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We study solutions to the Kadomtsev-Petviashvili equation whose underlying algebraic curves undergo tropical degenerations. Riemann's theta function becomes a finite exponential sum that is supported on a Delaunay polytope. We introduce the Hirota variety which parametrizes all tau functions arising from such a sum. We compute tau functions from points on the Sato Grassmannian that represent Riemann-Roch spaces and we present an algorithm that finds a soliton solution from a rational nodal curve., 21 pages, some edits based on referee comments

In this paper, we study the Stokes phenomenon of the cyclotomic Knizhnik-Zamolodchikov equation, and prove that its two types of Stokes matrices satisfy the Yang-Baxter and reflection equations respectively. We then discuss its isomonodromy deformation, and its relations with cyclotomic associators, twists, and quantum symmetric pairs., 16 pages

In this article, we derive exact analytical expressions for the spatial Fourier spectrum of the soliton family on a constant background. Also known as breathers, these solitons are exact solutions of the nonlinear Schr\"odinger equation and are considered as prototypes for rogue wave models. Depending on the periodicity in the spatial-temporal domain, the characteristics in the wavenumber-temporal domain may feature either a continuous or discrete spectrum., Comment: 16 pages, 10 figures, 22 references

The Painleve-IV equation has two families of rational solutions generated respectively by the generalized Hermite polynomials and the generalized Okamoto polynomials. We apply the isomonodromy method to represent all of these rational solutions by means of two related Riemann-Hilbert problems, each of which involves two integer-valued parameters related to the two parameters in the Painleve-IV equation. We then use the steepest-descent method to analyze the rational solutions in the limit that at least one of the parameters is large. Our analysis provides rigorous justification for formal asymptotic arguments that suggest that in general solutions of Painleve-IV with large parameters behave either as an algebraic function or an elliptic function. Moreover, the results show that the elliptic approximation holds on the union of a curvilinear rectangle and, in the case of the generalized Okamoto rational solutions, four curvilinear triangles each of which shares an edge with the rectangle; the algebraic approximation is valid in the complementary unbounded domain. We compare the theoretical predictions for the locations of the poles and zeros with numerical plots of the actual poles and zeros obtained from the generating polynomials, and find excellent agreement., Comment: 138 pages, 45 figures

We continue to investigate absolutely continuous spectrum of generalized indefinite strings. By following an approach of Deift and Killip, we establish stability of the absolutely continuous spectra of two more model examples of generalized indefinite strings under rather wide perturbations. In particular, one of these results allows us to prove that the absolutely continuous spectrum of the isospectral problem associated with the two-component Camassa-Holm system in a certain dispersive regime is essentially supported on the set $(-\infty,-1/2]\cup [1/2,\infty)$., Comment: 27 pages

We study integrability properties of the non-chiral intermediate long wave (ncILW) equation with periodic boundary conditions. The ncILW equation was recently introduced by the authors as a parity-invariant relative of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) use the Hirota method to construct the periodic multi-soliton solutions, (d) derive a B\"{a}cklund transformation, (e) use the B\"{a}cklund transformation to obtain an infinite number of conservation laws., Comment: 27 pages. This is a standalone paper based on Section 6 of arXiv:2005.10781v1

We study integrability properties of the non-chiral intermediate long wave equation recently introduced by the authors as a parity-invariant variant of the intermediate long wave equation. For this new equation we: (a) derive a Lax pair, (b) derive a Hirota bilinear form, (c) derive a B\"{a}cklund transformation, (d) use, separately, the B\"{a}cklund transformation and the Lax representation to obtain an infinite number of conservation laws., Comment: 29 pages

We reconsider the simple matrix model description of Hurwitz numbers proposed by S. Natanzon and A. Orlov, which uses the superintegrability property of the complex matrix model, and discuss a way of its possible supersymmetric extension to approach spin Hurwitz numbers., 6 pages

