Journal of Nonlinear Mathematical Physics

Using our previous work on reflectionless analytic difference operators and a nonlocal Toda equation, we introduce analytic versions of the Volterra and Kac-van Moerbeke lattice equations. The real-valued N-soliton solutions to our nonlocal equations corrspond to self-adjoint reflectionless analytic...

The boundary value problems for the two-dimensional, steady, irrotational flow of a frictionless, incompressible fluid past a wedge and a circular cylinder are considered. It is shown that by considering first the invariance of the boundary condition we are able to obtain a transformation group that...

The modified Kadomtsev–Petviashvili (mKP) equation is revisited from two 1 + 1-dimensional integrable equations whose compatible solutions yield a special solution of the mKP equation in view of a transformation. By employing the finite-order expansion of Lax matrix, the mKP equation is reduced to three...

Certain techniques to obtain properties of the zeros of polynomials satisfying second-order ODEs are reviewed. The application of these techniques to the classical polynomials yields formulas which were already known; new are instead the formulas for the zeros of the (recently identified, and rather...

For the first time we show that the quasiclassical limit of the symmetry constraint of the Sato operator for the KP hierarchy leads to the generalized Zakharov reduction of the Sato function for the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry constraint of the dKP...

Analogies in the spectral study of dissipative Schrödinger operator and Boltmann transport operator are analyzed. Scattering theory technique together with functional model approach are applied to construct spectral representtions for these operators.

Möbius invariant versions of the discrete Darboux, KP, BKP and CKP equations are derived by imposing elementary geometric constraints on an (irregular) lattice in a three-dimensional Euclidean space. Each case is represented by a fundamental theorem of plane geometry. In particular, classical theorems...

A new class of integrable Newton systems in Rn is presented. They are characterized by the existence of two quadratic integrals of motion of so-called cofactor type, and are therefore called cofactor pair systems. This class includes as special cases conservative systems separable in elliptic or parabolic...

A fully braided analog of the Faddeev-Reshetikhin-Takhtajan construction of a quasitriangular bialgebra A(X, R) is proposed. For a given pairing C, the factor-algebra A(X, R; C) is a dual quantum braided group. Corresponding inhomogeneous quantum group is obtained as a result of generalized bosonization....

The manuscript is devoted to nonisentropic solutions of simple wave type of the gas dynamics equations. For an isentropic flow these equations (in one-dimensional and steady two-dimensional cases) are reduced to the equations written in the Riemann invariants. The system written in the Riemann invariants...

In this work, we give a method to construct compatible Poisson structures on Lie groups by means of structure constants of their Lie algebras and some vector field. In this way we calculate some compatible Poisson structures on low-dimensional Lie groups. Then, using a theorem by Magri and Morosi, we...

In terms of the operator Nambu 3-bracket and the Lax pair (L, Bn) of the KP hierarchy, we propose the generalized Lax equation with respect to the Lax triple (L, Bn, Bm). The intriguing results are that we derive the KP equation and another integrable equation in the KP hierarchy from the generalized...

An invariant of three-dimensional orientable manifolds is built on the base of a slution of pentagon equation expressed in terms of metric characteristics of Euclidean tetrahedra.

The transition from Eulerian to Lagrangian coordinates is a nonlocal transformation. In general, isomorphism should not take place between basic Lie groups of studied equations. Besides, in the case of plane and rotational symmetric motion hydrodynamic equations in Lagrangian coordinates are partially...

For dynamical systems defined by vector fields over a compact invariant set, we introduce a new class of approximated first integrals based on finite time averages and satisfying an explicit first order partial differential equation. These approximated first integrals can be used as finite time indicators...

In this paper, we are concerned with a modified complex short pulse (mCSP) equation of defocusing type. Firstly, we show that the mCSP equation is linked to a complex coupled dispersionless equation of defocusing type via a hodograph transformation, thus, its Lax pair can be deduced. Then the bilinearization...

The problem of the classification of integrable truncations of the Toda chain is dicussed. A new example of the cutting off constraint is found.

The generalized Zakharov­Shabat systems with complex-valued Cartan elements and the systems studied A.V. Mikhailov, and later on by Caudrey, Beals and Coifman (CBC systems), and their gauge equivalent are studied. This includes: the properties of fundamental analytical solutions (FAS) for the gauge-equivalent...

We consider the class of autonomous systems x = f(x), where x R2n , f C1 (R2n ) whose phase portrait is a Cartesian product of n two-dimensional centres. We also consider perturbations of this system, namely x = f(x) + g(t, x), where g C1 (R × R2n ) and g is asymptotically small, that is g 0 as t + uniformly...

