arXiv subjects

Daisuke A. Takahashi

Publications and source records attributed to Daisuke A. Takahashi.

At least 19 recordsLinked to original sources

arXiv2026

Tsunami Solitons Emerging from Superconducting Gap

We propose a classical integrable system exhibiting tsunami-like solitons with a rocky-desert-like disordered stationary background. One of the Lax operators describing this system is interpretable as a Bogoliubov--de Gennes Hamiltonian in parity-mixed superconductors. The family of integrable equations is generated from this seed operator using Krichever's method, whose pure $s$-wave limit includes the coupled Schrödinger--Boussinesq hierarchy applied to plasma physics. A linearly unstable finite background with a superconducting gap supports the tsunami-soliton solution, where the propagation of the step structure turns back at a certain moment, accompanied with the oscillation on the opposite side. In addition, the equation allows inhomogeneous stationary solutions with an arbitrary number of bumps at arbitrary positions, which we term \textit{the Korteweg--de Vries (KdV) rocks}. In the Zakharov--Shabat scheme, the tsunami solitons are created from the Bogoliubov quasiparticles in the energy gap and the KdV rocks from normal electrons/holes. The unexpected large space of stationary solutions originates from the non-coprime Lax pair and the multivalued Baker--Akhiezer functions on the Riemann surface, formulated in terms of higher-rank holomorphic bundles by Krichever and Novikov. Furthermore, the concept of \textit{isodispersive phases} is introduced to characterize quasiperiodic multi-tsunami backgrounds and consider their classification.

nlin.PS↗

arXiv2025

Electric potentials and field lines for uniformly-charged tube and cylinder expressed by Appell's hypergeometric function and integration of $Z(u|m) \mathrm{sc}(u|m)$

The closed-form expressions of electric potentials and field lines for a uniformly-charged tube and cylinder are presented using elliptic integrals and Appell's hypergeometric functions, where field lines are depicted by introducing the concept of the field line potential in axisymmetric systems, whose contour lines represent electric field lines outside the charged region, thought of as an analog of the conjugate harmonic function in the presence of non-uniform metric. The field line potential for the tube shows a multi-valued behavior and enables us to define a topological charge. The integral of $Z(u|m)\operatorname{sc}(u|m)$, where $ Z $ and $ \operatorname{sc} $ are the Jacobi zeta and elliptic functions, is also expressed by Appell's hypergeometric function as a by-product, which was missing in classical tables of formulas. In the Addendum appended after the main article, several relevant references are provided and the decomposition of the solution by ``degrees of transcendence'' is proposed. The attached supplementary calculations provide detailed derivations of several formulas including the integral in the title and discuss the resemblance between the field line potential for tube and the electric potential for disk.

math-ph↗

arXiv2024

Basis of spinors expressed by differential forms and calculating its norm

The basis of spinors in three-dimensional Euclidean space is expressed by differential forms. Its expression is found from the spectral decomposition of the modified Hamiltonian describing Weyl semimetals where the wavenumber parameters are replaced by the differential forms. The generalization of the definitions of differential forms including not only fractional powers but also any algebraic functions is justified by algebraic extension of the symmetric Fock space isomorphic to multivariable polynomial ring. We furthermore define the inner product for these fractional-order differential forms written in two ways; the formal power series and the multiple integral. While the norms of the powers $\mathrm{d}s^{2ν}$ are reduced to the variant of Dyson's integral and have the value $2ν+1$, possibly related to the dimension of irreducible representation, the norm of two-component spinor is given by the integral \[I=\frac{2}π\int_{x^2+y^2\ge 1 \& 0\le x \le 1 \& 0 \le y \le 1}\frac{xy\mathrm{d}x\mathrm{d}y}{\sqrt{(1-x)(1-y)(x^2+y^2-1)}}=\frac{4\sqrt{2}}π\int_0^1\frac{(1+\sqrt{z})}{(1+z)^3}\big[2E(z)-(1-z)K(z)\big]\mathrm{d}z\simeq 1.774, \] where $K(z)$ and $E(z)$ are the complete elliptic integrals of the first and second kind, respectively. The product can also be generalized to include one parameter $p$ analogous to those in $L^p$ space, reducing to the original one when $p=2$. The norm with $p=\infty$ is considered as an example and the result for $ ν=-\frac{1}{2} $ is written by the Watson-Iwata integral. We also discuss the ambiguity of the definition of the spinor under coordinate rotation originating from Berry's phase, and point out that if we heuristically set $\mathrm{d}s=0$, the ambiguity disappears, though its implication remains unclear.

