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Yasuhiro Ohta

Publications and source records attributed to Yasuhiro Ohta.

At least 37 records · Page 2Linked to original sources
arXiv2014

Self-adaptive moving mesh schemes for short pulse type equations and their Lax pairs

Integrable self-adaptive moving mesh schemes for short pulse type equations (the short pulse equation, the coupled short pulse equation, and the complex short pulse equation) are investigated. Two systematic methods, one is based on bilinear equations and another is based on Lax pairs, are shown. Self-adaptive moving mesh schemes consist of two semi-discrete equations in which the time is continuous and the space is discrete. In self-adaptive moving mesh schemes, one of two equations is an evolution equation of mesh intervals which is deeply related to a discrete analogue of a reciprocal (hodograph) transformation. An evolution equations of mesh intervals is a discrete analogue of a conservation law of an original equation, and a set of mesh intervals corresponds to a conserved density which play an important role in generation of adaptive moving mesh. Lax pairs of self-adaptive moving mesh schemes for short pulse type equations are obtained by discretization of Lax pairs of short pulse type equations, thus the existence of Lax pairs guarantees the integrability of self-adaptive moving mesh schemes for short pulse type equations. It is also shown that self-adaptive moving mesh schemes for short pulse type equations provide good numerical results by using standard time-marching methods such as the improved Euler's method.

nlin.SI↗
arXiv2014

Integrable discretizations of the Dym equation

Integrable discretizations of the complex and real Dym equations are proposed. N-soliton solutions for both semi-discrete and fully discrete analogues of the complex and real Dym equations are also presented.

nlin.SI↗
arXiv2014

General rogue waves in the focusing and defocusing Ablowitz-Ladik equations

General rogue waves in the focusing and defocusing Ablowitz-Ladik equations are derived by the bilinear method. In the focusing case, it is shown that rogue waves are always bounded. In addition, fundamental rogue waves reach peak amplitudes which are at least three times that of the constant background, and higher-order rogue waves can exhibit patterns such as triads and circular arrays with different individual peaks. In the defocusing case, it is shown that rogue waves also exist. In addition, these waves can blow up to infinity in finite time.

nlin.SI↗
arXiv2012

On the tau-functions of the Degasperis-Procesi equation

The DP equation is investigated from the point of view of determinant-pfaffian identities. The reciprocal link between the Degasperis-Procesi (DP) equation and the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system is used to construct the N-soliton solution of the DP equation. The N-soliton solution of the DP equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations, the identities between determinants and pfaffians, and the $τ$-functions of the DP equation are obtained from the pseudo 3-reduction of the $C_{\infty}$ two-dimensional Toda system.

nlin.SI↗
arXiv2012

Dynamics of rogue waves in the Davey-Stewartson II equation

General rogue waves in the Davey-Stewartson-II equation are derived by the bilinear method, and the solutions are given through determinants. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background in a line profile and then retreat back to the constant background again. It is also shown that multi-rogue waves describe the interaction between several fundamental rogue waves, and higher-order rogue waves exhibit different dynamics (such as rising from the constant background but not retreating back to it). A remarkable feature of these rogue waves is that under certain parameter conditions, these rogue waves can blow up to infinity in finite time at isolated spatial points, i.e., exploding rogue waves exist in the Davey-Stewartson-II equation.

nlin.SI↗
arXiv2012

On the $τ$-functions of the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system

The reciprocal link between the reduced Ostrovsky equation and the $A_2^{(2)}$ two-dimensional Toda system is used to construct the $N$-soliton solution of the reduced Ostrovsky equation. The $N$-soliton solution of the reduced Ostrovsky equation is presented in the form of pfaffian through a hodograph (reciprocal) transformation. The bilinear equations and the $τ$-function of the reduced Ostrovsky equation are obtained from the period 3-reduction of the $B_{\infty}$ or $C_{\infty}$ two-dimensional Toda system, i.e., the $A_2^{(2)}$ two-dimensional Toda system. One of $τ$-functions of the $A_2^{(2)}$ two-dimensional Toda system becomes the square of a pfaffian which also become a solution of the reduced Ostrovsky equation. There is another bilinear equation which is a member of the 3-reduced extended BKP hierarchy. Using this bilinear equation, we can also construct the same pfaffian solution.

nlin.SI↗
arXiv2012

Rogue waves in the Davey-Stewartson equation

General rogue waves in the Davey-Stewartson-I equation are derived by the bilinear method. It is shown that the simplest (fundamental) rogue waves are line rogue waves which arise from the constant background with a line profile and then disappear into the constant background again. It is also shown that multi-rogue waves describe the interaction of several fundamental rogue waves. These multi-rogue waves also arise from the constant background and then decay back to it, but in the intermediate times, interesting curvy wave patterns appear. However, higher-order rogue waves are found to show more interesting features. Specifically, only part of the wave structure in the higher-order rogue waves rises from the constant background and then retreats back to it, and this transient wave exhibits novel patterns such as parabolas. But the other part of the wave structure comes from the far distance as a localized lump, which decelerates to the near field and interacts with the transient rogue wave, and is then reflected back and accelerates to the large distance again. These rogue-wave solutions have interesting implications for two-dimensional surface water waves in the ocean.

