Yasuhiro Ohta

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

General high-order rogue waves in the nonlinear Schrödinger 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. (Akhmediev et al. 2009 Phys. Rev. E 80, 026601 (doi:10.1103/PhysRevE.80.026601)) 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.

Rogue waves in the Davey-Stewartson I 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 multirogue waves describe the interaction of several fundamental rogue waves. These multirogue 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 exhibit different dynamics. 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 possesses 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.

Dynamics of rogue waves in the Davey?Stewartson II equation

General rogue waves in the Davey–Stewartson (DS)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). 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 DSII equation.

Hypergeometric solutions to the q-Painlevé equations

Hypergeometric solutions to seven q-Painleve equations in Sakais classification are constructed. Geometry of plane curves is used to reduce the q-Painleve equations to the three-term recurrence relations for q-hypergeometric functions.

10E9 solution to the elliptic Painlevé equation

A τ function formalism for Sakais elliptic Painleve equation is presented. This establishes the equivalence between the two formulations by Sakai and by Ohta–Ramani–Grammaticos. We also give a simple geometric description of the elliptic Painleve equation as a non-autonomous deformation of the addition formula on elliptic curves. By using these formulations, we construct a particular solution of the elliptic Painleve equation expressed in terms of the elliptic hypergeometric function 10E9.

Integrable discretizations of the short pulse equation

In this paper, we propose integrable semi-discrete and full-discrete analogues of the short pulse (SP) equation. The key construction is the bilinear form 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, and 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.

An integrable semi-discretization of the Camassa-Holm equation and its determinant solution

An integrable semi-discretization of the Camassa–Holm (CH) equation is presented. The keys of its construction are bilinear forms and determinant structure of solutions of the CH equation. Determinant formulas of N-soliton solutions of the continuous and semi-discrete Camassa–Holm equations are presented. Based on determinant formulas, we can generate multi-soliton, multi-cuspon and multi-soliton–cuspon solutions. Numerical computations using the integrable semi-discrete Camassa–Holm equation are performed. It is shown that the integrable semi-discrete Camassa–Holm equation gives very accurate numerical results even in the cases of cuspon–cuspon and soliton–cuspon interactions. The numerical computation for an initial value condition, which is not an exact solution, is also presented.

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.

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.

Quasideterminant solutions of a non-Abelian Hirota?Miwa equation

A non-Abelian version of the Hirota–Miwa equation is considered. In an earlier paper of Nimmo (2006 J. Phys. A: Math. Gen. 39 5053–65) it was shown how solutions expressed as quasideterminants could be constructed for this system by means of Darboux transformations. In this paper, we discuss these solutions from a different perspective and show that the solutions are quasi-Plucker coordinates and that the non-Abelian Hirota–Miwa equation may be written as a quasi-Plucker relation. The special case of the matrix Hirota–Miwa equation is also considered using a more traditional, bilinear approach and the techniques are compared.