We study the Bethe Ansatz Equations for the Quantum KdV model, which are also known to be solved by the spectral determinants of a specific family of anharmonic oscillators called monster potentials (ODE/IM correspondence). These Bethe Ansatz Equations depend on two parameters, identified with the momentum and the degree at infinity of the anharmonic oscillators. We provide a complete classification of the solutions with only real and positive roots -- when the degree is greater than 2 -- in terms of admissible sequences of holes. In particular, we prove that admissible sequences of holes are naturally parameterised by integer partitions, and we prove that they are in one-to-one correspondence with solutions of the Bethe Ansatz Equations if the momentum is large enough. Consequently, we deduce that the monster potentials are complete, in the sense that every solution of the Bethe Ansatz Equations coincides with the spectrum of a unique monster potential. This essentially (i.e. up to gaps in the previous literature) proves the ODE/IM correspondence for the Quantum KdV model/monster potentials -- which was conjectured by Dorey-Tateo and Bazhanov-Lukyanov-Zamolodchikov -- when the degree is greater than 2. Our approach is based on the transformation of the Bethe Ansatz Equations into a free-boundary nonlinear integral equation -- akin to the equations known in the physics literature as DDV or KBP or NLIE -- of which we develop the mathematical theory from the beginning., Comment: 49 pages, 3 figures. Minor revision. Accepted version. To appear on Communications in Mathematical Physics (2023)

We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter-Belavin $R$-matrix in the fundamental representation of ${\rm GL}_M$. In the scalar case $M=1$ these operators are the elliptic Macdonald-Ruijsenaars operators, while in the general case they can be viewed as anisotropic versions of the quantum spin Ruijsenaars Hamiltonians. We show that commutativity of the operators for any $M$ is equivalent to a set of $R$-matrix identities. The proof of identities is based on the properties of elliptic $R$-matrix including the quantum and the associative Yang-Baxter equations. As an application of our results, we introduce elliptic generalization of q-deformed Haldane-Shastry model., Comment: 38 pages, minor corrections

We discuss three manners to implement Lorentz symmetry breaking in a superfield theory formulated within the superfield formalism, that is, deformation of the supersymmetry algebra, introducing of an extra superfield whose components can depend on Lorentz-violating (LV) vectors (tensors), and adding of new terms proportional to LV vectors (tensors) to the superfield action. We illustrate these methodologies with examples of quantum calculations., Comment: Presented at the Ninth Meeting on CPT and Lorentz Symmetry, Bloomington, Indiana, May 17-26, 2022, 4 pages, minor corrections

We propose a novel semi-discrete Kadomtsev--Petviashvili equation with two discrete and one continuous independent variables, which is integrable in the sense of having the standard and adjoint Lax pairs, from the direct linearisation framework. By performing reductions on the semi-discrete Kadomtsev--Petviashvili equation, new semi-discrete versions of the Drinfel'd--Sokolov hierarchies associated with Kac--Moody Lie algebras $A_r^{(1)}$, $A_{2r}^{(2)}$, $C_r^{(1)}$ and $D_{r+1}^{(2)}$ are successfully constructed. A Lax pair involving the fraction of $\mathbb{Z}_\mathcal{N}$ graded matrices is also found for each of the semi-discrete Drinfel'd--Sokolov equations. Furthermore, the direct linearisation construction guarantees the existence of exact solutions of all the semi-discrete equations discussed in the paper, providing another insight into their integrability in addition to the analysis of Lax pairs., 18 pages

There are a few Lax matrices of the Clebsch system. Poles of the Baker-Akhiezer function determine the class of equivalent divisors on the corresponding spectral curves. According to the Riemann-Roch theorem, each class has a unique reduced representative. We discuss properties of such reduced divisor on the spectral curve of $3\times 3$ Lax matrix having a natural generalization to $gl^*(n)$ case., Comment: 14 pages, no figures, LaTeX with AMS fonts

In the context of the Cauchy problem for the Korteweg-de Vries equation we extend the inverse scattering transform to initial data that behave at plus infinity like a sum of Wigner-von Neumann type potentials with small coupling constants. Our arguments are based on the theory of Hankel operators., To appear in Nonlinearity

We introduce a class of reductions of the two-component KP hierarchy, which includes the Hirota-Ohta system hierarchy. The description of the reduced hierarchies is based on the Hirota bilinear identity and an extra bilinear relation characterising the reduction. We derive the reduction conditions in terms of the Lax operator and higher linear operators of the hierarchy, as well as in terms of the basic two-component KP system of equations., 12 pages, several misprints in the formulae, especially in the Section 3, corrected. To be published in Theoretical and Mathematical Physics

