All posts under category "Nonlinear Sciences / Nlin.SI"
Recent numerical work on the Zabusky--Kruskal experiment has revealed, amongst other things, the existence of hidden solitons in the wave profile. Here, using Osborne's nonlinear Fourier analysis, which is based on the periodic, inverse scattering transform, the hidden soliton hypothesis is corrobor
We consider two coupled phase oscillators in the presence of proportional ('common') and independent white noises. The global synchronization properties of the system are analytically studied via the Fokker-Planck equation. When the 'effective coupling' is big compared to the common noises, the form
Coupled second order nonlinear differential equations are of fundamental importance in dynamics. In this part of our study on the integrability and linearization of nonlinear ordinary differential equations we focus our attention on the method of deriving general solution of two coupled second order
Stochastic processes associated with traveling wave solutions of the sine-Gordon equation are presented. The structure of the forward Kolmogorov equation as a conservation law is essential in the construction and so is the traveling wave structure. The derived stochastic processes are analyzed numer
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)
Using a bidifferential graded algebra approach to integrable partial differential or difference equations, a unified treatment of continuous, semi-discrete (Ablowitz-Ladik) and fully discrete matrix NLS systems is presented. These equations originate from a universal equation within this framework,
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.
For the Harry Dym hierarchy, a non-standard Lax formulation is deduced from that of Korteweg-de Vries (KdV) equation through a reciprocal transformation. By supersymmetrizing this Lax operator, a new N=2 supersymmetric extension of the Harry Dym hierarchy is constructed, and is further shown to be l
We prove that in the Ryabov paper an application of the geometric Kharlamov method to the Goryachev system yields noncommutative 'new variables of separation' instead of the standard canonical variables of separation.
The history of structural optimization as an exact science begins possibly with the celebrated Lagrange problem: to find a curve which by its revolution about an axis in its plane determines the rod of greatest efficiency. The Lagrange hypothesis, that the optimal rod possesses the constant cross-se
The problem of a disc and a ball rolling on a horizontal plane without slipping is considered. Differential constrained equations are shown to be integrated when the trajectory of the point of contact is taken in a form of the natural equation, i.e. when the dependence of the curvature of the trajec
The generalized Bretherton equation is studied. The classification of the meromorphic traveling wave solutions for this equation is presented. All possible exact solutions of the generalized Brethenton equation are given.
This Letter presents a reduction of the lattice modified Korteweg-de-Vries equation that gives rise to a $q$-analogue of the sixth Painlev'e equation. This new approach allows us to give the first ultradiscrete Lax representation of an ultradiscrete analogue of the sixth Painlev'e equation.
The statistical distribution of levels of an integrable system is claimed to be a Poisson distribution. In this paper, we numerically generate an ensemble of N dimensional random diagonal matrices as a model for regular systems. We evaluate the corresponding nearest-neighbor spacing (NNS) distributi
Travelling waves and conservation laws are studied for a wide class of U(1)-invariant complex mKdV equations containing the two known integrable generalizations of the ordinary (real) mKdV equation. The main results on travelling waves include deriving new complex solitary waves and kinks that gener
The structure of integrable field theories in the presence of jump defects is discussed in terms of boundary functions under the Lagrangian formalism. Explicit examples of bosonic and fermionic theories are considered. In particular, the boundary functions for the N=1 and N=2 super sinh-Gordon model
We propose a modified condition of consistency on cubic lattices for some special classes of two-dimensional discrete equations and prove that the discrete nonlinear equations defined by determinants of matrices of orders N > 2 are consistent on cubic lattices in this sense.
In this short note, we construct mappings from one-dimensional integrable spinor BECs to matrix nonlinear Schr'odinger equation, and solve the Bogoliubov equation of these systems. A map of spin-$n$ BEC is constructed from the $2^n$-dimensional spinor representation of irreducible tensor operators o
Discrete Darboux-Manakov-Zakharov systems possess two distinct Hamiltonian forms. In the framework of discrete-differential geometry one Hamiltonian form appears in a geometry of circular net. In this paper a geometry of second form is identified.
We derive a Lagrangian based approach to study the compatible Hamiltonian structure of the dispersionless KdV and supersymmetric KdV hierarchies and claim that our treatment of the problem serves as a very useful supplement of the so-called r-matrix method. We suggest specific ways to construct resu
Motzkin numbers are derived from a special case of Random Domino Automaton - recently proposed toy model of earthquakes. An exact solution of the set of equations describing stationary state of Random Domino Automaton in 'inverse-power' case is presented. A link with Motzkin numbers allows to presen
In this paper, we report an interesting integrable equation that has both solitons and kink solutions. The integrable equation we study is $(frac{-u_{xx}}{u})_{t}=2uu_{x}$, which actually comes from the negative KdV hierarchy and could be transformed to the Camassa-Holm equation through a gauge tran
In the author's paper (Phys.Rev. D80, 041901(R) (2009)), the integrable structure of the symmetry reduced bosonic dynamics in the low energy heterotic string effective theory was presented. In that paper, for a complete system of massless bosonic fields which includes metric, dilaton field, antisymm
We consider two-component integrable generalizations of the dispersionless 2DTL hierarchy connected with non-Hamiltonian vector fields, similar to the Manakov-Santini hierarchy generalizing the dKP hierarchy. They form a one-parametric family connected by hodograph type transformations. Generating e
We consider non-vanishing boundary conditions (NVBC) for the NLS model [6,7,27] in the context of the hybrid dressing transformation and $tau$-function approach. In order to write the NLS model in a suitable form to deal with non-vanishing boundary conditions it is introduced a new spectral paramete
On a basis of theory of Riemann extension of the space of constant affine connection associated with the R'ossler system of equations relations between its parameters are investigated.