physics.gen-ph↗

arXiv2023

Multi-soliton solutions of the sine-Gordon equation with elliptic-function background

The multi-soliton solution of the sine-Gordon equation in the presence of elliptic-function background is derived by the inverse scattering method. The key tool in our formulation is the Lax pair written by $4\times4$ matrix differential operators given by Takhtadzhyan and Faddeev in 1974, which enables us to use the conventional form of the integral representation of the Jost solutions and Krichever's theory of commuting differential operators. As a by-product we also provide generalized orthogonality and completeness relations for eigenfunctions associated with indefinite inner product. The multi-soliton solution is expressed by a determinant of theta functions and the shift of the background lattice due to solitons is also determined using addition formula. One kink and one breather solutions are presented by animated gifs.

nlin.SI↗

arXiv2018

Multi-breathers and high order rogue waves for the nonlinear Schrödinger equation on the elliptic function background

We construct the multi-breather solutions of the focusing nonlinear Schrödinger equation (NLSE) on the background of elliptic functions by the Darboux transformation, and express them in terms of the determinant of theta functions. The dynamics of the breathers in the presence of various kinds of backgrounds such as dn, cn, and non-trivial phase-modulating elliptic solutions are presented, and their behaviors dependent on the effect of backgrounds are elucidated. We also determine the asymptotic behaviors for the multi-breather solutions with different velocities in the limit $t\to\pm\infty$, where the solution in the neighborhood of each breather tends to the simple one-breather solution. Furthermore, we exactly solve the linearized NLSE using the squared eigenfunction and determine the unstable spectra for elliptic function background. By using them, the Akhmediev breathers arising from these modulational instabilities are plotted and their dynamics are revealed. Finally, we provide the rogue-wave and higher-order rogue-wave solutions by taking the special limit of the breather solutions at branch points and the generalized Darboux transformation. The resulting dynamics of the rogue waves involves rich phenomena: depending on the choice of the background and possessing different velocities relative to the background. We also provide an example of the multi- and higher-order rogue wave solution.

nlin.SI↗

arXiv2018

Newton's equation of motion with quadratic drag force and Toda's potential as a solvable one

The family of exactly solvable potentials for Newton's equation of motion in the one-dimensional system with quadratic drag force has been determined completely. The determination is based on the implicit inverse-function solution valid for any potential shape, and hence exhaustive. This solvable family includes the exponential potential appearing in the Toda lattice as a special limit. The global solution is constructed by matching the solutions applicable for positive and negative velocity, yielding the piecewise analytic function with a cusp in the third-order derivative, i.e., the jerk. These procedures and features can be regarded as a generalization of Gorder's construction [Phys. Scr. 2015, {\bf 90}, 085208] to the energy-dissipating damped oscillators. We also derive the asymptotic formulae by solving the matching equation, and prove that the damping of the oscillation amplitude is proportional to $ t^{-1} $.

nlin.SI↗

arXiv2017

Two-dimensional Schrödinger symmetry and three-dimensional breathers and Kelvin-ripple complexes as quasi-massive-Nambu-Goldstone modes