nlin.SI↗
arXiv2011

Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves

We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms the $τ$ function are presented. Bäcklund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation.

nlin.SI↗
arXiv2011

General high-order rogue waves and their dynamics in the nonlinear Schroedinger equation

General high-order rogue waves in the nonlinear Schroedinger equation are derived by the bilinear method. These rogue waves are given in terms of determinants whose matrix elements have simple algebraic expressions. It is shown that the general N-th order rogue waves contain N-1 free irreducible complex parameters. In addition, the specific rogue waves obtained by Akhmediev et al. (Phys. Rev. E 80, 026601 (2009)) correspond to special choices of these free parameters, and they have the highest peak amplitudes among all rogue waves of the same order. If other values of these free parameters are taken, however, these general rogue waves can exhibit other solution dynamics such as arrays of fundamental rogue waves arising at different times and spatial positions and forming interesting patterns.

nlin.SI↗
arXiv2011

Motion and Bäcklund Transformations of Discrete Plane Curves

We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms the $τ$ function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.

nlin.SI↗
arXiv2011

Discrete Integrable Systems and Hodograph Transformations Arising from Motions of Discrete Plane Curves

We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation, and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.

nlin.SI↗
arXiv2010

N-Dark-Dark Solitons in the Generally Coupled Nonlinear Schroedinger Equations

N-dark-dark solitons in the generally coupled integrable NLS equations are derived by the KP-hierarchy reduction method. These solitons exist when nonlinearities are all defocusing, or both focusing and defocusing nonlinearities are mixed. When these solitons collide with each other, energies in both components of the solitons completely transmit through. This behavior contrasts collisions of bright-bright solitons in similar systems, where polarization rotation and soliton reflection can take place. It is also shown that in the mixed-nonlinearity case, two dark-dark solitons can form a stationary bound state.

nlin.PS↗
arXiv2010

A self-adaptive mesh method for the Camassa-Holm equation

A self-adaptive moving mesh method is proposed for the numerical simulations of the Camassa-Holm equation. It is an integrable scheme in the sense that it possesses the exact N-soliton solution. It is named a self-adaptive moving mesh method, because the non-uniform mesh is driven and adapted automatically by the solution. Once the non-uniform mesh is evolved, the solution is determined by solving a tridiagonal linear system. Due to these two superior features of the method, several test problems give very satisfactory results even if by using a small number of grid points.

nlin.SI↗
arXiv2010

Integrable discretizations for the short wave model of the Camassa-Holm equation

The link between the short wave model of the Camassa-Holm equation (SCHE) and bilinear equations of the two-dimensional Toda lattice (2DTL) is clarified. The parametric form of N-cuspon solution of the SCHE in Casorati determinant is then given. Based on the above finding, integrable semi-discrete and full-discrete analogues of the SCHE are constructed. The determinant solutions of both semi-discrete and fully discrete analogues of the SCHE are also presented.

nlin.SI↗
arXiv2009

Integrable discretizations of the short pulse equation

In the present paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key of the construction is the bilinear forms and determinant structure of solutions of the SP equation. We also give the determinant formulas of N-soliton solutions of the semi-discrete and full-discrete analogues of the SP equations, from which the multi-loop and multi-breather solutions can be generated. In the continuous limit, the full-discrete SP equation converges to the semi-discrete SP equation, then to the continuous SP equation. Based on the semi-discrete SP equation, an integrable numerical scheme, i.e., a self-adaptive moving mesh scheme, is proposed and used for the numerical computation of the short pulse equation.

nlin.SI↗
arXiv2008

Predicting Photometric and Spectroscopic Signatures of Rings around Transiting Extrasolar Planets

We present theoretical predictions for photometric and spectroscopic signatures of rings around transiting extrasolar planets. On the basis of a general formulation for the transiting signature in the stellar light curve and the velocity anomaly due to the Rossiter effect, we compute the expected signals analytically for a face-on ring system, and numerically for more general configurations. We study the detectability of a ring around a transiting planet located at a=3AU for a variety of obliquity and azimuthal angles, and find that it is possible to detect the ring signature both photometrically and spectroscopically unless the ring is almost edge-on (i.e., the obliquity angle of the ring θis much less than unity). We also consider the detectability of planetary rings around a close-in planet, HD 209458b ($θ\approx 90^\circ-i_{\rm orbit} \approx 3^\circ.32$), and Saturn ($θ\approx 26^\circ.7$) as illustrative examples. While the former is difficult to detect with the current precision (photometric precision of 10^-4 and radial velocity precision of 1 m/s), a marginal detection of the latter is possible photometrically. If the future precision of the radial velocity measurement reaches even below 0.1 m/s, they will be even detectable from the ground-based spectroscopic observations.

astro-ph↗