arXiv subjects

Kazuyuki Yagasaki

Publications and source records attributed to Kazuyuki Yagasaki.

32 records · Page 2Linked to original sources

arXiv2022

Nonintegrability of Forced Nonlinear Oscillators

In recent papers by the authors (S.~Motonaga and K.~Yagasaki, Obstructions to integrability of nearly integrable dynamical systems near regular level sets, submitted for publication, and K.~Yagasaki, Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits, submitted for publication), two different techniques which allow us to prove the real-analytic or complex-meromorphic nonintegrability of forced nonlinear oscillators having the form of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems were provided. Here the concept of nonintegrability in the Bogoyavlenskij sense is adopted and the first integrals and commutative vector fields are also required to depend real-analytically or complex-meromorphically on the small parameter. In this paper we review the theories and continue to demonstrate their usefulness. In particular, we consider the periodically forced damped pendulum and prove its nonintegrability in the above meaning.

math.DS↗

arXiv2021

Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits

In a recent paper by the author (K. Yagasaki, Nonintegrability of the restricted three-body problem, submitted for publication), a technique was developed for determining whether nearly integrable systems are not meromorphically Bogoyavlenskij-integrable such that the first integrals and commutative vector fields also depend meromorphically on the small parameter. Here we continue to demonstrate the technique for some classes of dynamical systems. In particular, we consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and discuss a relationship of the technique with the subharmonic Melnikov method, which enables us to detect the existence of periodic orbits and their stability. We illustrate the theory for the periodically forced Duffing oscillator and two more additional examples: second-order coupled oscillators and a two-dimensional system of pendulum-type subjected to a constant torque.

math.DS↗

arXiv2021

Bifurcations and Spectral Stability of Solitary Waves in Coupled Nonlinear Schrödinger Equations

We study bifurcations and spectral stability of solitary waves in coupled nonlinear Schrödinger equations (CNLS) on the line. We assume that the coupled equations possess a solution of which one component is identically zero, and call it a $\textit{fundamental solitary wave}$. By using a result of one of the authors and his collaborator, the bifurcations of the fundamental solitary wave are detected. We utilize the Hamiltonian-Krein index theory and Evans function technique to determine the spectral or orbital stability of the bifurcated solitary waves as well as as that of the fundamental one under some nondegenerate conditions which are easy to verify, compared with those of the previous results. We apply our theory to CNLS with a cubic nonlinearity and give numerical evidences for the theoretical results.

math.AP↗

arXiv2021

Obstructions to Integrability of Nearly Integrable Dynamical Systems near Regular Level Sets

We study the existence of real-analytic first integrals and real-analytic integrability for perturbations of integrable systems in the sense of Bogoyavlenskij including non-Hamiltonian ones. We especially assume that there exists a family of periodic orbits on a regular level set of the first integrals having a connected and compact component and give sufficient conditions for nonexistence of the same number of real-analytic first integrals in the perturbed systems as the unperturbed ones and for their real-analytic nonintegrability near the level set such that the first integrals and commutative vector fields depend analytically on the small parameter. We compare our results with classical results of Poincaré and Kozlov for systems written in action and angle coordinates and discuss their relationships with the subharmonic and homoclinic Melnikov methods for periodic perturbations of single-degree-of-freeedom Hamiltonian systems. We illustrate our theory for three examples containing the periodically forced Duffing oscillator.

math.DS↗

arXiv2021

Persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems

We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter $\varepsilon>0$ and give several necessary conditions for their persistence. Here we treat homoclinic orbits not only to equilibria but also to periodic orbits. We also discuss some relationships of these results with the standard subharmonic and homoclinic Melnikov methods for time-periodic perturbations of single-degree-of-freedom Hamiltonian systems, and with another version of the homoclinic Melnikov method for autonomous perturbations of multi-degree-of-freedom Hamiltonian systems. In particular, we show that a first integral which converges to the Hamiltonian or another first integral as the perturbation tends to zero does not exist near the unperturbed periodic or homoclinic orbits in the perturbed systems if the subharmonic or homoclinic Melnikov functions are not identically zero on connected open sets. We illustrate our theory for four examples: The periodically forced Duffing oscillator, two identical pendula coupled with a harmonic oscillator, a periodically forced rigid body and a three-mode truncation of a buckled beam.

math.DS↗

arXiv2021

Integrability of the Zakharov-Shabat systems by quadrature

We study the integrability of the general two-dimensional Zakharov-Shabat systems, which appear in application of the inverse scattering transform (IST) to an important class of nonlinear partial differential equations (PDEs) called integrable systems, in the meaning of differential Galois theory, i.e., their solvability by quadrature. It becomes a key for obtaining analytical solutions to the PDEs by using the IST. For a wide class of potentials, we prove that they are integrable in that meaning if and only if the potentials are reflectionless. It is well known that for such potentials particular solutions called n-solitons in the original PDEs are yielded by the IST.

nlin.SI↗

arXiv2021

Analytic and algebraic conditions for bifurcations of homoclinic orbits II: Reversible systems

Following Part~I, we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. First, we modify and extend arguments of Part~I to show in a form applicable to general systems discussed there that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory. We next extend the Melnikov method of Part~I to reversible systems and obtain theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones.

