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Ken-ichi Maruno

Publications and source records attributed to Ken-ichi Maruno.

At least 19 recordsLinked to original sources
arXiv2026

An integrable semi-discretization of the two-component Hunter-Saxton equation

In this paper, we propose an integrable semi-discretization of the two-component Hunter--Saxton (2-HS) equation, the short-wave limit of the two-component Camassa--Holm (2-CH) equation. At the continuous level, we show that the 2-HS equation can be derived from a new bilinear formulation, distinct from the conventional one in the literature, via a pseudo 2-reduction and a hodograph transformation. For the semi-discrete construction, we first discretize the underlying bilinear equations in the spatial direction. We then impose the pseudo 2-reduction and apply a discrete hodograph transformation to obtain the semi-discrete system in the physical variables. To the best of our knowledge, the resulting system is the first integrable semi-discretization that preserves the two-component structure of the 2-HS equation. We construct the N-soliton solutions of the continuous and semi-discrete systems in Wronskian and Casoratian forms, respectively. The integrability of the semi-discrete system is inherited from the underlying integrable hierarchy and is further verified by a Lax pair.

nlin.SI↗
arXiv2026

Integrable full discretization of the multi-component short pulse equation

We propose a new formulation of the multi-component short pulse (MCSP) equation that includes the coupled complex short pulse (CCSP) equation as a reduction. Using Hirota's bilinear method, we construct its $N$-soliton solutions in Pfaffian form. We then derive integrable semi-discrete and fully discrete analogues of the MCSP equation admitting Pfaffian $N$-soliton solutions. The resulting fully discrete system provides a practical self-adaptive moving mesh scheme for numerical simulations. For the parameter sets considered, numerical simulations demonstrate excellent agreement between the numerical and exact solutions, confirming the robustness and high accuracy of the proposed scheme.

nlin.SI↗
arXiv2026

Integrable self-adaptive moving mesh schemes for multi-component short pulse type equations with nonzero boundary values

In this paper, we construct integrable self-adaptive moving mesh schemes for multi-component modified short pulse and short pulse equations with nonzero boundary values by using the consistency condition with the hodograph transformation. The essential point is that the edge point $x_{0}$ of the hodograph transformation cannot be kept fixed when the boundary flux is nonzero. We derive the evolution equation for $x_{0}$ and incorporate it into the semi-discrete moving mesh scheme. This supplies a moving-edge mechanism that extends the previously fixed-edge schemes and, in particular, allows periodic computations with nonzero boundary values. These schemes automatically adjust the mesh intervals according to the solution profile. We also derive multi-soliton solutions in Pfaffian form for the proposed schemes, which preserve the integrable structure in the discrete scheme. Numerical experiments for one- and two-soliton solutions demonstrate that the proposed schemes achieve high accuracy even in regions with rapid variation, while maintaining stability over long-time simulations, with small relative errors near peak amplitudes.

nlin.SI↗
arXiv2026

A unified approach to the AKNS, DNLS, KP and mKP hierarchies in the anti-self-dual Yang-Mills reduction

We show a unified approach to the Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy and the unreduced derivative nonlinear Schrödinger (DNLS) hierarchies (including the Kaup-Newell, Chen-Lee-Liu, Gerdjikov-Ivanov and a generalized DNLS), together with their multi-component extensions, in the framework of the anti-self-dual Yang-Mills (ASDYM) reduction. By restricting the gauge group to GL(2), the Kadomtsev-Petviashvili (KP) and modified KP (mKP) hierarchies are formulated in the ASDYM reduction via squared eigenfunction symmetry constraints. In this case, the bilinearization of the generalized DNLS equations can also be understood through this reduction. Finally, Gram-type exact solutions for the relevant equations are presented in terms of quasi-determinants.

nlin.SI↗
arXiv2025

Construction of the Lax pairs for the delay Lotka-Volterra and delay Toda lattice equations and their reductions to delay Painleve equations

