arXiv subjects

Yuji Kodama

Publications and source records attributed to Yuji Kodama.

At least 37 records · Page 2Linked to original sources

arXiv2011

KP solitons, total positivity, and cluster algebras

Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that the Wronskian approach to the KP equation provides a method to construct soliton solutions. The regular soliton solutions that one obtains in this way come from points of the totally non-negative part of the Grassmannian. In this paper we explain how the theory of total positivity and cluster algebras provides a framework for understanding these soliton solutions to the KP equation. We then use this framework to give an explicit construction of certain soliton contour graphs, and solve the inverse problem for soliton solutions coming from the totally positive part of the Grassmannian.

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arXiv2010

Cohomology of real Grassmann manifold and KP flow

We consider a realization of the real Grassmann manifold Gr(k,n) based on a particular flow defined by the corresponding (singular) solution of the KP equation. Then we show that the KP flow can provide an explicit and simple construction of the incidence graph for the integral cohomology of Gr(k,n). It turns out that there are two types of graphs, one for the trivial coefficients and other for the twisted coefficients, and they correspond to the homology groups of the orientable and non-orientable cases of Gr(k,n) via the Poincare-Lefschetz duality. We also derive an explicit formula of the Poincare polynomial for Gr(k,n) and show that the Poincare polynomial is also related to the number of points on a suitable version of Gr(k,n) over a finite field $\F_q$ with q being a power of a prime. In particular, we find that the number of $\F_q$ points on Gr(k,n) can be computed by counting the number of singularities along the KP flow.

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arXiv2010

KP solitons in shallow water

The main purpose of the paper is to provide a survey of our recent studies on soliton solutions of the Kadomtsev-Petviashvili (KP) equation. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. Each soliton solution is then defined by a point of the totally non-negative Grassmann variety which can be parametrized by a unique derangement of the symmetric group of permutations. Our study also includes certain numerical stability problems of those soliton solutions. Numerical simulations of the initial value problems indicate that certain class of initial waves asymptotically approach to these exact solutions of the KP equation. We then discuss an application of our theory to the Mach reflection problem in shallow water. This problem describes the resonant interaction of solitary waves appearing in the reflection of an obliquely incident wave onto a vertical wall, and it predicts an extra-ordinary four-fold amplification of the wave at the wall. There are several numerical studies confirming the prediction, but all indicate disagreements with the KP theory. Contrary to those previous numerical studies, we find that the KP theory actually provides an excellent model to describe the Mach reflection phenomena when the higher order corrections are included to the quasi-two dimensional approximation. We also present laboratory experiments of the Mach reflection recently carried out by Yeh and his colleagues, and show how precisely the KP theory predicts this wave behavior.

nlin.SI↗

arXiv2010

Numerical study of the KP equation for non-periodic waves

The Kadomtsev-Petviashvili (KP) equation describes weakly dispersive and small amplitude waves propagating in a quasi-two dimensional situation. Recently a large variety of exact soliton solutions of the KP equation has been found and classified. Those soliton solutions are localized along certain lines in a two-dimensional plane and decay exponentially everywhere else, and they are called line-soliton solutions in this paper. The classification is based on the far-field patterns of the solutions which consist of a finite number of line-solitons. In this paper, we study the initial value problem of the KP equation with V- and X-shape initial waves consisting of two distinct line-solitons by means of the direct numerical simulation. We then show that the solution converges asymptotically to some of those exact soliton solutions. The convergence is in a locally defined $L^2$-sense. The initial wave patterns considered in this paper are related to the rogue waves generated by nonlinear wave interactions in shallow water wave problem.