math.DS↗

arXiv2021

Numerical Computations for Bifurcations and Spectral Stability of Solitary Waves in Coupled Nonlinear Schrödinger Equations

We numerically study solitary waves in the coupled nonlinear Schrödinger equations. We detect pitchfork bifurcations of the fundamental solitary wave and compute eigenvalues and eigenfunctions of the corresponding eigenvalue problems to determine the spectral stability of solitary waves born at the pitchfork bifurcations. Our numerical results demonstrate the theoretical ones which the authors obtained recently. We also compute generalized eigenfunctions associated with the zero eigenvalue for the bifurcated solitary wave exhibiting a saddle-node bifurcation, and show that it does not change its stability type at the saddle-node bifurcation point.

math.NA↗

arXiv2020

Nonintegrability of the unfoldings of codimension-two bifurcations

Codimension-two bifurcations are fundamental and interesting phenomena in dynamical systems. Fold-Hopf and double-Hopf bifurcations are the most important among them. We study the unfoldings of these two codimension-two bifurcations, and obtain sufficient conditions for their nonintegrability in the meaning of Bogoyavlenskij. We reduce the problems of the unfoldings to those of planar polynomial vector fields and analyze the nonintegrability of the planar vector fields, based on Ayoul and Zung's version of the Morales-Ramis theory. New useful criteria for nonintegrability of planar polynomial vector fields are also given. The approaches used here are applicable to many problems including circular symmetric systems.

math.DS↗

arXiv2019

Heteroclinic Orbits and Nonintegrability in Two-Degree-of-Freedom Hamiltonian Systems with Saddle-Centers

We consider a class of two-degree-of-freedom Hamiltonian systems with saddle-centers connected by heteroclinic orbits and discuss some relationships between the existence of transverse heteroclinic orbits and nonintegrability. By the Lyapunov center theorem there is a family of periodic orbits near each of the saddle-centers, and the Hessian matrices of the Hamiltonian at the two saddle-centers are assumed to have the same number of positive eigenvalues. We show that if the associated Jacobian matrices have the same pair of purely imaginary eigenvalues, then the stable and unstable manifolds of the periodic orbits intersect transversely on the same Hamiltonian energy surface when sufficient conditions obtained in previous work for real-meromorphic nonintegrability of the Hamiltonian systems hold; if not, then these manifolds intersect transversely on the same energy surface, have quadratic tangencies or do not intersect whether the sufficient conditions hold or not. Our theory is illustrated for a system with quartic single-well potential and some numerical results are given to support the theoretical results.

math.DS↗

arXiv2012

Families of solitons in Bragg supergratings

We study fundamental optical gap solitons in the model of a fiber Bragg grating (BG), which is subjected to a periodic modulation of the local reflectivity, giving rise to a supergrating. In addition, the local refractive index is also periodically modulated with the same period. It is known that the supergrating opens an infinite system of new bandgaps in the BG's spectrum. We use a combination of analytical and computational methods to show that each emerging bandgap is filled with gap solitons (GSs), including asymmetric ones and bound states of the GSs. In particular, bifurcations of the GSs created by the supergrating are studied in terms of a geometric analysis.

nlin.PS↗

arXiv2010

Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but two parameters are needed in general systems. We apply a version of Melnikov's method due to Gruendler to obtain saddle-node and pitchfork types of bifurcation results for homoclinic orbits. Furthermore we prove that if these bifurcations occur, then the variational equations around the homoclinic orbits are integrable in the meaning of differential Galois theory under the assumption that the homoclinic orbits lie on analytic invariant manifolds. We illustrate our theories with an example which arises as stationary states of coupled real Ginzburg-Landau partial differential equations, and demonstrate the theoretical results by numerical ones.

math.DS↗

arXiv2010

Galoisian approach for a Sturm-Liouville problem on the infinite interval

We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval. Here the eigenfunctions are nonzero solutions exponentially decaying at infinity. We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions. Our result is illustrated with two examples for a stationary Schroedinger equation having a generalized Hulthen potential and an eigenvalue problem for a traveling front in the Allen-Cahn equation.

math.DS↗

arXiv2005

Discrete embedded solitons

We address the existence and properties of discrete embedded solitons (ESs), i.e., localized waves existing inside the phonon band in a nonlinear dynamical-lattice model. The model describes a one-dimensional array of optical waveguides with both the quadratic (second-harmonic generation) and cubic nonlinearities. A rich family of ESs was previously known in the continuum limit of the model. First, a simple motivating problem is considered, in which the cubic nonlinearity acts in a single waveguide. An explicit solution is constructed asymptotically in the large-wavenumber limit. The general problem is then shown to be equivalent to the existence of a homoclinic orbit in a four-dimensional reversible map. From properties of such maps, it is shown that (unlike ordinary gap solitons), discrete ESs have the same codimension as their continuum counterparts. A specific numerical method is developed to compute homoclinic solutions of the map, that are symmetric under a specific reversing transformation. Existence is then studied in the full parameter space of the problem. Numerical results agree with the asymptotic results in the appropriate limit and suggest that the discrete ESs may be semi-stable as in the continuous case.

nlin.PS↗