The delay Lotka-Volterra and delay Toda lattice equations are delay-differential extensions of the well-known soliton equations, the Lotka-Volterra and Toda lattice equations, respectively. This paper investigates integrable properties of the delay Lotka-Volterra and delay Toda lattice equations, and study the relationships to the already known delay Painleve equations. First, Backlund transformations, Lax pairs and an infinite number of conserved quantities of these delay soliton equations are constructed. Then, applying spatial 2-periodic reductions to them, we show the known delay Painleve equations are derived. Using these reductions, we construct the N-soliton-type determinant solutions of the autonomous versions of delay Painleve equations, and the Casorati determinant solution of a higher order analogue of the discrete Painleve II equation.

nlin.SI↗
arXiv2024

A delay analogue of the box and ball system arising from the ultra-discretization of the delay discrete Lotka-Volterra equation

A delay analogue of the box and ball system (BBS) is presented. This new soliton cellular automaton is constructed by the ultra-discretization of the delay discrete Lotka-Volterra equation, which is an integrable delay analogue of the discrete Lotka-Volterra equation. Soliton patterns generated by this delay BBS are classified into normal solitons and abnormal solitons. Normal solitons have a clear relationship to the solitons of the BBS with K kinds of balls. On the other hand, abnormal solitons show various types of novel soliton patterns, which have not been observed in almost all known BBSs. We obtain them by numerical experiments, and then construct τ-functions of them analytically in 1-soliton cases.

nlin.SI↗
arXiv2023

Connection between the symmetric discrete AKP system and bilinear ABS lattice equations

In this paper, we show that all the bilinear Adler-Bobenko-Suris (ABS) equations (except Q2 and Q4) can be obtained from symmetric discrete AKP system by taking proper reductions and continuum limits. Among the bilinear ABS equations, a simpler bilinear form of the ABS H2 equation is given. In addition, an 8-point 3-dimensional lattice equation and an 8-point 4-dimensional lattice equation are obtained as by-products. Both of them can be considered as extensions of the symmetric discrete AKP equation.

nlin.SI↗
arXiv2023

Integrable discretizations of the SIR model

Structure-preserving discretizations of the SIR model are presented by focusing on the hodograph transformation and the conditions for integrability for their discrete SIR models are given. For those integrable discrete SIR models, we derive their exact solutions as well as conserved quantities. If we choose the parameter appropriately for one of our proposed discrete SIR models, it conserves the conserved quantities of the SIR model. We also investigate an ultradiscretizable discrete SIR model.

nlin.SI↗
arXiv2022

A systematic construction of integrable delay-difference and delay-differential analogues of soliton equations

We propose a systematic method for constructing integrable delay-difference and delay-differential analogues of known soliton equations such as the Lotka-Volterra, Toda lattice, and sine-Gordon equations and their multi-soliton solutions. It is carried out by applying a reduction and delay-differential limit to the discrete KP or discrete two-dimensional Toda lattice equations. Each of the delay-difference and delay-differential equations has the N-soliton solution, which depends on the delay parameter and converges to an N-soliton solution of a known soliton equation as the delay parameter approaches 0.

nlin.SI↗
arXiv2018

Isoperimetric deformations of curves on the Minkowski plane

We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the $τ$ functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class give regular solutions to the defocusing mKdV equation. Some pictures illustrating typical dynamics of the curves are presented.

math.DG↗
arXiv2018

High-order rogue waves of a long wave-short wave model

The long wave-short wave model describes the interaction between the long wave and the short wave. Exact higher-order rational solution expressed by determinants is calculated via the Hirota's bilinear method and the KP hierarchy reduction. It is found that the fundamental rogue wave for the short wave can be classified into three different patterns: bright, intermediate and dark ones, whereas the rogue wave for the long wave is always bright type. The higher-order rogue waves correspond to the superposition of fundamental rogue waves. The modulation instability analysis show that the condition of the baseband modulation instability where an unstable continuous-wave background corresponds to perturbations with infinitesimally small frequencies, coincides with the condition for the existence of rogue-wave solutions.