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arXiv2010

Mach reflection and KP solitons in shallow water

Reflection of an obliquely incident solitary wave onto a vertical wall is studied analytically and experimentally. We use the Kadomtsev-Petviashivili (KP) equation to analyze the evolution and its asymptotic state. Laboratory experiments are performed using the laser induced fluorescent (LIF) technique, and detailed features and amplifications at the wall are measured. We find that proper physical interpretation must be made for the KP solutions when the experimental results are compared with the theory under the assumptions of quasi-two dimensionality and weak nonlinearity. Due to the lack of physical interpretation of the theory, the numerical results were previously thought not in good agreement with the theory. With proper treatment, the KP theory provides an excellent model to predict the present laboratory results as well as the previous numerical results. The theory also indicates that the present laboratory apparatus must be too short to achieve the asymptotic state. The laboratory and numerical results suggest that the maximum of the predicted four-fold amplification would be difficult to be realized in the real-fluid environment. The reality of this amplification remains obscure.

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arXiv2009

The Pfaff lattice on symplectic matrices

The Pfaff lattice is an integrable system arising from the SR-group factorization in an analogous way to how the Toda lattice arises from the QR-group factorization. In our recent paper [{\it Intern. Math. Res. Notices}, (2007) rnm120], we studied the Pfaff lattice hierarchy for the case where the Lax matrix is defined to be a lower Hessenberg matrix. In this paper we deal with the case of a symplectic lower Hessenberg Lax matrix, this forces the Lax matrix to take a tridiagonal shape. We then show that the odd members of the Pfaff lattice hierarchy are trivial, while the even members are equivalent to the indefinite Toda lattice hierarchy defined in [Y. Kodama and J. Ye, {\it Physica D}, {\bf 91} (1996) 321-339]. This is analogous to the case of the Toda lattice hierarchy in the relation to the Kac-van Moerbeke system. In the case with initial matrix having only real or imaginary eigenvalues, the fixed points of the even flows are given by $2\times 2$ block diagonal matrices with zero diagonals. We also consider a family of skew-orthogonal polynomials with symplectic recursion relation related to the Pfaff lattice, and find that they are succinctly expressed in terms of orthogonal polynomials appearing in the indefinite Toda lattice.

nlin.SI↗

arXiv2009

Soliton solutions of the KP equation and application to shallow water waves

The main purpose of this paper is to give a survey of recent development on a classification of soliton solutions of the KP equation. The paper is self-contained, and we give a complete proof for the theorems needed for the classification. The classification is based on the Schubert decomposition of the real Grassmann manifold, Gr$(N,M)$, the set of $N$-dimensional subspaces in $\mathbb{R}^M$. Each soliton solution defined on Gr$(N,M)$ asymptotically consists of the $N$ number of line-solitons for $y\gg 0$ and the $M-N$ number of line-solitons for $y\ll 0$. In particular, we give the detailed description of those soliton solutions associated with Gr$(2,4)$, which play a fundamental role of multi-soliton solutions. We then consider a physical application of some of those solutions related to the Mach reflection discussed by J. Miles in 1977.

nlin.SI↗

arXiv2009

Soliton Solutions of the KP Equation with V-Shape Initial Waves

We consider the initial value problems of the Kadomtsev-Petviashvili (KP) equation for symmetric V-shape initial waves consisting of two semi-infinite line solitons with the same amplitude. Numerical simulations show that the solutions of the initial value problem approach asymptotically to certain exact solutions of the KP equation found recently in [Chakravarty and Kodama, JPA, 41 (2008) 275209]. We then use a chord diagram to explain the asymptotic result. We also demonstrate a real experiment of shallow water wave which may represent the solution discussed in this Letter.

nlin.SI↗

arXiv2008

Combinatorics of dispersionless integrable systems and universality in random matrix theory

It is well-known that the partition function of the unitary ensembles of random matrices is given by a tau-function of the Toda lattice hierarchy and those of the orthogonal and symplectic ensembles are tau-functions of the Pfaff lattice hierarchy. In these cases the asymptotic expansions of the free energies given by the logarithm of the partition functions lead to the dispersionless (i.e. continuous) limits for the Toda and Pfaff lattice hierarchies. There is a universality between all three ensembles of random matrices, one consequence of which is that the leading orders of the free energy for large matrices agree. In this paper, this universality, in the case of Gaussian ensembles, is explicitly demonstrated by computing the leading orders of the free energies in the expansions. We also show that the free energy as the solution of the dispersionless Toda lattice hierarchy gives a solution of the dispersionless Pfaff lattice hierarchy, which implies that this universality holds in general for the leading orders of the unitary, orthogonal, and symplectic ensembles. We also find an explicit formula for the two point function $F_{nm}$ which represents the number of connected ribbon graphs with two vertices of degrees n and m on a sphere. The derivation is based on the Faber polynomials defined on the spectral curve of the dispersionless Toda lattice hierarchy, and $\frac{1}{nm} F_{nm}$ are the Grunsky coefficients of the Faber polynomials.