Strong coupling of charge, spin, and lattice in solids brings about emergent elementary excitations with their intertwining and, in one dimension, solitons are known as such. The charge-transferred organic ferroelectric, TTF-CA, has been argued to host charge solitons; however, the existence of the charge solitons remains unverified. Here, we demonstrate that the charge-transport gap in the ionic Mott-Peierls insulating phase of TTF-CA is an order of magnitude smaller than expected from quasiparticle excitations, however, being entirely consistent with the charge soliton excitations. We further suggest that charge and spin solitons move with similar diffusion coefficients in accordance with their coexistence. These results provide a basis for the thermal excitations of the emergent solitons., 6 pages, 3 figures

We look into dark solitons in a quasi-1D dipolar Bose gas and in a quantum droplet. We derive the analytical solitonic solution of a Gross-Pitaevskii-like equation accounting for beyond mean-field effects. The results show there is a certain critical value of the dipolar interactions, for which the width of a motionless soliton diverges. Moreover, there is a peculiar solution of the motionless soliton with a non-zero density minimum. We also present the energy spectrum of these solitons with an additional excitation subbranch appearing. Finally, we perform a series of numerical experiments revealing the coexistence of a dark soliton inside a quantum droplet., Comment: 6+4 pages, 4+1 figures

We consider the Generalized Gibbs ensemble of the Ablowitz-Ladik lattice, and the Schur flow. We derive large deviations principles for the distribution of the empirical measures of the equilibrium measures for these ensembles. As a consequence, we deduce their almost sure convergence. Moreover, we are able to characterize their limit in terms of the equilibrium measure of the Circular, and the Jacobi beta ensemble respectively., 28 pages

We present a result which allows us to deform a Poisson-Nijenhuis manifold into a Poisson quasi-Nijenhuis manifold by means of a closed 2-form. Under an additional assumption, the deformed structure is also Poisson-Nijenhuis. We apply this result to show that the canonical Poisson-Nijenhuis structure on R^2n gives rise to both the Poisson-Nijenhuis structure of the open (or non periodic) n-particle Toda lattice, introduced by Das and Okubo [6], and the Poisson quasi-Nijenhuis structure of the closed (or periodic) n-particle Toda lattice, described in our recent work [7].

Generalizing a construction presented in [3], we show that the orbit space of $B_2$ less the image of coordinate lines under the quotient map is equipped with two Dubrovin-Frobenius manifold structures which are related respectively to the defocusing and the focusing nonlinear Schr\"odinger (NLS) equations. Motivated by this example, we study the case of $B_n$ and we show that the defocusing case can be generalized to arbitrary $n$ leading to a Dubrovin-Frobenius manifold structure on the orbit space of the group. The construction is based on the existence of a non-degenerate and non-constant invariant bilinear form that plays the role of the Euclidean metric in the Dubrovin-Saito standard setting. Up to $n=4$ the prepotentials we get coincide with those associated with constrained KP equations discussed in [20]., Comment: 37 pages, 1 figure

We consider the modulationally stable version of the Kaup-Boussinesq system which models propagation of nonlinear waves in various physical systems. It is shown that the Whitham modulation equations for this model have a new type of solutions which describe trigonometric shock waves. In the Gurevich-Pitaevskii problem of evolution of an initial discontinuity, these solutions correspond to a non-zero wave excitation on one of the sides of the discontinuity. Our analytical results are confirmed by numerical calculations.

Three decades ago, Inozemtsev found an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane-Shastry (HS) spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero-Sutherland model. Though Inozemtsev's spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev's `extended coordinate Bethe ansatz' and clarify various aspects of the model's exact spectrum and its limits. We identify quasimomenta in terms of which the $M$-particle energy is close to being (functionally) additive, as one would expect from the limiting models; our expression is additive iff the energy of the elliptic Calogero-Sutherland system is so. This enables us to rewrite the energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the $M=2$ particle sector and its limits in detail. We identify an $S$-matrix that is independent of positions. We show that the Bethe-ansatz equations reduce to those of Heisenberg in one limit and give rise to the `motifs' of HS in the other limit. We show that, as the interpolation parameter changes, the `scattering states' from Heisenberg become Yangian highest-weight states for HS, while bound states become ($\mathfrak{sl}_2$-highest weight versions of) affine descendants of the magnons from $M=1$. For bound states we find a generalisation of the known equation for the `critical length' for the Heisenberg spin chain. We discuss completeness for $M=2$ by passing to the elliptic curve. Our review of the two-particle sectors of the Heisenberg and HS spin chains may be of independent interest., 63 pages, 13 figures, 3 tables. v2: 65 pages, 13 figures, 4 tables. Minor changes. v3: 67 pages, 13 figures, 4 tables. Various minor changes; added discussion of bound states at large $L$, derivation of Haldane's Bethe-ansatz-like equations and discussion of Heisenberg limit of rationalisation

