Journal of Nonlinear Mathematical Physics

With the aid of symbolic computation by Maple, a class of third-order nonlinear evolution equations admitting invariant subspaces generated by solutions of linear ordinary differential equations of order less than seven is analyzed. The presented equations are either solved exactly or reduced to finite-dimensional...

A two-phase free boundary problem associated with nonlinear heat conduction is cosidered. The problem is mapped into two one-phase moving boundary problems for the linear heat equation, connected through a constraint on the relative motion of their moving boundaries. Existence and uniqueness of the solution...

The group invariant solution for the stream function and the effective viscosity of a two-dimensional turbulent free jet are derived. Prandtl’s hypothesis is not imposed. When the eddy viscosity is constant across the jet it is found that the mean velocity profile is the same as that of a laminar jet...

In this paper we construct a new solution which represents Pollard-like three-dimensional nonlinear geophysical internal water waves. The Pollard-like solution includes the effects of the rotation of Earth and describes the internal water wave which exists at all latitudes across Earth and propagates...

We present a detailed computation leading to an explicit formula for the fourth Hamitonian in the series of constants of motion with which any flow of the Camassa-Holm hierarchy is equipped, and explain the inherent difficulties in achieving such explicit expressions for invariants higher in the series.

The inheritance of symmetries of partial differential equations occurs in a different manner from that of ordinary differential equations. In particular, the Lie algebra of the symmetries of a partial differential equation is not sufficient to predict the symmetries that will be inherited by a resulting...

The Hirota equation is reduced to a pair of complex Finite-dimensional Hamiltonian Systems (FDHSs) with real-valued Hamiltonians, which are proven to be completely integrable in the Liouville sense. It turns out that involutive solutions of the complex FDHSs yield finite parametric solutions of the Hirota...

Let F(Sn ) be the space of tensor densities on Sn of degree . We consider this space as an induced module of the nonunitary spherical series of the group SO0(n+1, 1) and classify (so(n+1, 1), SO(n+1))-simple and unitary submodules of F(Sn ) as a function of .

The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-Bénard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-Bénard experiments with Prandtl number close to one, we prove the existence of...

Boundary value problems for integrable nonlinear evolution PDEs, like the modified KdV equation, formulated on the half-line can be analyzed by the so-called unified transform method. For the modified KdV equation, this method yields the solution in terms of the solution of a matrix Riemann-Hilbert problem...

It is known that many equations of interest in Mathematical Physics display solutions which are only asymptotically invariant under transformations (e.g. scaling and/or translations) which are not symmetries of the considered equation. In this note we extend the approach to asymptotic symmetries for...

Conditional symmetry of the nonlinear gas filtration equation is studied. The operators obtained enabled to constract ansatzes reducing this equation to ordinary differential equations and to obtain its exact solutions.

We study the effects of third order dispersion (TOD) on the collision of wavelength division multiplexed solitons in periodic dispersion maps. The analysis is based on a proposed ODE model obtained using the variational method which takes into account third order dispersion. The impact of TOD on the...

The classical water-wave problem is described, and two parameters (ε-amplitude; δ-long wave or shallow water) are introduced. We describe various nonlinear problems involving weak nonlinearity (ε → 0) associated with equations of integrable type (“soliton” equations), but with vorticity. The familiar...

Systems of two nonlinear ordinary differential equations of the first order admitting nonlinear superpositions are investigated using Lie’s enumeration of groups on the plane. It is shown that the systems associated with two-dimensional Vessiot–Guldberg–Lie algebras can be integrated by quadrature upon...

In this article we derive explicit solutions of the matrix integrable discrete nonlinear Schrödinger equation by using the inverse scattering transform and the Marchenko method. The Marchenko equation is solved by separation of variables, where the Marchenko kernel is represented in separated form, using...

In this paper, we apply Fokas unified method to study initial-boundary value problems for the two-component Gerdjikov-Ivanov equation formulated on the finite interval with 3×3 Lax pairs. The solution can be expressed in terms of the solution of a 3×3 Riemann-Hilbert problem. The relevant jump matrices...

