35 research outputs found
Soliton solutions of ultradiscrete integrable systems
In recent years, ultradiscrete integrable systems, in which both independent and dependent variables are discretized, have attracted much attention. In this thesis, we show how to obtain all line soliton solutions of (2+1)-dimensional ultradiscrete soliton systems from determinant solutions of discrete soliton systems by taking an ultradiscrete limit. Taking an ultradiscrete limit of determinant solutions with non-negativity, we obtain Casorati determinant-like solutions. Starting from Grammian (Gram-type determinant), we obtain another expression of Ï„-functions, which leads to a perturbed form after the expansion of Grammian. These two different forms are essentially equivalent, i.e., it is possible to transform a form into the other form by a simple combinatorics. In a Casorati determinant-like expression, there is a big advantage of constructing all possible line soliton solutions easily. Using ultradiscrete determinant-like solutions, we study the details of line soliton interactions of the ultradiscrete two-dimensional Toda lattice (2DTL) equation
Soliton solutions of the Kadomtsev-Petviashvili II equation
We study a general class of line-soliton solutions of the
Kadomtsev-Petviashvili II (KPII) equation by investigating the Wronskian form
of its tau-function. We show that, in addition to previously known line-soliton
solutions, this class also contains a large variety of new multi-soliton
solutions, many of which exhibit nontrivial spatial interaction patterns. We
also show that, in general, such solutions consist of unequal numbers of
incoming and outgoing line solitons. From the asymptotic analysis of the
tau-function, we explicitly characterize the incoming and outgoing
line-solitons of this class of solutions. We illustrate these results by
discussing several examples.Comment: 28 pages, 4 figure
Extensions of the General Solution to the Inverse Problem of the Calculus of Variations, and Variational, Perturbative and Reversible Systems Approaches to Regular and Embedded Solitary Waves
In the first part of this Dissertation, hierarchies of Lagrangians of degree two, three or four, each only partly determined by the choice of leading terms and with some coefficients remaining free, are derived. These have significantly greater freedom than the most general differential geometric criterion currently known for the existence of a Lagrangian and variational formulation since our existence conditions are for individual coefficients in the Lagrangian. For different choices of leading coefficients, the resulting variational equations could also represent traveling waves of various nonlinear evolution equations. Families of regular and embedded solitary waves are derived for some of these generalized variational ODEs in appropriate parameter regimes. In the second part, an earlier approach based on soliton perturbation theory is significantly generalized to obtain an analytical formula for the tail amplitudes of nonlocal solitary waves of a perturbed generalized fifth-order Korteweg-de Vries (FKdV) equation. On isolated curves in the parameter space, these tail amplitudes vanish, producing families of localized embedded solitons in large regions of the space. Off these curves, the tail amplitudes of the nonlocal waves are shown to be exponentially small in the small wavespeed limit. These seas of delocalized solitary waves are shown to be entirely distinct from those derived in that earlier work. These perturbative results are also discussed within the framework of known reversible systems results for various families of homoclinic orbits of the corresponding traveling-wave ordinary differential equation of our generalized FKdV equation. The third part considers a variety of dynamical behaviors in a multiparameter nonlinear Mathieu equation with distributed delay. A slow flow is derived using the method of averaging, and the predictions from that are then tested against direct numerical simulations of the nonlinear Mathieu system. Both areas of agreement and disagreement between the averaged and full numerical solutions are considered
Integrable Floquet dynamics
We discuss several classes of integrable Floquet systems, i.e. systems which
do not exhibit chaotic behavior even under a time dependent perturbation. The
first class is associated with finite-dimensional Lie groups and
infinite-dimensional generalization thereof. The second class is related to the
row transfer matrices of the 2D statistical mechanics models. The third class
of models, called here "boost models", is constructed as a periodic interchange
of two Hamiltonians - one is the integrable lattice model Hamiltonian, while
the second is the boost operator. The latter for known cases coincides with the
entanglement Hamiltonian and is closely related to the corner transfer matrix
of the corresponding 2D statistical models. We present several explicit
examples. As an interesting application of the boost models we discuss a
possibility of generating periodically oscillating states with the period
different from that of the driving field. In particular, one can realize an
oscillating state by performing a static quench to a boost operator. We term
this state a "Quantum Boost Clock". All analyzed setups can be readily realized
experimentally, for example in cod atoms.Comment: 18 pages, 2 figures; revised version. Submission to SciPos
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.Comment: 50 pages, 25 figure