In this paper, first we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras, Manin triples of Leibniz algebras and Leibniz bialgebras are equivalent. Then we introduce the notion of a (relative) Rota-Baxter operator on a Leibniz algebra and construct the graded Lie algebra that characterizes relative Rota-Baxter operators as Maurer-Cartan elements. By these structures and the twisting theory of twilled Leibniz algebras, we further define the classical Leibniz Yang-Baxter equation, classical Leibniz r-matrices and triangular Leibniz bialgebras. Finally, we construct solutions of the classical Leibniz Yang-Baxter equation using relative Rota-Baxter operators and Leibniz-dendriform algebras., 28 pages, comments are welcome. J. Noncommutative Geom. 2022

We investigate higher-order breathers of the cubic nonlinear Schr\"odinger equation on an elliptic background. We find that, beyond first-order, any arbitrarily constructed breather is a single-peaked solitary wave on a disordered background. These "quasi-rogue waves" are also common on periodic backgrounds. We assume the higher-order breather is constructed out of constituent first-order breathers with commensurate periods (i.e., higher-order harmonic waves). In that case, one obtains "quasi-periodic" breathers with distorted side-peaks. Fully periodic breathers are obtained when their wavenumbers are harmonic multiples of the background and each other. They are truly rare, requiring finely-tuned parameters. Thus, on a periodic background, we arrive at the paradoxical conclusion that the apparent higher-order rogue waves are rather common, while the truly periodic breathers are exceedingly rare., Comment: 14 pages, 8 figures

We study dynamics of multi-soliton solutions of anti-self-dual Yang-Mills equations for G=GL(2,C) in four-dimensional spaces. The one-soliton solution can be interpreted as a codimension-one soliton in four-dimensional spaces because the principal peak of action density localizes on a three-dimensional hyperplane. We call it the soliton wall. We prove that in the asymptotic region, the n-soliton solution possesses n isolated localized lumps of action density, and interpret it as n intersecting soliton walls. More precisely, each action density lump is essentially the same as a soliton wall because it preserves its shape and "velocity" except for a position shift of principal peak in the scattering process. The position shift results from the nonlinear interactions of the multi-solitons and is called the phase shift. We calculate the phase shift factors explicitly and find that the action densities can be real-valued in three kind of signatures. Finally, we show that the gauge group can be G=U(2) in the Ultrahyperbolic space (the split signature (+, +, -, -)). This implies that the intersecting soliton walls could be realized in all region in N=2 string theories. It is remarkable that quasideterminants dramatically simplify the calculations and proofs., Comment: 21 pages; v3: minor changes, proof of unitarity simplified, version to appear in JHEP

The integrability of a four-dimensional sixth-order bilinear equation associated with the exceptional affine Lie algebra $D_4^{(1)}$ is studied by means of the singularity analysis. This equation is shown to pass the Painlev\'{e} test in three distinct cases of its coefficients, exactly when the equation is effectively a three-dimensional one, equivalent to the BKP equation., Comment: 8 pages; v2: one sentence corrected

The Moutard transform is constructed for the solutions of the Davey-Stewartson II equation. It is geometrically interpreted using the spinor (Weierstrass) representation of surfaces in four-dimensional Euclidean space. Using the Moutard transformation and minimal surfaces, examples of solutions are constructed that have smooth, rapidly decreasing initial data and lose their regularity in a finite time., Comment: 18 pages