A new class of integro-partial differential equation models is derived for the prediction of granular flow dynamics. These models are obtained using a novel limiting averaging method (inspired by techniques employed in the derivation of infinite-dimensional dynamical systems models) on the Newtonian...

Under investigation in this paper, with symbolic computation, is the Whitham–Broer–Kaup (WBK) system for the dispersive long waves in the shallow water small-amplitude regime. N-fold Darboux transformation (DT) for a spectral problem associated with the WBK system is constructed. Odd-soliton solutions...

A solution of the KP-hierarchy can be given by the -function or the Baker function associated to an element of the Grassmannian Gr(L2 (S1 )) consisting of some subspaces of the space L2 (S1 ) of square-integrable functions on the unit circle S1 . The Krichever map associates an element W Gr(L2 (S1 ))...

Magnetic curves with respect to the canonical contact structure of the space Sol3 are investigated.

We study the problem of non-integrability (integrability) of cosmological dynamcal systems which are given in the Hamiltonian form with indefinite kinetic energy form T = 1 2 g(v, v), where g is a two-dimensional pseudo-Riemannian metric with a Lorentzian signature (+, -), and v TxM is a tangent vector...

In this paper, we present all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space ℝn of arbitrary finite dimension n. Using the invariance of the Yang-Mills equations under the orthogonal transformations of coordinates...

We consider the Hamiltonian function defined by the cubic polynomial H=12(px2+py2)+12(x2+y2)+A3x3+Bxy2+Dx2y , where A, B, D ∈ ℝ are parameters and so H is an extension of the well known Hénon-Heiles problem. Our main contribution for D ≠ 0, A + B ≠ 0 and other technical restrictions are in three...

Bäcklund transformations, which are relations among solutions of partial differential equations­usually nonlinear­have been found and applied mainly for systems with two independent variables. A few are known for equations like the Kadomtsev-Petviashvili equation [1], which has three independent variables,...

We describe the derivation of a formalism in the context of the governing equations for two-dimensional water waves propagating over a flat bed in a flow with non-vanishing vorticity. This consists in providing a Hamiltonian structure in terms of two variables which are scalar functions.

The Hilbert space representations of a class of commutation relations associated with a Möbius transformation is studied using results on convergence of continued fractions.

Analyzing the spectrum of the Schrödinger-Pauli Hamiltonian for a particle of spin s > 1/2 we find that some energy levels are degenerated while the other are not. We investigate the symmetry (which is neither super- nor parasymmetry) causing this specific degeneration.

We present the Lie point symmetries admitted by third order partial differential equations (PDEs) which model the pressure of a visco-elastic liquid with relaxation which filtrates through a porous medium. The symmetries are used to construct reductions of the PDEs to ordinary differential equations...

We consider spacetime to be a connected real 4-manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is quadratic in curvature...

A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is based on an extension of the original construction of Gelfand and Dorfman which gave examples of Novikov algebras in terms of structures defined...

New Cellular Automata associated with the Schroedinger discrete spectral problem are derived. These Cellular Automata possess an infinite (countable) set of constants of motion.

We use a method inspired by the Jacobi last multiplier [M.C. Nucci, Jacobi last multiplier and Lie symmetries: a novel application of an old relationship, J. Nonlinear Math. Phys. 12, 284-304 (2005)] in order to find Lie symmetries of a Painlev ?e-type equation without Lie point symmetries.

In this second paper on the method of deriving linearizing transformations for nonlinear ODEs, we extend the method to a set of two coupled second-order nonlinear ODEs. We show that beside the conventional point, Sundman and generalized linearizing transformations one can also find a large class of mixed...

In this paper one considers the problem of finding solutions to a number of Todtype hierarchies. All of them are associated with a commutative subalgebra of the k×k-matrices. The first one is formulated in terms of upper triangular Z×Z-matrices, the second one in terms of lower triangular ones and the...

For systems of particles with singular magnetic interation for special choice of a selfadjoint extension of the Hamiltoniam equilibrium reduced density matrices are calculated in the thermodynamic limit for simplest pair magnetic potentials.

The three ansatzes are constructed for the nonlinear Dirac equation.

A supersymmetric integrable equation in (2+1) dimensions is constructed by means of the approach of the homogenous space of the super Lie algebra, where the super Lie algebra osp(3/2) is considered. For this (2+1) dimensional integrable equation, we also derive its Bäcklund transformation.

