Nonlinear Sciences

A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and delayed time reversal is derived. Various exact solutions of the VCmKdV equation, including elliptic periodic waves, solitary waves and interaction solutions between solitons and periodic waves are obtained and analyzed graphically. As an illustration, an approximate solution of the original nonlocal two-layer fluid system is also given.

The celebrated (1+1)-dimensional Korteweg de-Vries (KdV) equation and its (2+1)-dimensional extention, the Kadomtsev-Petviashvili (KP) equation, are two of the most important models in physical science. The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation. A novel (2+1)-dimensional KdV extension, the cKP3-4 equation, is obtained by combining the third member (KP3, the usual KP equation) and the fourth member (KP4) of the KP hierarchy. The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair. The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable. Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations. For instance, the soliton molecules and the missing D'Alembert type solutions (the arbitrary travelling waves moving in one direction with a fixed model dependent velocity) including periodic kink molecules, periodic kink-antikink molecules, few cycle solitons and envelope solitons are existed for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.

We present a novel 8-parameter integrable map in R 4 . The map is measure-preserving and possesses two functionally independent 2-integrals, as well as a measure-preserving 2-symmetry.

Discrete Painlevé equations are nonlinear, nonautonomous difference equations of second-order. They have coefficients that are explicit functions of the independent variable n and there are three different types of equations according to whether the coefficient functions are linear, exponential or elliptic functions of n . In this paper, we focus on the elliptic type and give a review of the construction of such equations on the E 8 lattice. The first such construction was given by Sakai \cite{SakaiH2001:MR1882403}. We focus on recent developments giving rise to more examples of elliptic discrete Painlevé equations.

We study the simple-looking scalar integrable equation f xxt −3( f x f t −1)=0 , which is related (in different ways) to the Novikov, Hirota-Satsuma and Sawada-Kotera equations. For this equation we present a Lax pair, a Bäcklund transformation, soliton and merging soliton solutions (some exhibiting instabilities), two infinite hierarchies of conservation laws, an infinite hierarchy of continuous symmetries, a Painlevé series, a scaling reduction to a third order ODE and its Painlevé series, and the Hirota form (giving further multisoliton solutions).

As a generalization of the integrable extended Toda hierarchy and a reduction of the extended multicomponent Toda hierarchy, from the point of a commutative subalgebra of gl(2,C) , we construct a strongly coupled extended Toda hierarchy(SCETH) which will be proved to possess a Virasoro type additional symmetry by acting on its tau-function. Further we give the multi-fold Darboux transformations of the strongly coupled extended Toda hierarchy.

We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation is linearizable if it admits an abelian Lie symmetry algebra that is of dimension greater than or equal to five or of dimension greater than or equal to three with rank one.

In previous work, we have considered Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. Previously our Hamiltonians have represented free motion, but here we consider the problem of adding potential functions in the presence of symmetry. Separable potentials in the 3 dimensional space reduce to 3 or 4 parameter potentials for Darboux-Koenigs Hamiltonians. Other 3D coordinate systems reveal connections between Darboux-Koenigs and other well known super-integrable Hamiltonians, such as the Kepler problem and isotropic oscillator.

We study the symmetry and integrability of a Generalized Modified Camassa-Holm Equation (GMCH) of the form u t − u xxt +2n u x ( u 2 − u 2 x ) n−1 (u− u xx ) 2 +( u 2 − u 2 x ) n ( u x − u xxx )=0. We observe that for increasing values of n∈N , N denotes natural number, the above equation gives a family of equations in which nonlinearity is rapidly increasing as n increases. However, this family has similar form of symmetries except the values of n . Interestingly the resultant second-order nonlinear ODE which is to be obtained from GMCH equation has eight dimensional symmetries. Hence the second-order nonlinear ODE is linearizable. Finally we conclude that the resultant second-order nonlinear ordinary differential equation which is obtained from the family of GMCH passes the Painlevé Test also it posses the similar form of leading order, resonances and truncated series solution too.

In the article a classification method for nonlinear integrable equations with three independent variables is discussed based on the notion of the integrable reductions. We call the equation integrable if it admits a large class of reductions being Darboux integrable systems of hyperbolic type equations with two independent variables. The most natural and convenient object to be studied within the frame of this scheme is the class of two dimensional lattices generalizing the well-known Toda lattice. In the present article we deal with the quasilinear lattices of the form u n,xy =α( u n+1 , u n , u n−1 ) u n,x u n,y +β( u n+1 , u n , u n−1 ) u n,x +γ( u n+1 , u n , u n−1 ) u n,y +δ( u n+1 , u n , u n−1 ) . We specify the coefficients of the lattice assuming that there exist cutting off conditions which reduce the lattice to a Darboux integrable hyperbolic type system of the arbitrarily high order. Under some extra assumption of nondegeneracy we described the class of the lattices integrable in the sense indicated above. There are new examples in the obtained list of chains.