In this paper, the Fokas-Lenells equations are investigated via bilinear approach. We bilinearize the unreduced Fokas-Lenells system, derive double Wronskian solutions, and then, by means of a reduction technique we obtain variety of solutions of the reduced equations. This enables us to have a full profile of solutions of the classical and nonlocal Fokas-Lenells equations. Some obtained solutions are illustrated based on asymptotic analysis. As a notable new result, we obtain solutions to the Fokas-Lenells equation, which are related to real discrete eigenvalues and not reported before in the analytic approaches. These solutions behave like (multi-)periodic waves or solitary waves with algebraic decay. In addition, we also obtain solutions to the two-dimensional massive Thirring model from those of the Fokas-Lenells equation., Comment: 32 pages, 11 figures, appendixes and references updated

We review and give detailed description for ${\rm gl}_{NM}$ Gaudin models related to holomorphic vector bundles of rank $NM$ and degree $N$ over elliptic curve with $n$ punctures. Then we introduce their generalizations constructed by means of $R$-matrices satisfying the associative Yang-Baxter equation. A natural extension of the obtained models to the Schlesinger systems is given as well., Comment: 25 pages, minor changes

We consider the Cauchy problem for the classical Hirota equation on the line with decaying initial data. Based on the spectral analysis of the Lax pair of the Hirota equation, we first expressed the solution of the Cauchy problem in terms of the solution of a Riemann-Hilbert problem. Further we apply nonlinear steepest descent analysis to obtain the long-time asymptotics of the solution in the critical transition region $|\frac{x}{t} - \frac{\alpha^2}{3\beta}|t^{2/3} \leq M$, $M$ is a positive constant. Our result shows that the long time asymptotics of the Hirota equation can be expressed in terms of the solution of Painlev\'{e} $\mathrm{II}$ equation. Keywords: Hirota equation, steepest descent method, Painlev\'{e} $\mathrm{II}$ equation, long-time asymptotics., Comment: 17 pages

We study the Gauge/Bethe correspondence for two-dimensional $\mathcal{N}=(2,2)$ supersymmetric quiver gauge theories associated with toric Calabi-Yau three-folds, whose BPS algebras have recently been identified as the quiver Yangians. We start with the crystal representations of the quiver Yangian, which are placed at each site of the spin chain. We then construct integrable models by combining the single-site crystals into crystal chains by a coproduct of the algebra, which we determine by a combination of representation-theoretical and gauge-theoretical arguments. For non-chiral quivers, we find that the Bethe ansatz equations for the crystal chain coincide with the vacuum equation of the quiver gauge theory, thus confirming the corresponding Gauge/Bethe correspondence. For more general chiral quivers, however, we find obstructions to the $R$-matrices satisfying the Yang-Baxter equations and the unitarity conditions, and hence to their corresponding Gauge/Bethe correspondence. We also discuss trigonometric (quantum toroidal) versions of the quiver BPS algebras, which correspond to three-dimensional $\mathcal{N}=2$ gauge theories and arrive at similar conclusions. Our findings demonstrate that there are important subtleties in the Gauge/Bethe correspondence, often overlooked in the literature., 109 pages, 4 figures

We propose a Hamiltonian formalism for $N$ periodic dressing chain with the even number $N$. The formalism is based on Dirac reduction applied to the $N+1$ periodic dressing chain with the odd number $N+1$ for which the Hamiltonian formalism is well known. The Hamilton dressing chain equations in the $N$ even case depend explicitly on a pair of conjugated Dirac constraints and are equivalent to $A^{(1)}_{N-1}$ invariant symmetric Painlev\'e equations., Comment: 13 pages, comment added in subsection 3.1

Nonlinearities in lattices with topological band structures can induce topological interfaces in the bulk of structures and give rise to bulk solitons in the topological bandgaps. Here we study a photonic Chern insulator with saturable nonlinearity and show the existence of topological bulk solitons. The fundamental bulk solitons exhibit as semi-vortex solitons, where only one pseudospin component has a nonzero vorticity. The bulk solitons have equal angular momentum at different valleys. This phenomenon is a direct outcome of the topology of the linear host lattice and the angular momentum can be changed by switching the sign of the nonlinearity. The bulk solitons bifurcate from the linear bulk band edge and terminate when their powers saturate. We find that these bulk solitons are stable within the whole spectrum range. Moreover, these bulk solitons are robust against lattice disorders both from on-site energies and hopping amplitudes. Our work extends the study of Chern insulators into the nonlinear regime and highlights the interplay between topology and nonlinearity.

