arXiv subjects

Yoshimasa Matsuno

Publications and source records attributed to Yoshimasa Matsuno.

At least 19 recordsLinked to original sources

arXiv2024

A direct approach for solving the cubic Szegö equation

We study the cubic Szegö equation which is an integrable nonlinear non-dispersive and nonlocal evolution equation. In particular, we present a direct approach for obtaining the multiphase and multisoliton solutions as well as a special class of periodic solutions. Our method is substantially different from the existing one which relies mainly on the spectral analysis of the Hankel operator. We show that the cubic Szegö equation can be bilinearized through appropriate dependent variable transformations and then the solutions satisfy a set of bilinear equations. The proof is carried out within the framework of an elementary theory of determinants. Furthermore, we demonstrate that the eigenfunctions associated with the multiphase solutions satisfy the Lax pair for the cubic Szegö equation, providing an alternative proof of the solutions. Last, the eigenvalue problem for a periodic solution is solved exactly to obtain the analytical expressions of the eigenvalues.

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arXiv2023

Multiphase solutions and their reductions for a nonlocal nonlinear Schrödinger equation with focusing nonlinearity

A nonlocal nonlinear Schrödinger equation with focusing nonlinearity is considered which has been derived as a continuum limit of the Calogero-Sutherland model in an integrable classical dynamical system. The equation is shown to stem from the compatibility conditions of a system of linear PDEs, assuring its complete integrability. We construct a nonsingular $N$-phase solution ($N$: positive integer) of the equation by means of a direct method. The features of the one- and two-phase solutions are investigated in comparison with the corresponding solutions of the defocusing version of the equation. We also provide an alternative representation of the $N$-phase solution in terms of solutions of a system of nonlinear algebraic equations. Furthermore, the eigenvalue problem associated with the $N$-phase solution is discussed briefly with some exact results. Subsequently, we demonstrate that the $N$-soliton solution can be obtained simply by taking the long-wave limit of the $N$-phase solution. The similar limiting procedure gives an alternative representation of the $N$-soliton solution as well as the exact results related to the corresponding eigenvalue problem.

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arXiv2021

Integrability, conservation laws and solitons of a many-body dynamical system associated with the half-wave maps equation

We consider the half-wave maps (HWM) equation which is a continuum limit of the classical version of the Haldane-Shastry spin chain. In particular, we explore a many-body dynamical system arising from the HWM equation under the pole ansatz. The system is shown to be completely integrable by demonstrating that it exhibits a Lax pair and relevant conservation lows. Subsequently, the analytical multisoliton solutions of the HWM equation are constructed by means of the pole expansion method. The properties of the one- and two-soliton solutions are then investigated in detail as well as their pole dynamics. Last, an asymptotic analysis of the $N$-soliton solution reveals that no phase shifts appear after the collision of solitons. This intriguing feature is worth noting since it is the first example observed in the head-on collision of rational solitons. A number of problems remain open for the HWM equation, some of which are discussed in concluding remarks.

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arXiv2020

Parametric solutions of the generalized short pulse equations

We consider three novel PDEs associated with the integrable generalizations of the short pulse equation classified recently by Hone {\it et al} (2018 {\it Lett. Math. Phys.} {\bf 108} 927-947). In particular, we obtain a variety of exact solutions by means of a direct method analogous to that used for solving the short pulse equation. The main results reported here are the parametric representations of the multisoliton solutions. These solutions include cusp solitons, unbounded solutions with finite slope and breathers. In addition, the cusped periodic wave solutions are constructed from the cusp solitons by means of a simple procedure. As for non-periodic solutions, smooth breather solutions are of particular interest from the perspective of applications to real physical phenomena. The cycloid reduced from the periodic traveling wave with cusps is also worth remarking in connection with Gerstner's trochoidal solution in deep gravity waves. A number of works are left for future study, some of which will be addressed in concluding remarks.