Bose-Einstein condensates (BECs) confined in a two-dimensional (2D) harmonic trap are known to possess a hidden 2D Schrödinger symmetry, that is, the Schrödinger symmetry modified by a trapping potential. Spontaneous breaking of this symmetry gives rise to a breathing motion of the BEC, whose oscillation frequency is robustly determined by the strength of the harmonic trap. In this paper, we demonstrate that the concept of the 2D Schrödinger symmetry can be applied to predict the nature of three dimensional (3D) collective modes propagating along a condensate confined in an elongated trap. We find three kinds of collective modes whose existence is robustly ensured by the Schrödinger symmetry, which are physically interpreted as one breather mode and two Kelvin-ripple complex modes, i.e., composite modes in which the vortex core and the condensate surface oscillate interactively. We provide analytical expressions for the dispersion relations (energy-momentum relation) of these modes using the Bogoliubov theory [D. A. Takahashi and M. Nitta, Ann. Phys. 354, 101 (2015)]. Furthermore, we point out that these modes can be interpreted as "quasi-massive-Nambu-Goldstone (NG) modes", that is, they have the properties of both quasi-NG and massive NG modes: quasi-NG modes appear when a symmetry of a part of a Lagrangian, which is not a symmetry of full a Lagrangian, is spontaneously broken, while massive NG modes appear when a modified symmetry is spontaneously broken.

cond-mat.quant-gas↗

arXiv2017

Integrable model for density-modulated quantum condensates: solitons passing through a soliton lattice

An integrable model possessing inhomogeneous ground states is proposed as an effective model of non-uniform quantum condensates such as supersolids and Fulde--Ferrell--Larkin--Ovchinnikov superfluids. The model is a higher-order analog of the nonlinear Schrödinger equation. We derive an $n$-soliton solution via the inverse scattering theory with elliptic-functional background, and reveal various kinds of soliton dynamics such as dark soliton billiards, dislocations, gray solitons, and envelope solitons. We also provide the exact bosonic and fermionic quasiparticle eigenstates and clarify their tunneling phenomena. The solutions are expressed by a determinant of theta functions.

cond-mat.quant-gas↗

arXiv2016

Exhaustive derivation of static self-consistent multi-soliton solutions in the matrix Bogoliubov-de Gennes systems

The matrix-generalized Bogoliubov-de Gennes systems have recently been considered by the present author [arXiv:1509.04242, Phys. Rev. B 93, 024512 (2016)], and time-dependent and self-consistent multi-soliton solutions have been constructed based on the ansatz method. In this paper, restricting the problem to the static case, we exhaustively determine the self-consistent solutions using the inverse scattering theory. Solving the gap equation, we rigorously prove that the self-consistent potential must be reflectionless. As a supplementary topic, we elucidate the relation between the stationary self-consistent potentials and the soliton solutions in the matrix nonlinear Schrödinger equation. Asymptotic formulae of multi-soliton solutions for sufficiently isolated solitons are also presented.

math-ph↗

arXiv2016

Bogoliubov-de Gennes Soliton Dynamics in Unconventional Fermi Superfluids

Exact self-consistent soliton dynamics based on the Bogoliubov-de Gennes (BdG) formalism in unconventional Fermi superfluids/superconductors possessing an $SU(d)$-symmetric two-body interaction is presented. The derivation is based on the ansatz having the similar form as the Gelfand-Levitan-Marchenko equation in the inverse scattering theory. Our solutions can be regarded as a multicomponent generalization of the solutions recently derived by Dunne and Thies [Phys. Rev. Lett. 111, 121602 (2013)]. We also propose superpositions of occupation states, which make it possible to realize various filling rates even in one-flavor systems, and include Dirac and Majorana fermions. The soliton solutions in the $ d=2 $ systems, which describe the mixture of singlet $s$-wave and triplet $p$-wave superfluids, exhibit a variety of phenomena such as SU(2)-DHN breathers, Majorana triplet states, $s$-$p$ mixed dynamics, and so on. These solutions are illustrated by animations, where order parameters are visualized by spherical harmonic functions. The full formulation of the BdG theory is also supported, and the double-counting problem of BdG eigenstates and $N$-flavor generalization are discussed.