We study properties of Hamiltonian integrable systems with random initial data by considering their Lax representation. Specifically, we investigate the spectral behaviour of the corresponding Lax matrices when the number $N$ of degrees of freedom of the system goes to infinity and the initial data is sampled according to a properly chosen Gibbs measure. We give an exact description of the limit density of states for the exponential Toda lattice and the Volterra lattice in terms of the Laguerre and antisymmetric Gaussian $\beta$-ensemble in the high temperature regime. For generalizations of the Volterra lattice to short range interactions, called INB additive and multiplicative lattices, the focusing Ablowitz--Ladik lattice and the focusing Schur flow, we derive numerically the density of states. For all these systems, we obtain explicitly the density of states in the ground states., Comment: 35 pages, 8 figures, 1 table

We study complex integrable systems on quiver varieties associated with the cyclic quiver, and prove their superintegrability by explicitly constructing first integrals. We interpret them as rational Calogero-Moser systems endowed with internal degrees of freedom called spins. They encompass the usual systems in type $A_{n-1}$ and $B_n$, as well as generalisations introduced by Chalykh and Silantyev in connection with the multicomponent KP hierarchy. We also prove that superintegrability is preserved when a harmonic oscillator potential is added., v2: 15 pages, accepted in Nonlinearity

In this article we consider two particular examples of general construction proposed in arXiv:2012.15529. We consider the integrable extensions of the classical elliptic Calogero-Moser model of N particles with spin and the integrable Euler-Arnold top related to the group SL(N,C). The extended systems has additional N-1 degrees of freedom and can be described in terms of the Darboux variables., Comment: 14 pages. Contribution in the TMP volume dedicated to the ninetieth anniversary of M.K. Polivanov's birth

Dynamics of the Fermi-Pasta-Ulam (FPU) system on a two-dimensional square lattice is considered in the limit of small-amplitude long-scale waves with slow transverse modulations. In the absence of transverse modulations, dynamics of such waves, even at an oblique angle with respect to the square lattice, is known to be described by the Korteweg-de Vries (KdV) equation. For the three basic directions (horizontal, vertical, and diagonal), we prove that the modulated waves are well described by the Kadomtsev-Petviashvili (KP-II) equation. The result was expected long ago but proving rigorous bounds on the approximation error turns out to be complicated due to the nonlocal terms of the KP-II equation and the vector structure of the FPU systems on two-dimensional lattices. We have obtained these error bounds by extending the local well-posedness result for the KP-II equation in Sobolev spaces and by controlling the error terms with energy estimates. The bounds are useful in the analysis of transverse stability of solitary and periodic waves in two-dimensional FPU systems due to many results available for the KP-II equation., 30 pages, 2 figures

We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Korteweg-de Vries (KdV) equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons' statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport., Comment: 48 pages, 7 figures

We propose a novel jet substructure observable of boosted tops that is related to the linear polarization of the $W$ boson in boosted top quark decay, which results in a $\cos2\phi$ angular correlation between the $t\to bW$ and $W\to f\bar{f'}$ decay planes. We discuss in detail the origin of such linear polarization by applying Wigner's little group transformation. We show that the unique $\cos2\phi$ angular correlation only exists in the boosted regime but not in the top quark rest frame. We construct an experimental observable for such correlation based on the transverse energy deposition asymmetry in the top jet that does not require the reconstruction of $W$ decay products. The degree of this asymmetry can be used to measure the longitudinal polarization of the top quark, which is an important probe of new physics that couples to the top sector, and can discriminate a boosted top quark jet from its background events, such as QCD jets. A numerical simulation is also performed and found to agree well with the analytic prediction of the Standard Model., Comment: 6 pages, 3 figures. Matches the journal accepted version in PRL

We propose realizations of the Poisson structures for the Lax representations of three integrable $n$-body peakon equations, Camassa--Holm, Degasperis--Procesi and Novikov. The Poisson structures derived from the integrability structures of the continuous equations yield quadratic forms for the $r$-matrix representation, with the Toda molecule classical $r$-matrix playing a prominent role. We look for a linear form for the $r$-matrix representation. Aside from the Camassa--Holm case, where the structure is already known, the two other cases do not allow such a presentation, with the noticeable exception of the Novikov model at $n=2$. Generalized Hamiltonians obtained from the canonical Sklyanin trace formula for quadratic structures are derived in the three cases., Comment: 19 pages

