The ILW hierarchy

In this paper, we present a new hierarchy which includes the intermediate long wave (ILW) equation at the lowest order. This hierarchy is thought to be a novel reduction of the 1st modified KP type hierarchy. The framework of our investigation is Sato theory.Comment: 8 pages, LaTeX2

A Class of Special Solutions for the Ultradiscrete Painlev\'e II Equation

A class of special solutions are constructed in an intuitive way for the ultradiscrete analog of $q$-Painlev\'e II ($q$-PII) equation. The solutions are classified into four groups depending on the function-type and the system parameter

Ideas about the Economic Advantages of Colonial Maritime War and their Impact on British Politics and Naval Policy, 1701-1729

In early modern England (after 1707, Britain), there was an argument that war at sea, especially war in Spanish America, was an ideal means of warfare for England. This argument, whose origin can be traced back to the glorious memory of Elizabethan maritime war, revived at the beginning of the War of the Spanish Succession. This thesis examines this pro-maritime war argument, by focusing on its connection with its supposed economic advantages, and investigates its impact on British politics and naval policy during the war, and changes after the war. It reveals that this argument received support from politicians of different political stances because of its alleged economic advantages; colonial maritime war was expected to damage enemy financial resources while enriching Britain, and help to recover the Spanish American market where French merchants were making a rapid advance. At the same time, it makes clear that different political affiliations of the supporters created two types of pro-maritime arguments with different political functions. The thesis also shows that the supporters of colonial maritime war in the government as well as in the opposition tried to implement pro-maritime war policy by naval operations such as capture of Spanish silver fleets and colonial expeditions, and by legislation such as the American Act of 1708. However, their attempts were frustrated by diplomatic considerations, incapacity of naval administration, and by conflicting interests between several groups concerned in the West Indian colonies and Spanish American trade. After the South Sea expedition planned by the South Sea Company in 1712 did not materialise due to similar difficulties, the government focused on protection of the Spanish American trade, and refrained from taking aggressive action against Spanish colonies partly because of considerations for the interests of the company which started the Asiento trade. On the other hand, by the late 1720s the opposition, which championed the interests of private merchants, gradually came to advocate pro-maritime war policy, which eventually led up to propaganda campaigns against the Walpole ministry in the period of the War of Jenkins’s Ear.University of Exete

Casorati Determinant Solution for the Relativistic Toda Lattice Equation

The relativistic Toda lattice equation is decomposed into three Toda systems, the Toda lattice itself, B\"acklund transformation of Toda lattice and discrete time Toda lattice. It is shown that the solutions of the equation are given in terms of the Casorati determinant. By using the Casoratian technique, the bilinear equations of Toda systems are reduced to the Laplace expansion form for determinants. The $N$-soliton solution is explicitly constructed in the form of the Casorati determinant.Comment: 19 pages in plain Te

Casorati Determinant Solutions for the Discrete Painlev\'e III Equation

The discrete Painlev\'e III equation is investigated based on the bilinear formalism. It is shown that it admits the solutions expressed by the Casorati determinant whose entries are given by the discrete Bessel function. Moreover, based on the observation that these discrete Bessel functions are transformed to the $q$-Bessel functions by a simple variable transformation, we present a $q$-difference analogue of the Painlev\'e III equation.Comment: 16 pages in LaTe