nlin.SI↗

arXiv2008

The Finite Non-periodic Toda Lattice: A Geometric and Topological Viewpoint

In 1967, Japanese physicist Morikazu Toda published the seminal papers exhibiting soliton solutions to a chain of particles with nonlinear interactions between nearest neighbors. In the decades that followed, Toda's system of particles has been generalized in different directions, each with its own analytic, geometric, and topological characteristics that sets it apart from the others. These are known collectively as the Toda lattice. This survey describes and compares several versions of the finite non-periodic Toda lattice from the perspective of their geometry and topology.

nlin.SI↗

arXiv2008

A generating function for the N-soliton solutions of the Kadomtsev-Petviashvili II equation

This work describes a classification of the $N$-soliton solutions of the Kadomtsev-Petviashvili II equation in terms of chord diagrams of N chords joining pairs of 2N points. The different classes of N-solitons are enumerated by the distribution of crossings of the chords. The generating function of the chord diagrams is expressed as a continued fraction, special cases of which are moment generating functions for certain kinds of $q$-orthogonal polynomials.

nlin.SI↗

arXiv2007

On the Whitham Equations for the Defocusing Complex Modified KdV Equation

We study the Whitham equations for the defocusing complex modified KdV (mKdV) equation. These Whitham equations are quasilinear hyperbolic equations and they describe the averaged dynamics of the rapid oscillations which appear in the solution of the mKdV equation when the dispersive parameter is small. The oscillations are referred to as dispersive shocks. The Whitham equations for the mKdV equation are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the solutions of the Whitham equations when the initial values are given by a step function. We also compare the results with those of the defocusing nonlinear Schrodinger (NLS) equation. For the NLS equation, the Whitham equations are strictly hyperbolic and genuinely nonlinear. We show that the weak hyperbolicity of the mKdV-Whitham equations is responsible for an additional structure in the dispersive shocks which has not been found in the NLS case.

nlin.SI↗

arXiv2007

Classification of the line-soliton solutions of KPII

In the previous papers (notably, Y. Kodama, J. Phys. A 37, 11169-11190 (2004), and G. Biondini and S. Chakravarty, J. Math. Phys. 47 033514 (2006)), we found a large variety of line-soliton solutions of the Kadomtsev-Petviashvili II (KPII) equation. The line-soliton solutions are solitary waves which decay exponentially in $(x,y)$-plane except along certain rays. In this paper, we show that those solutions are classified by asymptotic information of the solution as $|y| \to \infty$. Our study then unravels some interesting relations between the line-soliton classification scheme and classical results in the theory of permutations.

nlin.SI↗

arXiv2005

On the Whitham equations for the defocusing nonlinear Schrodinger equation with step initial data

The behavior of solutions of the finite-genus Whitham equations for the weak dispersion limit of the defocusing nonlinear Schrodinger equation is investigated analytically and numerically for piecewise-constant initial data. In particular, the dynamics of constant-amplitude initial conditions with one or more frequency jumps (i.e., piecewise linear phase) are considered. It is shown analytically and numerically that, for finite times, regions of arbitrarily high genus can be produced; asymptotically with time, however, the solution can be divided into expanding regions which are either of genus-zero, genus-one or genus-two type, their precise arrangement depending on the specifics of the initial datum given. This behavior should be compared to that of the Korteweg-deVries equation, where the solution is devided into the regions which are either genus-zero or genus-one asymptotically. Finally, the potential application of these results to the generation of short optical pulses is discussed: the method proposed takes advantage of nonlinear compression via appropriate frequency modulation, and allows control of both the pulse amplitude and its width, as well as the distance along the fiber at which the pulse is produced.