We construct new large classes of BPS Wilson hyperloops in three-dimensional ${\cal N}=4$ quiver Chern-Simons-matter theory on $S^3$. The main strategy is to start with the 1/2 BPS Wilson loop of this theory, choose any linear combination of the supercharges it preserves, and look for deformations built out of the matter fields that still preserve that supercharge. This is a powerful generalization of a recently developed approach based on deformations of 1/4 and 1/8 BPS bosonic loops, which itself was far more effective at discovering new operators than older methods relying on complicated ansatze. We discover many new moduli spaces of BPS hyperloops preserving varied numbers of supersymmetries and varied subsets of the symmetries of the 1/2 BPS operator. In particular, we find new bosonic operators preserving 2 or 3 supercharges as well as new families of loops that do not share supercharges with any bosonic loops, including subclasses of both 1/8 and 1/4 BPS loops that are conformal., 42 pages; v2: 43 pages, expanded Sec. 6.3.3, other minor changes

We revisit the symmetry structure of integrable PDEs, looking at the specific example of the KdV equation. We identify 4 nonlocal symmetries of KdV depending on a parameter, which we call generating symmetries. We explain that since these are nonlocal symmetries, their commutator algebra is not uniquely determined, and we present three possibilities for the algebra. In the first version, 3 of the 4 symmetries commute; this shows that it is possible to add further (nonlocal) commuting flows to the standard KdV hierarchy. The second version of the commutator algebra is consistent with Laurent expansions of the symmetries, giving rise to an infinite dimensional algebra of hidden symmetries of KdV. The third version is consistent with asymptotic expansions for large values of the parameter, giving rise to the standard commuting symmetries of KdV, the infinite hierarchy of ``additional symmetries'', and their traditionally accepted commutator algebra (though this also suffers from some ambiguity as the additional symmetries are nonlocal). We explain how the 3 symmetries that commute in the first version of the algebra can all be regarded as infinitesimal double B\"acklund transformations. The 4 generating symmetries incorporate all known symmetries of the KdV equation, but also exhibit some remarkable novel structure, arising from their nonlocality. We believe this structure to be shared by other integrable PDEs., Comment: 22 pages

We develop an analytical description of static ferrodark solitons, the $\mathbb{Z}_2$ topological defects in the magnetic order, in the easy-plane phase of ferromagnetic spin-1 Bose-Einstein condensates. We find that the type-I ferrodark soliton has a single width while the type-II ferrodark soliton exhibits two characteristic length scales. The proposed ansatzes show excellent agreement with numerical results. Spin-singlet amplitudes, nematic tensor densities and nematic currents of ferrodark solitons are also discussed. The $\mathbb{Z}_2$ topological defects in the mass superfluid order, dark-dark-dark vector solitons, are obtained exactly in the parameter regime where exact ferrodark solitons exist. The dark-dark-dark vector soliton has higher excitation energy than ferrodark solitons., 8 pages, 10 figures

This article surveys the Weierstrass representation of surfaces in the three- and four-dimensional spaces, with an emphasis on its relation to the Willmore functional. We also describe an application of this representation to constructing a new type of solutions to the Davey-Stewartson II equation. They have regular initial data, gain one-point singularities at certain moments of time, and extend to smooth solutions for the remaining times., 18 pages

A family of deformed models of the sine-Gordon-type can be generated by twisting the sine-Gordon model. As a particular case, the 3-sine-Gordon model is here addressed, whose differential configurational entropy and the differential configurational complexity of three topological sectors are discussed, using two complementary approaches. Stability aspects are also discussed., 30 pages, 23 figs

Solitons are self-reinforcing localized wave packets arising from a balance of linear and nonlinear effects. This definition encompasses the interplay of nonlinear gain and loss, leading to the concept of dissipative solitons that has been instrumental in understanding the wide variety of mode locking phenomena in ultrafast optics. To date, most studies have involved the group velocity dispersion as a key ingredient for soliton generation. Here, we report on a novel kind of soliton, both theoretically and experimentally, which builds up in spectrally confined cavities when dispersion is practically absent. Precisely, the interplay between the Kerr nonlinearity and spectral filtering results in an infinite hierarchy of eigenfunctions which, combined with optical gain, allow for the generation of stable dispersion-less dissipative solitons in a previously uncharted regime. When the filter order tends to be infinite, we find an unexpected link between dissipative and conservative solitons, in the form of Nyquist-pulse-like solitons endowed with an ultra-flat spectrum. In contrast to the dispersion-enabled solitons, these dispersion-less Nyquist solitons build on a fully confined spectrum and their energy scaling is not constrained by the pulse duration. This study broadens the fundamental scope of dissipative soliton physics and opens new avenues for engineering optical solitons endowed with superior temporal and spectral features., Comment: 25 pages, 16 figures