It is shown here that the possibility of the existence of new (2 + 1) dimensional intgrable equations of the modified KdV equation using the Painlevé test.

A dispersionless integrable system with repeated eigenvalues is presented. For N 3 components the system has no local Hamiltonian structure. Infinitely many simple compatible non-local Hamiltonian structures are given, using a result of Ferapontov.

In this paper (as in previous ones) we identify and investigate polynomials pn(ν)(x) featuring at least one additional parameter ν besides their argument x and the integer n identifying their degree. They are orthogonal (provided the parameters they generally feature fit into appropriate ranges)...

In this article, given a regular Lagrangian system L on the phase space TM of the configuration manifold M and a 1-parameter group G of transformations of M whose lifting to TM preserve the canonical symplectic dynamics associated to L, we determine conditions so that its infinitesimal generator produces...

An isochronous dynamical system is characterized by the existence of an open domain of initial data such that all motions evolving from it are completely periodic with a fixed period (independent of the initial data). Taking advantage of a recently introduced trick, two new Hamiltonian classes of such...

We describe Jacobi's method for integrating the Hamilton-Jacobi equation and his discovery of elliptic coordinates, the generic separable coordinate systems for real and complex constant curvature spaces. This work was an essential precursor for the modern theory of second-order superintegrable systems...

A new approach to discrete KP equation is considered, starting from the GelfanZakhharevich theory for the research of Casimir function for Toda Poisson pencil. The link between the usual approach through the use of discrete Lax operators, is emphsized. We show that these two different formulations of...

This paper presents a new approach to the optimization problem for the bilinear system x = {x, } (1) based on the well-known method of continuous parametric group reconstruction using of its structure constants defined by the Brockett equation z = {z, }. (2) Here x is the system state vector, {·, ·}...

Recently using a Madelung fluid description a connection between envelope-like solutions of NLS-type equations and soliton-like solutions of KdV-type equations was found and investigated by R. Fedele et al. (2002). A similar discussion is given for the class of derivative NLS-type equations. For a motion...

We characterize the equations in the class 𝒜 of the second-order ordinary differential equations ẍ = M(t, x, ẋ) which have first integrals of the form A(t, x)ẋ + B(t, x). We give an intrinsic characterization of the equations in 𝒜 and an algorithm to calculate explicitly such first integrals. Although...

The nonlinear Schrödinger equation is studied for a periodic sequence of delta-potentials (a delta-comb) or narrow Gaussian potentials. For the delta-comb the time-independent nonlinear Schrödinger equation can be solved analytically in terms of Jacobi elliptic functions and thus provides useful insight...

The recursion operators and symmetries of nonautonomous, (1 + 1) dimensional itegrable evolution equations are considered. It has been previously observed that the symmetries of the integrable evolution equations obtained through their recursion oerators do not satisfy the symmetry equations. There have...

We construct nonlinear representations of the Poincaré, Galilei, and conformal algebras on a set of the vector-functions = (E, H). A nonlinear complex equation of Euler type for the electromagnetic field is proposed. The invariance algebra of this equation is found.

Lie group analysis is applied to a core group model for sexually transmitted disease formulated by Hadeler and Castillo-Chavez [Hadeler K P and Castillo-Chavez C, A core group model for disease transmission, Math. Biosci. 128 (1995), 41­55]. Several instances of integrability even linearity are found...

We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity symmetries. The computtion of first integrals gives in the most general...

It is shown that invertible linearizing transformations of the one-dimensional Ito stochastic differential equations cast the associated Fokker-Planck equation to the heat equation. This leads to the time-dependent exact solutions to the Fokker-Planck equations via inverse transformations. To obtain...

This study focuses on the analysis of Ramsey dynamical model with current Hamiltonian defining an optimal control problem in a neoclassical growth model by utilizing Lie group theory. Lie point symmetries of coupled nonlinear first-order ordinary differential equations corresponding to first-order conditions...

We consider the cubic and quartic Hénon-Heiles Hamiltonians with additional inverse square terms, which pass the Painlevé test for only seven sets of coefficients. For all the not yet integrated cases we prove the singlevaluedness of the general solution. The seven Hamiltonians enjoy two properties:...

The Laplacian growth problem in the limit of zero surface tension is proved to be equivalent to finding a particular solution to the dispersionless Toda lattice hierarchy. The hierarchical times are harmonic moments of the growing domain. The Laplacian growth equation itself is the quasiclassical version...

Switching model with one predator and two prey species is considered. The prey species have the ability of group defence. Therefore, the predator will be attracted towards that habitat where prey are less in number. The stability analysis is carried out for two equilibrium values. The theoretical results...

