks proves very favorable in a wide parametric range. The gradual deviations between (approximate) theory and numerical computation, as amplitude parameters of the solution increase are quantified and discussed.
  • "Inverse scattering transform for two-level systems with one-sided nonzero background"
    A. Abeya, G. Biondini, G. Kovacic and B. Prinari
    Communications in Mathematical Physics 405, 192 (2024) (Abstract)
    We formulate the inverse scattering transform for the scalar Maxwell-Bloch system of equations describing the resonant interaction of light and active optical media in the case when the light intensity does not vanish at infinity. We show that pure background states in general do not exist with a nonzero background field. We then use the formalism to compute explicitly the soliton solutions of this system. We discuss the initial population of atoms and show that the pure soliton solutions do not correspond to a pure state initially. We obtain a representation for the soliton solutions in determinant form and explicitly write down the one-soliton solutions. We next derive periodic solutions and rational solutions from the one-soliton solutions. We then analyze the properties of these solutions, including discussion of the sharp-line and small-amplitude limits, and thereafter show that the two limits do not commute. Finally, we investigate the behavior of general solutions, showing that solutions are stable (i.e., the radiative parts of solutions decay) only when initially atoms in the ground state dominate, i.e., initial population inversion is negative.
  • "Experimental realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates"
    A. Romero-Ros, G. C. Katsimiga, S. I. Mistakidis, S. Mossman, P. Schmelcher, G. Biondini, P. Engels and P. G. Kevrekidis
    Physical Review Letters 132, 033402 (2024) (Abstract)
    We experimentally realize the Peregrine soliton in a highly particle-imbalanced two-component repulsive Bose-Einstein condensate in the immiscible regime. The effective focusing dynamics and resulting modulational instability of the minority component provide the opportunity to dynamically create a Peregrine soliton with the aid of an attractive potential well that seeds the initial dynamics. The Peregrine soliton formation is highly reproducible, and our experiments allow us to separately monitor the minority and majority components, and to compare with the single component dynamics in the absence or presence of the well with varying depths. We showcase the centrality of each of the ingredients leveraged herein. Numerical corroborations and a theoretical basis for our findings are provided through three-dimensional simulations emulating the experimental setting and via a one-dimensional analysis further exploring its evolution dynamics.
  • "Whitham modulation theory for the Zakharov-Kuznetsov equation and transverse stability of its periodic traveling wave solutions"
    G. Biondini and A. Chernyavsky
    Studies in Applied Mathematics 152, 596 (2024) (Abstract)
    We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.
  • "Two-dimensional reductions of the Whitham modulation system for the Kadomtsev-Petviashvili equation"
    G. Biondini, A. J. Bivolcic, M. A. Hoefer and A. Moro
    Nonlinearity 37, 025012 (2024) (Abstract)
    Two-dimensional reductions of the Kadomtsev–Petviashvili(KP)–Whitham system, namely the overdetermined Whitham modulation system for five dependent variables that describe the periodic solutions of the KP equation, are studied and characterized. Three different reductions are considered corresponding to modulations that are independent of x, independent of y, and of t (i.e. stationary), respectively. Each of these reductions still describes dynamic, two-dimensional spatial configurations since the modulated cnoidal wave, generically, has a nonzero speed and a nonzero slope in the xy plane. In all three of these reductions, the integrability of the resulting systems of equations is proven, and various other properties are elucidated. Compatibility with conservation of waves yields a reduction in the number of dependent variables to two, three and four, respectively. As a byproduct of the stationary case, the Whitham modulation system for the classical Boussinesq equation is explicitly obtained. 2023:

2023:

  • "Elliptic finite-band potentials of a non-self-adjoint Dirac operator"
    G. Biondini, X.-D. Luo, J. Oregero and A. Tovbis
    Adv. Math. 429, 109188 (2023) (Abstract)