Models describing long wave–short wave resonant interactions have many physical applications, from fluid dynamics to plasma physics. We consider here the Yajima–Oikawa–Newell (YON) model, which was recently introduced, combining the interaction terms of two long wave–short wave, integrable models, one proposed by Yajima–Oikawa, and the other one by Newell. The new YON model contains two arbitrary coupling constants and it is still integrable—in the sense of possessing a Lax pair—for any values of these coupling constants. It reduces to the Yajima–Oikawa or the Newell systems for special choices of these two parameters. We construct families of periodic and solitary wave solutions, which display the generation of very long waves. We also compute the explicit expression of a number of conservation laws.

A series of new soliton solutions are presented for the inhomogeneous variable coefficient Hirota equation by using the Riemann Hilbert method and transformation relationship. First, through a standard dressing procedure, the N-soliton matrix associated with the simple zeros in the Riemann Hilbert problem for the Hirota equation is constructed. Then the N-soliton matrix of the inhomogeneous variable coefficient Hirota equation can be obtained by a special transformation relationship from the N-soliton matrix of the Hirota equation. Next, using the generalized Darboux transformation, the high-order soliton solutions corresponding to the elementary high-order zeros in the Riemann Hilbert problem for the Hirota equation can be derived. Similarly, employing the transformation relationship mentioned above can lead to the high-order soliton solutions of the inhomogeneous variable coefficient Hirota equation. In addition, the collision dynamics of Hirota and inhomogeneous variable coefficient Hirota equations are analyzed; the asymptotic behaviors for multi-solitons and long-term asymptotic estimates for the high-order one-soliton of the Hirota equation are concretely calculated. Most notably, by analyzing the dynamics of the multi-solitons and high-order solitons of the inhomogeneous variable coefficient Hirota equation, we discover numerous new waveforms such as heart-shaped periodic wave solutions, O-shaped periodic wave solutions etc. that have never been reported before, which are crucial in theory and practice., Comment: arXiv admin note: text overlap with arXiv:2010.09417

The Lagrange formalism is developed for Bateman oscillators, which include both damped and amplified systems, and a novel method to derive the Caldirola-Kanai and null Lagrangians is presented. For the null Lagrangians, corresponding gauge functions are obtained. It is shown that the gauge functions can be used to convert the undriven Bateman oscillators into the driven ones. Applications of the obtained results to quantizatation of the Bateman oscillators are briefly discussed., Comment: 10 pages

The interaction of an oblique line soliton with a one-dimensional dynamic mean flow is analyzed using the Kadomtsev-Petviashvili II (KPII) equation. Building upon previous studies that examined the transmission or trapping of a soliton by a slowly varying rarefaction or oscillatory dispersive shock wave in one space and one time dimension, this paper allows for the incident soliton to approach the changing mean flow at a nonzero oblique angle. By deriving invariant quantities of the soliton-mean flow modulation equations$-$a system of three (1+1)-dimensional quasilinear, hyperbolic equations for the soliton and mean flow parameters$-$and positing the initial configuration as a Riemann problem in the modulation variables, it is possible to derive quantitative predictions regarding the evolution of the line soliton within the mean flow. It is found that the interaction between an oblique soliton and a changing mean flow leads to several novel features not observed in the (1+1)-dimensional reduced problem. Many of these interesting dynamics arise from the unique structure of the modulation equations that are nonstrictly hyperbolic, including a well-defined multivalued solution interpreted as a solution of the (2+1)-dimensional soliton-mean modulation equations, in which the soliton interacts with the mean flow and then wraps around to interact with it again. Finally, it is shown that the oblique interactions between solitons and dispersive shock wave solutions for the mean flow give rise to all three possible types of 2-soliton solutions of the KPII equation. The analytical findings are quantitatively supported by direct numerical simulations., 27 pages, 19 figures