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arXiv2019

Multisoliton formulas for the multi-component Fokas-Lenells equation with nonzero boundary conditions

The multi-component Fokas-Lenells equation is considered. In particular, we present the multisoliton formulas for the system with plane-wave boundary conditions, as well as with mixed zero and plane-wave boundary conditions. A direct approach is employed to construct solutions, showing that for both boundary conditions, the multisoliton solutions have compact determinantal expressions.

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arXiv2019

The multi-component modified nonlinear Schrödinger system with nonzero boundary conditions

In a previous paper (Matsuno 2011 {\it J. Phys. A: Math. Theore.} {\bf 44} 495202), we have presented a determinantal expression of the bright $N$-soliton solution for a multi-component modified nonlinear Schrödinger (NLS) system with zero boundary conditions. The present paper provides the dark $N$-soliton solution for the same system of equations with plane-wave boundary conditions, as well as the bright-dark $N$-soliton solution with mixed zero and plane-wave boundary conditions. The proof of the $N$-soliton solutions is performed by means of an elementary theory of determinants in which acobi's identity plays the central role. The $N$-soliton formulas obtained here include as special cases the existing soliton solutions of the NLS and derivative NLS equations and their integrable multi-omponent analogs. The new features of soliton solutions are discussed. In particular, it is shown that the $N$ constraints must be imposed on the amplitude parameters of solitons in constructing the dark $N$-soliton solution with plane-wave boundary conditions. This makes it difficult to analyze the solutions as the number of components increases. For mixed type boundary conditions, the structure of soliton solutions is found to be more explicit than that of dark soliton solutions since no constraints are imposed on the soliton parameters. Last, we comment on an integrable multi-component system associated with the first negative flow of the multi-component derivative NLS hierarchy.

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arXiv2017

Multisoliton solutions of the two-component Camassa-Holm system and their reductions

We develop a systematic procedure for constructing soliton solutions of an integrable two-component Camassa-Holm (CH2) system. The parametric representation of the multisoliton solutions is obtained by using a direct method combined with a reciprocal transformation. The properties of the solutions are then investigated in detail focusing mainly on the smooth one- and two-soliton solutions. The general $N$-soliton case is described shortly. Subsequently, we show that the CH2 system reduces to the CH equation and the two-component Hunter-Saxton (HS2) system by means of appropriate limiting procedures. The corresponding expressions of the multisoliton solutions are presented in parametric forms, reproducing the existing results for the reduced equations. Last, we discuss the reduction from the HS2 system to the HS equation.

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arXiv2016

Integrable multi-component generalization of a modified short pulse equation

We propose a multi-component generalization of the modified short pulse (SP) equation which was derived recently as a reduction of Feng's two-component SP equation. Above all, we address the two-component system in depth. We obtain the Lax pair, an infinite nember of conservation laws and multisoliton solutions for the system, demonstrating its integrability. Subsequently, we show that the two-component system exhibits cusp solitons and breathers for which the detailed analysis is performed. Specifically, we explore the interaction process of two cusp solitons and derive the formula for the phase shift. While cusp solitons are singular solutions, smooth breather solutions are shown to exist, provided that the parameters characterizing the solutions satisfy certain condition. Last, we discuss the relation between the proposed system and existing two-component SP equations.

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arXiv2016

Hamiltonian structure for two-dimensional extended Green-Naghdi equations

The two-dimensional Green-Naghdi (GN) shallow-water model for surface gravity waves is extended to incorporate the arbitrary higher-order dispersive effects. This can be achieved by developing a novel asymptotic analysis applied to the basic nonlinear water wave problem. The linear dispersion relation for the extended GN system is then explored in detail. In particular, we use its characteristics to discuss the well-posedness of the linearized problem. As illustrative examples of approximate model equations, we derive a higher-order model that is accurate to the fourth power of the dispersion parameter in the case of a flat bottom topography, and address the related issues such as the linear dispersion relation, conservation laws and the pressure distribution at the fluid bottom on the basis of this model. The original GN model is then briefly described in the case of an uneven bottom topography. Subsequently, the extended GN system presented here is shown to have the same Hamiltonian structure as that of the original GN system. Last, we demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

