Compactons dispersive structures for variants of the K(n,n) and the KP equations

In this paper we discuss two generalized forms of the K(n,n) and the KP equations that exhibit compactons: solitons with the absence of infinite wings, and solitary patterns solutions having infinite slopes or cusps. The variants are extended to include nonlinear dispersion to support compactons structures and solitary patterns in higher dimensions. Two distinct general formulas for compact and noncompact solutions, that are of substantial interest, are formally developed.     

Travelling waves of the KP equations with transverse modulations

Kadomtsev-Petviashvili (KP) equations arise genetically in modelling nonlinear wave propagation for primarily unidirectional long waves of small amplitude with weak transverse dependence. In the case when transverse dispersion is positive (such as for water waves with large surface tension) we investigate the existence of transversely modulated travelling waves near one-dimensional solitary waves. Using bifurcation theory we show the existence of a unique branch of periodically modulated solitary waves. Then, we briefly discuss the case when the transverse dispersion is negative (such as for water waves with zero surface tension).     

Simple connection between the geometric and the Hamiltonian representations of integrable nonlinear equations

One gives a simple and general derivation of the well-known connection between the geometric and the Hamiltonian approaches in the classical method of the inverse problem. Namely, for the case of a two-dimensional auxiliary problem and periodic boundary conditions it is explicitly shown how the existence of the classical -matrix for the integrable equations leads to their representation in the form of the condition of zero curvature.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 264–273, 1982.     

The compact and noncompact structures for two types of generalized Camassa–Holm–KP equations

The objective of this paper is to investigate two types of generalized nonlinear Camassa–Holm–KP equations in (2+1) dimensional space. Compactons, solitons, solitary patterns, periodic solutions and algebraic travelling wave solutions are expressed analytically under various circumstances. The conditions that cause the qualitative change in the physical structures of the solutions are emphasized.     

The Kadomtsev–Petviashvili (KP), MKP,and coupled KP equations for two-ion-temperature dusty plasmas

It is well known that the Korteweg–de Vires (KdV) equation can describe small but finite amplitude dust acoustic waves in a dusty plasmas. In this paper, we use the reductive perturbation method and derive a Kadomtsev–Petviashvili (KP) equation, a modified KP (MKP) equation and a coupled KP equation for unmagnetized, collisionless, cold, and two-ion-temperature dusty plasmas with N different species of dust grains. We find that if a solitary wave exist in this system, the smaller grains have larger velocities and propagate longer distances than that of larger particles. The comparisons are given between the dusty plasma composed by different dust particles and the mono-sized dusty plasma.     

The Chinese connection between the pascal triangle and the solution of numerical equations of any degree

One of the significant contributions of Chinese mathematicians is the method of solving numerical equations of higher degree. A number of scholarly works have established similarities between ancient and medieval Chinese root-extraction procedures and Horner's method of solving a numerical equation of any degree. The conceptual development of the Chinese methods, which began with the procedures of extracting square and cube roots during the Han dynasty, culminated in the solution of higher numerical equations in the 13th and the beginning of the 14th centuries. This paper attempts to show the intimate role played by the triangular array of numbers (known in the West as the Pascal triangle) in the process of the development of the Chinese methods, especially when the original geometrical concept was being replaced by the more general algebraic method.     

Autonomous self-similar ordinary differential equations and the Painlevé connection

We demonstrate an intimate connection between nonlinear higher-order ordinary differential equations possessing the two symmetries of autonomy and self-similarity and the leading-order behaviour and resonances determined in the application of the Painlevé Test. Similar behaviour is seen for systems of first-order differential equations. Several examples illustrate the theory. In an integrable case of the ABC system the singularity analysis reveals a positive and a negative resonance and the method of leading-order behaviour leads naturally to a Laurent expansion containing both.     

Atom structures and Sahlqvist equations

This paper addresses the question for which varieties of boolean algebras with operators membership of an atomic algebra is determined by its atom structure . We prove a positive answer for conjugated Sahlqvist varieties; we also show that the conjugation condition is necessary. As a corollary to the positive result and a recent result by I. Hodkinson, we prove that the variety RRA of representable relation algebras, although canonical, cannot be axiomatized by Sahlqvist equations. Received February 21, 1996; accepted in final form October 1, 1997.     

A hybrid type of soliton equations with self-consistent sources: KP and Toda cases

Source generation procedure is applied to construct a hybrid type of soliton equations with self-consistent sources (SESCSs). The examples include the KP equation with self-consistent sources (KPESCS) and two-dimensional TodaESCS. One typical feature for this hybrid type of SESCSs is that soliton solutions of these new systems contain arbitrary functions of a linear combination of two independent variables, which is different from the normal SESCSs where soliton solutions only contain arbitrary functions of one independent variable. What's more, the obtained two hybrid SESCSs can be reduced to two different simpler SESCSs respectively.     

Scelernomic analysis of structures considering connection slip

A challenge in structural engineering of space and ground radio and light wave reflectors is to retain surface geometry to a high precision in the context of pinned connections and changing loads. This paper describes a direct approach to assessing worst-case geometric degradation of these structures for hypothesized slots in the members.The paper presents the structural model, an analysis procedure, and illustrative results. The model addresses articulated structures with slots small compared with member lengths. The analysis process implies stepwise linear behavior. The examples encompass determinate and indeterminate two-dimensional trusses.The approach results in an analysis process capable of predicting the extremes of slip accumulation. Like limit analysis, it provides behavior prediction in a finite number of steps with guaranteed success. Data from the sample analyses suggest that fabrication tolerances should be much less than allowable member elongations if long-time high precision is desired.     

Noncommutative Constrained KP Hierarchy and Multi-component Noncommutative Constrained KP Hierarchy

In this paper, the authors define the noncommutative constrained Kadomtsev-Petviashvili (KP) hierarchy and multi-component noncommutative constrained KP hierarchy. Then they give the recursion operators for the noncommutative constrained KP (NcKP) hierarchy and multi-component noncommutative constrained KP (NmcKP) hierarchy. The authors hope these studies might be useful in the study of D-brane dynamics whose noncommutative coordinates emerge from limits of the M theory and string theory.     

Connection between the kinetic equations and the equations of hydrodynamics

One investigates the Cauchy problem for the nonlinear Boltzmann equation

Quasi-symmetric functions and the KP hierarchy

Quasi-symmetric functions arise in an approach to solve the Kadomtsev-Petviashvili (KP) hierarchy. This moreover features a new nonassociative product of quasi-symmetric functions that satisfies simple relations with the ordinary product and the outer coproduct. In particular, supplied with this new product and the outer coproduct, the algebra of quasi-symmetric functions becomes an infinitesimal bialgebra. Using these results we derive a sequence of identities in the algebra of quasi-symmetric functions that are in formal correspondence with the equations of the KP hierarchy.     

Recursion and group structures of soliton equations

The recursion operator method for nonlinear evolution equations integrable by the inverse spectral transform method is discussed. This method enables us to present the integrable equations in a compact and convenient form and to construct the infinite-dimensional groups of general Bäcklund transformations and the infinite-dimensional symmetry groups for these equations. Adjoint representation of the spectral problems plays a central role in the recursion operator method. Nonlinear integrable equations in 1+1 and 1+2 dimensions are considered.     

Hamiltonian structures of the generalized cylindrical KdV equations

Hamiltonian structures of the cylindrical Korteweg-deVries equation and its higher order equations are found. The connection between the generalized C-KdV equation and nonisospectral problem is pointed out.Projects supported by the Science Fund of the Chinese Academy of Sciences.