cond-mat.supr-con↗

arXiv2015

Nambu-Goldstone Modes Propagating along Topological Defects: Kelvin and Ripple Modes from Small to Large Systems

Nambu-Goldstone modes associated with (topological) defects such as vortices and domain walls in (super)fluids are known to possess quadratic/non-integer dispersion relations in finite/infinite-size systems. Here, we report interpolating formulas connecting the dispersion relations in finite- and infinite-size systems for Kelvin modes along a quantum vortex and ripplons on a domain wall in superfluids. Our method can provide not only the dispersion relations but also the explicit forms of quasiparticle wavefunctions $(u,v)$. We find a complete agreement between the analytical formulas and numerical simulations. All these formulas are derived in a fully analytical way, and hence not empirical ones. We also discuss common structures in the derivation of these formulas and speculate on the general procedure.

cond-mat.other↗

arXiv2015

Quasi-Nambu-Goldstone modes in nonrelativistic systems

When a continuous symmetry is spontaneously broken in nonrelativistic systems, there appear either type-I or type-II Nambu-Goldstone modes (NGMs) with linear or quadratic dispersion relation, respectively. When equation of motion or the potential term has an enhanced symmetry larger than that of Lagrangian or Hamiltonian, there can appear quasi-NGMs if it is spontaneously broken. We construct a theory to count the numbers of type-I and type-II quasi-NGMs and NGMs, when the potential term has a symmetry of a non-compact group. We show that the counting rule based on the Watanabe-Brauner matrix is valid only in the absence of quasi-NGMs because of non-hermitian generators, while that based on the Gram matrix [DT & MN, arXiv:1404.7696, Ann. Phys. 354, 101 (2015)] is still valid in the presence of quasi-NGMs. We show that there exist two types of type-II gapless modes, a genuine NGM generated by two conventional zero modes (ZMs) originated from the Lagrangian symmetry, and quasi-NGM generated by a coupling of one conventional ZMand one quasi-ZM, which is originated from the enhanced symmetry, or two quasi-ZMs. We find that, depending on the moduli, some NGMs can change to quasi-NGMs and vice versa with preserving the total number of gapless modes. The dispersion relations are systematically calculated by a perturbation theory. The general result is illustrated by the complex linear $O(N)$ model, containing the two types of type-II gapless modes and exhibiting the change between NGMs and quasi-NGMs.

hep-th↗

arXiv2015

An inequality for spinor Bose-Einstein condensates

An inequality for spin-$F$ Bose-Einstein condensates (BECs) $ F^2(ρ^2-|Θ|^2)-\boldsymbol{M}^2\ge0 $ is reported, where $ρ$, $Θ$, and $\boldsymbol{M}$ represent the density, singlet pair amplitude, and magnetization vector, respectively. The distribution of high-symmetry spinors in the allowed region by the inequality is elucidated with using the Majorana representation. The result is illustrated by the example of spin-2 BECs.

cond-mat.quant-gas↗

arXiv2015

Counting rule of Nambu-Goldstone modes for internal and spacetime symmetries: Bogoliubov theory approach

When continuous symmetry is spontaneously broken, there appear Nambu-Goldstone modes (NGMs) with linear and quadratic dispersion relations, which are called type-I and type-II, respectively. We propose a framework to count these modes including the coefficients of the dispersion relations with applying the standard Gross-Pitaevskii-Bogoliubov theory. Our method is mainly based on (i) zero-mode solutions of the Bogoliubov equation originated from spontaneous symmetry breaking and (ii) their generalized orthogonal relations, which naturally arises from well-known Bogoliubov transformations and is referred to as "$σ$-orthogonality" in this paper. Unlike previous works, our framework is applicable without any modification to the cases where there are additional zero modes, which do not have a symmetry origin, such as quasi-NGMs, and/or where spacetime symmetry is spontaneously broken in the presence of a topological soliton or a vortex. As a by-product of the formulation, we also give a compact summary for mathematics of bosonic Bogoliubov equations and Bogoliubov transformations, which becomes a foundation for any problem of Bogoliubov quasiparticles. The general results are illustrated by various examples in spinor Bose-Einstein condensates (BECs). In particular, the result on the spin-3 BECs includes new findings such as a type-I-type-II transition and an increase of the type-II dispersion coefficient caused by a linearly-independent pair of zero modes.