We consider a version of random motion of hard core particles on the semi-lattice $ 1, 2, 3,...$, where in each time instant one of three possible events occurs, viz., (a) a randomly chosen particle hops to a free neighboring site, (b) a particle is created at the origin (namely, at site 1) provided
We study a (2+1)-dimensional system that can be viewed as an infinite number of O(3) sigma-fields coupled by a nearest-neighbour Heisenberg-like interaction. We reduce the field equations of this model to an integrable system that is closely related to the two-dimensional relativistic Toda chain and
Particular solutions of the Benney equations are constructed. Their properties are discussed.
We establish a connection between the hyperbolic relativistic Calogero-Moser systems and a class of soliton solutions to the Tzitzeica equation (aka the Dodd-Bullough-Zhiber-Shabat-Mikhailov equation). In the 6N-dimensional phase space $Omega$ of the relativistic systems with 2N particles and $N$ an
We establish unique continuation for various discrete nonlinear wave equations. For example, we show that if two solutions of the Toda lattice coincide for one lattice point in some arbitrarily small time interval, then they coincide everywhere. Moreover, we establish analogous results for the Toda,
In this paper, we propose a two-component generalization of the generalized Hunter-Saxton equation obtained in cite{BLG2008}. We will show that this equation is a bihamiltonian Euler equation, and also can be viewed as a bi-variational equation.
We study localized modes on a single Ablowitz-Ladik impurity embedded in the bulk or at the surface of a one-dimensional linear lattice. Exact expressions are obtained for the bound state profile and energy. Dynamical excitation of the localized mode reveals exponentially-high amplitude oscillations
In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP) and Harry-Dym under $n$-reduction. This shows a new inherent relationship between them. To illustr
We investigate a tropical analogue of the Hessian group $G_{216}$, the group of linear automorphisms acting on the Hesse pencil. Through the procedure of ultradiscretization, the group law on the Hesse pencil reduces to that on the tropical Hesse pencil. We then show that the dihedral group $mathcal
We present algebraic construction of Darboux matrices for 1+1-dimensional integrable systems of nonlinear partial differential equations with a special stress on the nonisospectral case. We discuss different approaches to the Darboux-Backlund transformation, based on different lambda-dependencies of
We develop the Baxterization approach to (an extension of) the quantum group GL_q(2). We introduce two matrices which play the role of spectral parameter dependent L-matrices and observe that they are naturally related to two different comultiplications. Using these comultiplication structures, we f
The symmetry approach is used for classification of integrable isotropic vector Volterra lattices on the sphere. The list of integrable lattices consists mainly of new equations. Their symplectic structure and associated PDE of vector NLS-type are discussed.
We study dynamics of two coupled periodically driven oscillators. Important example of such a system is a dynamic vibration absorber which consists of a small mass attached to the primary vibrating system of a large mass. Periodic solutions of the approximate effective equation are determined with
When written in terms of $vartheta$-functions, the classical Frobenius-Stickelberger pseudo-addition formula takes a very simple form. Generalizations of this functional identity are studied, where the functions involved are derivatives (including derivatives with respect to the modular parameter) o
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to overcome such shortcomings, a fractional variational iteratio
The 'pseudodual' of Ward's modified chiral model is a dispersionless limit of the matrix Kadomtsev-Petviashvili (KP) equation. This relation allows to carry solution techniques from KP over to the former model. In particular, lump solutions of the su(m) model with rather complex interaction patterns
We consider a system of birational functional equations (BFEs) (or finite-difference equations at w=m in Z) for functions y(w) of the form: y(w+1)=F_n(y(w)), y(w):C to C^N, n=deg(F_n(y)), F_n in (bf Bir}(C^N), where the map F_n is a given birational one of the group of all automorphisms of C^N to C^
The link between the short wave model of the Camassa-Holm equation (SCHE) and bilinear equations of the two-dimensional Toda lattice (2DTL) is clarified. The parametric form of N-cuspon solution of the SCHE in Casorati determinant is then given. Based on the above finding, integrable semi-discrete a
This paper explores integrable structures of a generalized melting crystal model that has two $q$-parameters $q_1,q_2$. This model, like the ordinary one with a single $q$-parameter, is formulated as a model of random plane partitions (or, equivalently, random 3D Young diagrams). The Boltzmann weigh
The superposition of the Kepler-Coulomb potential on the 3D Euclidean space with three centrifugal terms has recently been shown to be maximally superintegrable [Verrier P E and Evans N W 2008 J. Math. Phys. 49 022902] by finding an additional (hidden) integral of motion which is quartic in the mome
The proposed model modifies option pricing formulas for the basic case of log-normal probability distribution providing correspondence to formulated criteria of efficiency and completeness. The model is self-calibrating by historic volatility data; it maintains the constant expected value at maturit
Differential-difference equation $frac{d}{dx}t(n+1,x)=f(x,t(n,x),t(n+1,x),frac{d}{dx}t(n,x))$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$ is studied. We call an equation of such kind Darboux integrable, if there exist two functions (called integrals) $F$ and $I$
We give a simple algebraic proof that the two different Lax pairs for the Kac-van Moerbeke hierarchy, constructed from Jacobi respectively super-symmetric Dirac-type difference operators, give rise to the same hierarchy of evolution equations. As a byproduct we obtain some new recursions for computi
We investigate the spatial asymptotics of decaying solutions of the Toda hierarchy and show that the asymptotic behaviour is preserved by the time evolution. In particular, we show that the leading asymptotic term is time independent. Moreover, we establish infinite propagation speed for the Toda la
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