    We present an explicit two-parameter family of finite-band Jacobi elliptic potentials given by q ≡ A dn(x; m), where m ∈ (0, 1\) and A can be taken to be positive without loss of generality, for a non-self-adjoint Dirac operator L, which connects two well-known limiting cases of the plane wave (m = 0\) and of the sech potential (m = 1). We show that, if A ∈ N, then the spectrum consists of R plus 2A Schwarz symmetric segments (bands) on iR. This characterization of the spectrum is obtained by relating the periodic and antiperiodic eigenvalue problems for the Dirac operator to corresponding eigenvalue problems for tridiagonal operators acting on Fourier coefficients in a weighted Hilbert space, and to appropriate connection problems for Heun’s equation. Conversely, if A is not an integer, the spectrum of L consists of infinitely many bands in C. When A ∈ N, the corresponding potentials generate finite-genus solutions for all the positive and negative flows associated with the focusing nonlinear
  • "Whitham modulation theory for the nonlinear Schrodinger equation in two and three spatial dimensions"
    A. Abeya, G. Biondini and M. A. Hoefer
    J. Phys. A 56, 025701 (2023) (Abstract)
    The Whitham modulation equations for the defocusing nonlinear Schrödinger (NLS) equation in two, three and higher spatial dimensions are derived using a two-phase ansatz for the periodic traveling wave solutions and by period-averaging the conservation laws of the NLS equation. The resulting Whitham modulation equations are written in vector form, which allows one to show that they preserve the rotational invariance of the NLS equation, as well as the invariance with respect to scaling and Galilean transformations, and to immediately generalize the calculations from two spatial dimensions to three. The transformation to Riemann-type variables is described in detail; the harmonic and soliton limits of the Whitham modulation equations are explicitly written down; and the reduction of the Whitham equations to those for the radial NLS equation is explicitly carried out. Finally, the extension of the theory to higher spatial dimensions is briefly outlined. The multidimensional NLS-Whitham equations obtained here may be used to study large amplitude wavetrains in a variety of applications including nonlinear photonics and matter waves.
  • "On the spectrum of the focusing Zakharov-Shabat operator with periodic potentials"
    G. Biondini, J. Oregero and A. Tovbis
    J. Spectral Theory 12, 939--992 (2022) (Abstract)
    The spectrum of the focusing Zakharov–Shabat operator on the circle is studied, and its explicit dependence on the presence of a semiclassical parameter is also considered. Several new results are obtained. In particular: (i) it is proved that the resolvent set is comprised of two connected components; (ii) new bounds on the location of the Floquet and Dirichlet spectra are obtained, some of which depend explicitly on the value of the semiclassical parameter; (iii) it is proved that the spectrum localizes to a “cross” in the spectral plane in the semiclassical limit. The results are illustrated by discussing several examples in which the spectrum is computed analytically or numerically. 2022:

    [The journal volume lists 2022 as the year, but the actual publication date of the issue was April 2023.]

2022:

  • "p-star models, mean field random networks and the heat hierarchy"
    G. Biondini, A. Moro, B. Prinari and O. Senkevich
    Phys. Rev. E 105, 014306 (2022) (Abstract)
    We consider the mean-field analog of the p-star model for homogeneous random networks, and we compare its behavior with that of the p-star model and its classical mean-field approximation in the thermodynamic regime. We show that the partition function of the mean-field model satisfies a sequence of partial differential equations known as the heat hierarchy, and the models connectance is obtained as a solution of a hierarchy of nonlinear viscous PDEs. In the thermodynamic limit, the leading-order solution develops singularities in the space of parameters that evolve as classical shocks regularized by a viscous term. Shocks are associated with phase transitions and stable states are automatically selected consistently with the Maxwell construction. The case p = 3 is studied in detail. Monte Carlo simulations show an excellent agreement between the p-star model and its mean-field analog at the macroscopic level, although significant discrepancies arise when local features are compared.