A list of twenty five integrable vectorial evolutionary equations of the third order is presented. Each equation from the list possesses higher symmerties and higher conservation laws.

We consider the Riemann–Hilbert method for initial problem of the vector Gerdjikov–Ivanov equation, and obtain the formula for its N-soliton solution, which is expressed as a ratio of (N + 1) × (N + 1) determinant and N × N determinant. Furthermore, by applying asymptotic analysis, the simple elastic...

In our recent paper [1], we gave a complete description of symmetry reduction of four Lax-integrable (i.e., possessing a zero-curvature representation with a non-removable parameter) 3-dimensional equations. Here we study the behavior of the integrability features of the initial equations under the reduction...

This paper contains a list of known integrable systems. It gives their recursion-, Hamiltonian-, symplectic- and cosymplectic operator, roots of their symmetries and their scaling symmetry.

A review of a new separability theory based on degenerated Poisson pencils and the scalled separation curves is presented. This theory can be considered as an alternative to the Sklyanin theory based on Lax representations and the so-called spectral curves.

In the literature, the generalized Sundman transformation has been used for obtaining necessary and sufficient conditions for a single second- and third-order ordinary differential equation to be equivalent to a linear equation in the Laguerre form. As far as we are aware, the generalized Sundman transformation...

The symmetry reduction algorithm for ordinary differential equations due to Sophus Lie is revisited using the method of equivariant moving frames. Using the recurrence formulas provided by the theory of equivariant moving frames, computations are performed symbolically without relying on the coordinate...

We consider discontinuous flows of relativistic magnetic fluid with a general equation of state that is not supposed to be normal in the sense of Bethe and Weyl. The criteria of admissibility of the shock waves without a supposition of the relativistic version of the convexity condition are obtained....

Let us consider the multidimensional nonlinear system of heat equations u0 = f(v)u; v0 = u, (1) where u = u(x) R1, v = v(x) R1, x = (x0, x) R1+3, is the Laplace operator, f(v) is an arbitrary differentiable function. In this paper the classification of symmetry properties of equations (1) is investigated...

We present a geometric description, based on the affine Weyl group E (1) 6 , of two discrete analogues of the Painlevé VI equation, known as the asymmetric q-PV and asymmetric d-PIV. This approach allows us to describe in a unified way the evolution of the mapping along the independent variable and along...

Using a complex representation of planar motions, we show that the dynamical conserved quantities associated to the isotropic harmonic oscillator (Fradkin–Jauch–Hill tensor) and to the Kepler's problem (Laplace–Runge–Lenz vector) find a very simple and natural interpretation. In this frame we also...

In this paper we consider the Holm-Staley b-family of equations in the Sobolev spaces Hs(ℝ) for s>3/2. Using a geometric approach we show that, for any value of the parameter b, the corresponding solution map,u(0)↦u(T ), is nowhere locally uniformly continuous.

In this talk I am going to present a brief review of some key ideas and methods which were given start and were developed in Kyiv, at the Institute of Mathematics of National Academy of Sciences of Ukraine during recent years.

The present paper concerns the study of a Riemann problem for the conservation law ut + [ϕ(u)]x = kδ(x − vt) where x, t, k, v and u = u(x, t) are real numbers. We consider ϕ an entire function taking real values on the real axis and δ stands for the Dirac measure. Within a convenient space of distributions...

Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and...

In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows. By introducing an auxiliary parameter, whose flow corresponds to the so-called squared eigenfunction symmetry of KP...

We give a complete description of differential operators generating a given bracket. In particular we consider the case of Jacobi-type identities for odd operators and brackets. This is related with homotopy algebras using the derived bracket construction.

The singularity analysis is carried out for a system of four first-order quadratic ODEs with a parameter, which was proposed recently by Golubchik and Sokolov. A transfomation of dependent variables is revealed by the analysis, after which the transformed system possesses the Painlevé property and does...

Over algebraically closed fields of characteristic 2, the analogs of the orthogonal, symplectic, Hamiltonian, Poisson, and contact Lie superalgebras are described. The number of non-isomorphic types, and several properties of these algebras are unexpected, for example, interpretation in terms of exterior...

The nonlinear Lorentz-Dirac equation of motion for charged particle, if one takes into account radiative friction can be written in dimensionless variables. Then, there is a possibility of introducing the constant of fine structure and following approximate solving it. This result may be used for more...