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arXiv2015

Hamiltonian formulation of the extended Green-Naghdi equations

A novel method is developed for extending the Green-Naghdi (GN) shallow-water model equation to the general system which incorporates the arbitrary higher-order dispersive effects. As an illustrative example, we derive a model equation which is accurate to the fourth power of the shallowness parameter while preserving the full nonlinearity of the GN equation, and obtain its solitary wave solutions by means of a singular perturbation analysis. We show that the extended GN equations have the same Hamiltonian structure as that of the GN equation. We also demonstrate that Zakharov's Hamiltonian formulation of surface gravity waves is equivalent to that of the extended GN system by rewriting the former system in terms of the momentum density instead of the velocity potential at the free surface.

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arXiv2014

Smooth and singular multisoliton solutions of a modified Camassa-Holm equation with cubic nonlinearity and linear dispersion

We develop a direct method for solving a modified Camassa-Holm equation with cubic nonlinearity and linear dispersion under the rapidly decreasing vanishing boundary condition. We obtain a compact parametric representation for the multisoliton solutions and investigate their properties. We show that the introduction of a linear dispersive term exhibits various new features in the structure of solutions. In particular, we find the smooth solitons whose characteristics are different from those of the Camassa-Holm equation, as well as the novel types of singular solitons. A remarkable feature of the soliton solutions is that the underlying structure of the associated tau-functions is the same as that of a model equation for shallow-water waves introduced by Ablowitz {\it et al}(1974 {\it Stud. Appl. Math.} {\bf 53} 249-315). Finally, we demonstrate that the short-wave limit of the soliton solutions recovers the soliton solutions of the short pulse equation which describes the propagation of ultra-short optical pulses in nonlinear media.

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arXiv2013

Smooth multisoliton solutions and their peakon limit of Novikov's Camassa-Holm type equation with cubic nonlinearity

We consider Novikov's Camassa-Holm type equation with cubic nonlinearity. In particular, we present a compact parametric representation of the smooth bright multisolution solutions on a constant background and investigate their structure. We find that the tau-functions associated with the solutions are closely related to those of a model equation for shallow-water waves (SWW) introduced by Hirota and Satsuma. This novel feature is established by applying the reciprocal transformation to the Novikov equation. We also show by specifying a complex phase parameter that the smooth soliton is converted to a novel singular soliton with single cusp and double peaks. We demonstrate that both the smooth and singular solitons converge to a peakon as the background field tends to zero whereby we employ a method that has been developed for performing the similar limiting procedure for the multisoliton solutions of the Camassa-Holm equation. In the subsequent asymptotic analysis of the two- and $N$-soliton solutions, we confirm their solitonic behaviour. Remarkably, the formulas for the phase shifts of solitons as well as their peakon limits coincide formally with those of the Degasperis-Procesi equation. Last, we derive an infinite number of conservation laws of the Novikov equation by using a relation between solutions of the Novikov equation and those of the SWW equation. In appendix, we prove various bilinear identities associated with the tau-functions of the multisoliton solutions of the SWW equation.

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arXiv2013

Bäcklund transformation and smooth multisoliton solutions for a modified Camassa-Holm equation with cubic nonlinearity

We present a compact parametric representation of the smooth bright multisolution solutions for the modified Camassa-Holm (mCH) equation with cubic nonlinearity. We first transform the mCH equation to an associated mCH equation through a reciprocal transformation and then find a novel Bäcklund transformation between solutions of the associated mCH equation and a model equation for shallow-water waves (SWW) introduced by Ablowitz {\it at al}. We combine this result with the expressions of the multisoliton solutions for the SWW and modified Korteweg-de Vries equations to obtain the multisoliton solutions of the mCH equation. Subsequently, we investigate the properties of the one- and two-soliton solutions as well as the general multisoliton solutions. We show that the smoothness of the solutions is assured only if the amplitude parameters of solitons satisfy certain conditions. We also find that at a critical value of the parameter beyond which the solution becomes singular, the soliton solution exhibits a different feature from that of the peakon solution of the CH equation. Then, by performing an asymptotic analysis for large time, we obtain the formula for the phase shift and confirm the solitonic nature of the multisoliton solutions. Last, we use the Bäcklund transformation to derive an infinite number of conservation laws of the mCH equation.