  • "Soliton resonance and web structure in the Davey-Stewartson system"
    G. Biondini, D. Kireyev and K.-i. Maruno
    J. Phys. A 55, 305701 (2022) (Abstract)
    We write down and characterize a large class of nonsingular multi-soliton solutions of the defocusing Davey–Stewartson II equation. In particular we study their asymptotics at space infinities as well as their interaction patterns in the xy-plane, and we identify several subclasses of solutions. Many of these solutions describe phenomena of soliton resonance and web structure. We identify a subclass of solutions that is the analogue of the soliton solutions of the Kadomtsev–Petviashvili II equation. In addition to this subclass, however, we show that more general solutions exist, describing phenomena that have no counterpart in the Kadomtsev–Petviashvili equation, including V-shape solutions and soliton reconnection.
  • "On the periodic solutions of the Davey-Stewartson systems"
    G. Biondini and D. Kireyev
    East Asian J. Appl. Math. 12, 1 (2022) (Abstract)
    The periodic, traveling wave solutions of all four versions of the Davey-Stewartson system (namely the focusing and the defocusing cases of both the Davey-Stewartson I and the Davey-Stewartson II equations) are derived and classified. For all four versions, these solutions are described in terms of elliptic functions. Special reductions and limiting cases, including harmonic limits, soliton limits, and one-dimensional solutions, are also explicitly discussed.
  • "Mach reflection, resonant line solitons and modulation theory"
    S. Ryskamp, M. A. Hoefer and G. Biondini
    Roy. Soc. Proc. A 478, 20210823 (2022) (Abstract)
    Resonant Y-shaped soliton solutions to the Kadomtsev–Petviashvili II (KPII) equation are modelled as shock solutions to an infinite family of modulation conservation laws. The fully two-dimensional soliton modulation equations, valid in the zero dispersion limit of the KPII equation, are demonstrated to reduce to a one-dimensional system. In this same limit, the rapid transition from the larger Y soliton stem to the two smaller legs limits to a travelling discontinuity. This discontinuity is a multivalued, weak solution satisfying modified Rankine–Hugoniot jump conditions for the one-dimensional modulation equations. These results are applied to analytically describe the dynamics of the Mach reflection problem, V-shaped initial conditions that correspond to a soliton incident upon an inward oblique corner. Modulation theory results show excellent agreement with direct KPII numerical simulation.
  • "Manakov system with parity symmetry on nonzero background and related boundary value problems"
    A. Abeya, G. Biondini and B. Prinari
    J. Phys. A 55, 254001 (2022) (Abstract)
    We characterize initial value problems for the defocusing Manakov system (coupled two-component nonlinear Schrödinger equation) with nonzero background and well-defined spatial parity symmetry (i.e., when each of the components of the solution is either even or odd), corresponding to boundary value problems on the half line with Dirichlet or Neumann boundary conditions at the origin. We identify the symmetries of the eigenfunctions arising from the spatial parity of the solution, and we determine the corresponding symmetries of the scattering data (reflection coefficients, discrete spectrum and norming constants). All parity induced symmetries are found to be more complicated than in the scalar (i.e., one-component) case. In particular, we show that the discrete eigenvalues giving rise to dark solitons arise in symmetric quartets, and those giving rise to dark–bright solitons in symmetric octets. We also characterize the differences between the purely even or purely odd case (in which both components are either even or odd functions of x) and the ‘mixed parity’ cases (in which one component is even while the other is odd). Finally, we show how, in each case, the spatial symmetry yields a constraint on the possible existence of self-symmetric eigenvalues, corresponding to stationary solitons, and we study the resulting behavior of solutions.
