arXiv subjects

K. Toda

Publications and source records attributed to K. Toda.

5 recordsLinked to original sources

arXiv2024

Inelastic Scattering, Emergent Interactions of Solitons in the Zakharov-Kuznetsov Equation through Conservative and non-Conservative Physics-Informed Neural Networks

The Zakharov-Kuznetsov equation, originally a three dimensional mathematical model of plasma with a uniform magnetic field, is a direct extension of the KdV equation into higher dimensions and is a typical quasi-integrable system. Physics-Informed Neural Networks (PINNs) are used to study the collision of soliton solutions in the 2+1 dimensional Zakharov-Kuznetsov equation. PINNs are able to successfully solve the equations in the forward process, and the solutions are obtained using a mesh-free approach and automatic differentiation, taking into account conservation laws. In the inverse process, the proper form of the equation can be successfully derived from a given training data. However, the situation becomes intractable in the collision process. The forward analysis result no longer adheres to the laws of conservation, and is better described as a dynamically incompatible field configuration (DIFC) than a solution to the system. Conservative PINNs have thus been introduced for this purpose, and in this paper we succeed in obtaining solutions that satisfy conservation laws. The inverse analysis suggests a different equation in which the coefficients exhibit significant changes, implying an emergence of temporary interactions. With these modulated coefficients, we recalculate the equation and confirm that the adherence to the laws of conservation has unquestionably improved.

nlin.SI↗

arXiv2024

Discovery of Quasi-Integrable Equations from traveling-wave data using the Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for analyzing nonlinear partial differential equations and identifying governing equations from observational data. In this study, we apply PINNs to investigate vortex-type solutions of quasi-integrable equations in two spatial dimensions, specifically the Zakharov-Kuznetsov (ZK) and the Regularized Long-Wave (RLW) equations. These equations are toy models for geostrophic shallow water dynamics in planetary atmospheres. We first demonstrate that PINNs can successfully solve these equations in the forward process using a mesh-free approach with automatic differentiation. However, in the inverse process, substantial misidentification occurs due to the structural similarities between the ZK and the RLW equations. To address this issue, we then introduce conservation law-enhanced PINNs, initial condition variations, and a friction-based perturbation approach to improve identification accuracy. Our results show that incorporating small perturbations while preserving conservation laws significantly enhances the resolution of equation identification. These findings may contribute to the broader goal of using deep learning techniques for discovering governing equations in complex fluid dynamical systems, such as Jupiter's Great Red Spot.

physics.flu-dyn↗

arXiv2012

Some vortex solutions in the extended Skyrme-Faddeev model

Analytical and numerical vortex solutions for the extended Skyrme-Faddeev model in a (3+1) dimensional Minkowski space-time are investigated. The extension is obtained by adding to the Lagrangian a quartic term, which is the square of the kinetic term, and a potential which breaks the SO(3) symmetry down to SO(2). The construction of the solutions has been done in twofold: one makes use of an axially symmetric ansatz and solves the resulting ODE by an analytical and a numerical way. The analytical vortices are obtained for special form of the potentials, and the numerical ones are computed using the successive over relaxation method for wider choice of the potentials. Another is based on a simulational technique named the simulated annealing method which is available to treat the non-axisymmetric shape of solutions. The crucial thing for determining the structure of vortices is the type of the potential.

hep-th↗

arXiv1998

N-Soliton Solutions to a New (2 + 1) Dimensional Integrable Equation

We give explicitly N-soliton solutions of a new (2 + 1) dimensional equation, $ϕ_{xt} + ϕ_{xxxz}/4 + ϕ_x ϕ_{xz} + ϕ_{xx} ϕ_z/2 + \partial_x^{-1} ϕ_{zzz}/4 = 0$. This equation is obtained by unifying two directional generalization of the KdV equation, composing the closed ring with the KP equation and Bogoyavlenskii-Schiff equation. We also find the Miura transformation which yields the same ring in the corresponding modified equations.

solv-int↗

arXiv1998

N Soliton Solutions to The Bogoyavlenskii-Schiff Equation and A Quest for The Soliton Solution in (3 + 1) Dimensions

We study the integrable systems in higher dimensions which can be written not by the Hirota's bilinear form but by the trilinear form. We explicitly discuss about the Bogoyavlenskii-Schiff(BS) equation in (2 + 1) dimensions. Its analytical proof of multi soliton solution and a new feature are given. Being guided by the strong symmetry, we also propose a new equation in (3 + 1) dimensions.

solv-int↗