In the present work we discuss arguments in favour of the view that massive fermions represent dislocations (i.e. topological solitons) in discrete space-time, with Burgers vectors parallel to the axis of time. If we assume that the symmetrical parts of tensors of distortions (i.e. derivatives of atomic...

We apply the similarity method based on a Lie group to a nonlinear model of the heat equation and find its Lie algebra.The optimal system of the model is contructed from the Lie algebra. New classes of similarity solutions are obtained.

Among simple Z-graded Lie superalgebras of polynomial growth, there are several which have no Cartan matrix but, nevertheless, have a quadratic even Casimir element C2: these are the Lie superalgebra kL (1|6) of vector fields on the (1|6)-dimensional supercircle preserving the contact form, and the series:...

A family of asymptotic solutions at infinity for a system of ordinary differential equations is considered. Existence of exact solutions which have these asymptotics is proved.

New completely integrable generalized C. Neumann systems on several symplectic submanifolds are presented, and the relations between the generalized C. Neumann systems and the spectral problems are further discussed in this paper. In particular, a new eigenvalue problem is proposed in Part 3.3.

In this article, we explain how to extend the Lie symmetry analysis method for n-coupled system of fractional ordinary differential equations in the sense of Riemann-Liouville fractional derivative. Also, we systematically investigated how to derive Lie point symmetries of scalar and coupled fractional...

The existence of a new type of cusped solitary wave, which decays algebraically at infinity, for a nonlinear equation modeling the free surface evolution of moderate amplitude waves in shallow water is established by employing qualitative analysis for differential equations. Furthermore, the exact parametric...

The multiplication in the Virasoro algebra [ep, eq] = (p - q)ep+q + p3 - p p+q, p, q Z, [, ep] = 0, comes from the commutator [ep, eq] = ep eq - eq ep in a quasiassociative algebra with the multiplication ep eq = q(1 + q)

For two-dimensional lattice equations the standard definition of integrability is that it should be possible to extend the map consistently to three dimensions, i.e., that it is "consistent around a cube" (CAC). Recently Adler, Bobenko and Suris conducted a search based on this principle, together with...

In this paper we further investigate some applications of Nambu mechanics in hydrdynamical systems. Using the Euler equations for a rotating rigid body Névir and Blender [J. Phys. A 26 (1993), L1189­L1193] had demonstrated the connection btween Nambu mechanics and noncanonical Hamiltonian mechanics....

It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quagraphs has been recently discovered by A. Bobenko and...

An influence of the quantum potential on the Chern­Simons solitons leads to quatization of the statistical parameter = me2 /g, and the quantum potential strenght s = 1 - m2 . A new type of exponentially localized Chern­Simons solitons for the Bloch electrons near the hyperbolic energy band boundary are...

The family of simple quasilinear systems in one space variable is partitioned into classes of commuting flows, i.e., symmetry classes. The systems in a symmetry class have the same zeroth order conserved densities and the same Hamiltonian structure. The zeroth and first order conservation laws and the...

We study the general applicability of the Clarkson­Kruskal's direct method, which is known to be related to symmetry reduction methods, for the similarity solutions of nonlinear evolution equations (NEEs). We give a theorem that will, when satisfied, immediately simplify the reduction procedure or ansatz...

The modified discrete KP equation is the Bäcklund transformation for the Hirota’s discrete KP equation or the Hirota-Miwa equation. We construct the modified discrete KP equation with self-consistent sources via source generation procedure and clarify the algebraic structure of the resulting coupled...

Approximate Lie symmetries of the Navier-Stokes equations are used for the applica- tions to scaling phenomenon arising in turbulence. In particular, we show that the Lie symmetries of the Euler equations are inherited by the Navier-Stokes equations in the form of approximate symmetries that allows to...

Quasiclassical generalized Weierstrass representation for highly corrugated surfaces in R3 with slow modulation is proposed. Integrable deformations of such surfaces are described by the dispersionless modified Veselov-Novikov hierarchy.

Boundary value problems for the nonlinear Schrödinger equation formulated on the half-line can be analyzed by the Fokas method. For the Dirichlet problem, the most difficult step of this method is the characterization of the unknown Neumann boundary value. For the case that the Dirichlet datum consists...

It is well-known that the basic difficulty in studying the initial boundary value prolems for linear and nonlinear PDEs is the presence, in any method of solution, of unknown boundary values. In the first part of this paper we review two spectral methods in which the above difficulty is faced in different...

Data from many sources indicate that the Earth ecological crisis might not wait till distant future. To avert it, some difficult truth must be accepted and adequate steps taken. One of them is the strict protection of the world forests, even at the cost of the short term economic growth.