nlin.SI↗

arXiv2005

Toda lattice, cohomology of compact Lie groups and finite Chevalley groups

In this paper, we describe a connection that exists among (a) the number of singular points along the trajectory of Toda flow, (b) the cohomology of a compact subgroup $K$, and (c) the number of points of a Chevalley group $K({\mathbb F}_q)$ related to $K$ over a finite field ${\mathbb F}_q$. The Toda lattice is defined for a real split semisimple Lie algebra $\mathfrak g$, and $K$ is a maximal compact Lie subgroup of $G$ associated to $\mathfrak g$. Relations are also obtained between the singularities of the Toda flow and the integral cohomology of the real flag manifold $G/B$ with $B$ the Borel subgroup of $G$ (here we have $G/B=K/T$ with a finite group $T$). We also compute the maximal number of singularities of the Toda flow for any real split semisimple algebra, and find that this number gives the multiplicity of the singularity at the intersection of the varieties defined by the zero set of Schur polynomials.

math.AT↗

arXiv2005

Spectral stability and time evolution of N-solitons in KdV hierarchy

This paper concerns spectral stability and time evolution of $N$-solitons in the KdV hierarchy with mixed commuting time flows. Spectral stability problem is analyzed by using a pair of self-adjoint operators with finite numbers of negative eigenvalues. We show that the absence of unstable eigenvalues in the stability problem is related to the absence of negative eigenvalues of these operators in the constrained function spaces. Time evolution of $N$-solitons is uniquely characterized from the inverse scattering transform technique.

nlin.SI↗

arXiv2004

Young diagrams and N-soliton solutions of the KP equation

We consider $N$-soliton solutions of the KP equation, (-4u_t+u_{xxx}+6uu_x)_x+3u_{yy}=0 . An $N$-soliton solution is a solution $u(x,y,t)$ which has the same set of $N$ line soliton solutions in both asymptotics $y\to\infty$ and $y\to -\infty$. The $N$-soliton solutions include all possible resonant interactions among those line solitons. We then classify those $N$-soliton solutions by defining a pair of $N$-numbers $({\bf n}^+,{\bf n}^-)$ with ${\bf n}^{\pm}=(n_1^{\pm},...,n_N^{\pm}), n_j^{\pm}\in\{1,...,2N\}$, which labels $N$ line solitons in the solution. The classification is related to the Schubert decomposition of the Grassmann manifolds Gr$(N,2N)$, where the solution of the KP equation is defined as a torus orbit. Then the interaction pattern of $N$-soliton solution can be described by the pair of Young diagrams associated with $({\bf n}^+,{\bf n}^-)$. We also show that $N$-soliton solutions of the KdV equation obtained by the constraint $\partial u/\partial y=0$ cannot have resonant interaction.

nlin.SI↗

arXiv2004

Compactification of the isospectral varieties of nilpotent Toda lattices

The paper concerns a compactification of the isospectral varieties of nilpotent Toda lattices for real split simple Lie algebras. The compactification is obtained by taking the closure of unipotent group orbits in the flag manifolds. The unipotent group orbits are called the Peterson varieties and can be used in the complex case to describe the quantum cohomology of Grassmannian manifolds. We construct a chain complex based on a cell decomposition consisting of the subsystems of Toda lattices. Explicit formulae for the incidence numbers of the chain complex are found, and encoded in a graph containing an edge whenever an incidence number is non-zero. We then compute rational cohomology, and show that there are just three different patterns in the calculation of Betti numbers. Although these compactified varieties are singular, they resemble certain smooth Schubert varieties e.g. they both have a cell decomposition consiting of unipotent group orbits of the same dimensions. In particular, for the case of a Lie algebra of type $A$ the rational homology/cohomology obtained from the compactified isospectral variety of the nilpotent Toda lattice equals that of the corresponding Schubert variety.

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