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arXiv2012

A direct method of solution for the Fokas-Lenells derivative nonlinear Schrödinger equation: II. Dark soliton solutions

In a previous study (Matsuno Y, J. Phys. A: Math. Theor. 45(2012)23202), we have developed a systematic method for obtaining the bright soliton solutions of the Fokas-Lenells derivative nonlinear Schrödinger equation (FL equation shortly) under vanishing boundary condition. In this paper, we apply the method to the FL equation with nonvanishing boundary condition. In particular, we deal with a more sophisticated problem on the dark soliton solutions with a plane wave boundary condition. We first derive the novel system of bilinear equations which is reduced from the FL equation through a dependent variable transformation and then construct the general dark $N$-soliton solution of the system, where $N$ is an arbitrary positive integer. In the process, a trilinear equation derived from the system of bilinear equations plays an important role. As a byproduct, this equation gives the dark $N$-soliton solution of the derivative nonlinear Schrödinger equation on the background of a plane wave. We then investigate the properties of the one-soliton solutions in detail, showing that both the dark and bright solitons appear on the nonzero background which reduce to algebraic solitons in specific limits. Last, we perform the asymptotic analysis of the two- and $N$-soliton solutions for large time and clarify their structure and dynamics.

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arXiv2012

A direct method of solution for the Fokas-Lenells derivative nonlinear Schrödinger equation: I. Bright soliton solutions

We develop a direct method of solution for finding the bright $N$-soliton solution of the Fokas-Lenells derivative nonlinear Schrödinger equation. The construction of the solution is performed by means of a purely algebraic procedure using an elementary theory of determinants and does not rely on the inverse scattering transform method. We present two different expressions of the solution both of which are expressed as a ratio of determinants. We then investigate the properties of the solutions and find several new features. Specifically, we derive the formula for the phase shift caused by the collisions of bright solitons.

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arXiv2011

A novel multi-component generalization of the short pulse equation and its multisoliton solutions

We propose a novel multi-component system of nonlinear equations that generalizes the short pulse (SP) equation describing the propagation of ultra-short pulses in optical fibers. By means of the bilinear formalism combined with a hodograph transformation, we obtain its multi-soliton solutions in the form of a parametric representation. Notably, unlike the determinantal solutions of the SP equation, the proposed system is found to exhibit solutions expressed in terms of pfaffians. The proof of the solutions is performed within the framework of an elementary theory of determinants. The reduced 2-component system deserves a special consideration. In particular, we show by establishing a Lax pair that the system is completely integrable. The properties of solutions such as loop solitons and breathers are investigated in detail, confirming their solitonic behavior. A variant of the 2-component system is also discussed with its multisoliton solutions.

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arXiv2011

The bright $N$-soliton solution of a multi-component modified nonlinear Schrödinger equation

A direct method is developed for constructing the bright $N$-soliton solution of a multi-component modified nonlinear Schrödinger equation. Specifically, the two different expressions of the solution are obtained both of which are expressed as a rational function of determinants. A simple relation is found between them by employing the properties of the Cauchy matrix. The proof of the solution reduces to the bilinear equations among the bordered determinants in which Jacobi's identity and related formulas play a central role. Last, the bright $N$-soliton solution is presented for a (2+1)-dimensional nonlocal model equation arising from the multi-component system as the number of dependent variables tends to infinity.

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arXiv2011

The N-soliton solution of a two-component modified nonlinear Schrödinger equation

The $N$-soliton solution is presented for a two-component modified nonlinear Schrödinger equation which describes the propagation of short pulses in birefringent optical fibers. The solution is found to be expressed in terms of determinants. The proof of the solution is carried out by means of an elementary theory of determinants. The generalization of the 2-component system to the multi-component system is discussed as well as a (2+1)-dimensional nonlocal equation arising from its continuum limit.

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