  • "Inverse scattering transform for the defocusing Manakov system with non-parallel boundary conditions at infinity"
    A. Abeya, G. Biondini and B. Prinari
    East Asian J. Appl. Math. 12, 715 (2022) (Abstract)
    The inverse scattering transform (IST) for the defocusing Manakov system is developed with non-zero boundary conditions at infinity comprising non-parallel boundary conditions — i.e., asymptotic polarization vectors. The formalism uses a uniformization variable to map two copies of the spectral plane into a single copy of the complex plane, thereby eliminating square root branching. The “adjoint” Lax pair is also used to overcome the problem of non-analyticity of some of the Jost eigenfunctions. The inverse problem is formulated in term of a suitable matrix Riemann-Hilbert problem (RHP). The most significant difference in the IST compared to the case of parallel boundary conditions is the asymptotic behavior of the scattering coefficients, which affects the normalization of the eigenfunctions and the sectionally meromorphic matrix in the RHP. When the asymptotic polarization vectors are not orthogonal, two different methods are presented to convert the RHP into a set of linear algebraic-integral equations. When the asymptotic polarization vectors are orthogonal, however, only one of these methods is applicable. Finally, it is shown that, both in the case of orthogonal and non-orthogonal polarization vectors, no reflectionless potentials can exist, which implies that the problem does not admit pure soliton solutions.
  • "Realization of the Peregrine soliton in repulsive two-component Bose-Einstein condensates"
    A. Romero-Ros, G. C. Katsmiga, S. I. Mistakidis, B. Prinari, G. Biondini, P. Schmelcher and P. G. Kevrekidis
    Phys. Rev. A 105, 053306 (2022) (Abstract)
    The present work is motivated by the recent experimental realization of the Townes soliton in an effective two-component Bose-Einstein condensate by B. Bakkali-Hassan et al. [Phys. Rev. Lett. 127, 023603 (2021)]. Here, we use a similar multicomponent platform to exemplify theoretically and numerically, within the mean-field Gross-Pitaevskii framework, the potential toward the experimental realization of a different fundamental wave structure, namely the Peregrine soliton. Leveraging the effective attractive interaction produced within the mixture’s minority species in the immiscible regime, we illustrate how initialization of the condensate with a suitable power-law decaying spatial density pattern yields the robust emergence of the Peregrine wave in the absence and in the presence of a parabolic trap. We then showcase the spontaneous emergence of the Peregrine soliton via a suitably crafted wide Gaussian initialization, again both in the homogeneous case and in the trap scenario. It is also found that narrower wave packets may result in periodic revivals of the Peregrine soliton, while broader ones give rise to a cascade of Peregrine solitons arranged in a so-called Christmas-tree structure. Strikingly, the persistence of these rogue-wave structures is demonstrated in certain temperature regimes as well as in the presence of transversal excitations through three-dimensional computations in a quasi-one-dimensional regime. This proof-of-principle illustration is expected to represent a practically feasible way to generate and observe this rogue wave in realistic current ultracold atom experimental settings.
  • "On-demand generation of dark-bright soliton trains in Bose-Einstein condensates"
    A. Romero-Ros, G. C. Katsmiga, B. Prinari, G. Biondini, P. Schmelcher and P. G. Kevrekidis
    Phys. Rev. A 105, 023325 (2022) (Abstract)
    The controlled creation of dark-bright (DB) soliton trains in multicomponent Bose-Einstein condensates (BECs) is a topic of ongoing interest. In this work we generalize earlier findings on the creation of dark soliton trains in single-component BECs [A. Romero-Ros et al., Phys. Rev. A 103, 023329 (2021)] to two-component BECs. By choosing suitable filled box-type initial configurations (FBTCs) and solving the direct scattering problem for the defocusing vector nonlinear Schrödinger equation with nonzero boundary conditions we obtain analytical expressions for the DB soliton solutions produced by a general FBTC. It is found that the size of the initial box and the amount of filling directly affect the number, size, and velocity of the solitons, while the initial phase determines the parity (even or odd) of the solutions. Our analytical results are compared to direct numerical integration of the coupled Gross-Pitaevskii equations, both in the absence and in the presence of a harmonic trap, and an excellent agreement between the two is demonstrated.

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