cond-mat.quant-gas↗

arXiv2013

On reflectionless nature of self-consistent multi-soliton solutions in Bogoliubov-de Gennes and chiral Gross-Neveu models

Recently the most general static self-consistent multi-soliton solutions in Bogoliubov-de Gennes and chiral Gross-Neveu systems are derived by the present authors [D. A. Takahashi and M. Nitta, Phys. Rev. Lett. 110, 131601 (2013)]. Here we show a few complementary results, which were absent in the previous our work. We prove directly from the gap equation that the self-consistent solutions need to have reflectionless potentials. We also give the self-consistent condition for the system consisting of only right-movers, which is more used in high-energy physics.

cond-mat.supr-con↗

arXiv2013

Self-consistent multiple complex-kink solutions in Bogoliubov-de Gennes and chiral Gross-Neveu systems

We exhaust all exact self-consistent solutions of complex-valued fermionic condensates in the 1+1 dimensional Bogoliubov-de Gennes and chiral Gross-Neveu systems under uniform boundary conditions. We obtain $n$ complex (twisted) kinks, or grey solitons, with $2n$ parameters corresponding to their positions and phase shifts. Each soliton can be placed at an arbitrary position while the self-consistency requires its phase shift to be quantized by $π/N$ for $N$ flavors.

cond-mat.supr-con↗

arXiv2012

Fermionic solutions of chiral Gross-Neveu and Bogoliubov-de Gennes systems in nonlinear Schrödinger hierarchy

The chiral Gross-Neveu model or equivalently the linearized Bogoliubov-de Gennes equation has been mapped to the nonlinear Schrödinger (NLS) hierarchy in the Ablowitz-Kaup-Newell-Segur formalism by Correa, Dunne and Plyushchay. We derive the general expression for exact fermionic solutions for all gap functions in the arbitrary order of the NLS hierarchy. We also find that the energy spectrum of the n-th NLS hierarchy generally has n+1 gaps. As an illustration, we present the self-consistent two-complex-kink solution with four real parameters and two fermion bound states. The two kinks can be placed at any position and have phase shifts. When the two kinks are well separated, the fermion bound states are localized around each kink in most parameter region. When two kinks with phase shifts close to each other are placed at distance as short as possible, the both fermion bound states have two peaks at the two kinks, i.e., the delocalization of the bound states occurs.

cond-mat.supr-con↗

arXiv2012

Soliton-phonon scattering problem in 1D nonlinear Schrödinger systems with general nonlinearity

A scattering problem (or more precisely, a transmission-reflection problem) of linearized excitations in the presence of a dark soliton is considered in a one-dimensional nonlinear Schrödinger system with a general nonlinearity: $ \mathrm{i}\partial_t ϕ= -\partial_x^2 ϕ+ F(|ϕ|^2)ϕ$. If the system is interpreted as a Bose-Einstein condensate, the linearized excitation is a Bogoliubov phonon, and the linearized equation is the Bogoliubov equation. We exactly prove that the perfect transmission of the zero-energy phonon is suppressed at a critical state determined by Barashenkov's stability criterion [Phys. Rev. Lett. 77, (1996) 1193.], and near the critical state, the energy-dependence of the reflection coefficient shows a saddle-node type scaling law. The analytical results are well supported by numerical calculation for cubic-quintic nonlinearity. Our result gives an exact example of scaling laws of saddle-node bifurcation in time-reversible Hamiltonian systems. As a by-product of the proof, we also give all exact zero-energy solutions of the Bogoliubov equation and their finite energy extension.

cond-mat.quant-gas↗