nlin.SI↗
arXiv2017

Integrable Discrete Model for One-dimensional Soil Water Infiltration

We propose an integrable discrete model of one-dimensional soil water infiltration. This model is based on the continuum model by Broadbridge and White, which takes the form of nonlinear convection-diffusion equation with a nonlinear flux boundary condition at the surface. It is transformed to the Burgers equation with a time-dependent flux term by the hodograph transformation. We construct a discrete model preserving the underlying integrability, which is formulated as the self-adaptive moving mesh scheme. The discretization is based on linearizability of the Burgers equation to the linear diffusion equation, but the naïve discretization based on the Euler scheme which is often used in the theory of discrete integrable systems does not necessarily give a good numerical scheme. Taking desirable properties of a numerical scheme into account, we propose an alternative discrete model that produces solutions with similar accuracy to direct computation on the original nonlinear equation, but with clear benefits regarding computational cost.

nlin.SI↗
arXiv2017

Breather to the Yajima-Oikawa system

The Yajima-Oikawa (YO) system describes the resonant interaction between long and short waves under certain condition. In this paper, through the KP hierarchy reduction, we construct the breather solutions for the YO system in one- and two-dimensional cases. Similar to Akhmediev and Kuznetsov-Ma breather solutions (the wavenumber k_i->ik_i) for the nonlinear Schrodinger equation, is shown that the YO system have two kinds ofbreather solutions with the relations p_{2k-1}->ip_{2k-1}, p_{2k}->-ip_{2k}, q_{2k-1}->iq_{2k-1} and q_{2k}->-iq_{2k}, in which the homoclinic orbit and dark soliton solutions are two special cases respectively. Furthermore, taking the long wave limit, we derive the rational and rational and rational-exp solutions which contain lump, line rogue wave, soliton and their mixed cases. By considering the further reduction, such solutions can be reduced to one-dimensional YO system.

nlin.SI↗
arXiv2017

General high-order rogue waves of the (1+1)-dimensional Yajima-Oikawa system

General high-order rogue wave solutions for the (1+1)-dimensional Yajima-Oikawa (YO) system are derived by using Hirota's bilinear method and the KP-hierarchy reduction technique. These rogue wave solutions are presented in terms of determinants in which the elements are algebraic expressions. The dynamics of first and higher-order rogue wave are investigated in details for different values of the free parameters. It is shown that the fundamental (first-order) rogue waves can be classified into three different patterns: bright, intermediate and dark ones. The high-order rogue waves correspond to the superposition of fundamental rogue waves. Especially, compared with the nonlinear Schodinger equation, there exists an essential parameter αto control the pattern of rogue wave for both first- and high-order rogue waves since the YO system does not possess the Galilean invariance.

nlin.SI↗
arXiv2016

Geometric formulation and multi-dark soliton solution to the defocusinig complex short pulse equation

In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space $\mathbf{R}^{2,1}$, then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the the defocusing CSP equation from the single-component extended KP hierarchy by the reduction method. As a by-product, the $N$-dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.

nlin.SI↗
arXiv2016

A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue

In the present paper, we propose a two-component generalization of the reduced Ostrovsky equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis-Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and $N$-soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bäcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. Especially, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their $N$-soliton solutions in terms of pffafians are also provided.

nlin.SI↗
arXiv2015

An integrable semi-discrete Degasperis-Procesi equation

Based on our previous work to the Degasperis-Procesi equation (J. Phys. A 46 045205) and the integrable semi-discrete analogue of its short wave limit (J. Phys. A 48 135203), we derive an integrable semi-discrete Degasperis-Procesi equation by Hirota's bilinear method. Meanwhile, $N$-soliton solution to the semi-discrete Degasperis-Procesi equation is provided and proved. It is shown that the proposed semi-discrete Degasperis-Procesi equation, along with its $N$-soliton solution converge to ones of the original Degasperis-Procesi equation in the continuous limit.

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