We prove reality of the spectrum for a class of PT - symmetric, non self-adjoint quantum nonlinear oscillators of the form H = p2 + P(q) + igQ(q). Here P(q) is an even polynomial of degree 2p positive at infinity, Q(q) an odd polynomial of degree 2r - 1, and the conditions p > 2r, |g| 0 hold.

We derive and discuss equations of motion of infinitesimal affinely-rigid body moving in Riemannian spaces. There is no concept of extended rigid and affinely rigid body in a general Riemannian space. Therefore the gyroscopes with affine degrees of freedom are described as moving bases attached to the...

In the space of complex-valued smooth functions on S2 × S1, we explicitly realize a Weil representation of the real Lie algebra sp(4) by means of differential generators. This representation is a rare example of highest weight irreducible representation of sp(4) all whose weight spaces are 1-dimensional....

Multi-scales method is used to analyze a nonlinear differential-difference equation. In the order 3 the NLS eq. is found to determine the space-time evolution of the leading amplitude. In the next order this has to satisfy a complex mKdV eq. (the next in the NLS hierarchy) in order to eliminate secular...

The different natures of approximate symmetries and their corresponding first intgrals/invariants are delineated in the contexts of both Lie symmetries of ordinary differential equations and Noether symmetries of the Action Integral. Particular note is taken of the effect of taking higher orders of the...

The cubic Hénon-Heiles system contains parameters, for most values of which, the system is not integrable. In such parameter regimes, the general solution is expressible in formal expansions about arbitrary movable branch points, the so-called psi-series expansions. In this paper, the convergence of...

Orthogonal separability of finite-dimensional Hamiltonians is characterized by using various geometrical concepts, including Killing tensors, moving frames, the Nijehuis tensor, bi-Hamiltonian and quasi-bi-Hamiltonian representations. In addition, a complete classification of separable metrics defined...

In this paper we give a method to obtain Darboux transformations (DTs) of integrable equations. As an example we give a DT of the dispersive water wave equation. Using the Miura map, we also obtain the DT of the Jaulent-Miodek equation.

We present a geometric version of the Lie algebra 2-cocycle connected to quasi- geostrophic motion in the beta-plane approximation. We write down an Euler equation for the fluid velocity, corresponding to the evolution equation for the stream function in quasigeostrophic motion.

In this paper, we study the periodic Hunter–Saxton equation with weak dissipation. We first establish the local existence of strong solutions, blow-up scenario and blow-up criteria of the equation. Then, we investigate the blow-up rate for the blowing-up solutions to the equation. Finally, we prove that...

In this paper we use deep ideas in complex geometry that proved to be very powerful in unveiling the Polyakov measure on the moduli space of Riemann surfaces and lead to obtain the partition function of perturbative string theory for 2, 3, 4 loops. Indeed a geometrical interpretation of the conformal...

We observe a correspondence between the zero modes of superconformal algebras S (2, 1) and W(4) ([8]) and the Lie superalgebras formed by classical operators apearing in the Kähler and hyper-Kähler geometry.

The algebra A ~P (1, 3) invariants were found. These invariants allowed to reduce the Born-Infeld equation. After the reduction some solutions of the equation were found.

In this paper, we introduce a q-analogue of the Tricomi expansion for the incomplete q-gamma function. A general method is described for converting a power series into an expansion in incomplete q-gamma function. Also, we use the q-Tricomi expansion for giving a formal proof of the relation between the...

We present a study of discrete Painlevé equations which do not have any parameter, apart from those that can be removed by the appropriate scaling. We find four basic equations of this type as well as several more related to the basic ones by Miura transformations, which we derive explicitly. We obtain...

We propose a way of discretization for the soliton equations associated with the toroidal Lie algebra based on the direct method. By the discretization, the symetry of the system is modified so that the discrete time evolutions are no longer compatible with the original continuous ones. The solutions...

The paper is devoted to the Lie group analysis of a nonlinear equation arising in metallurgical applications of Magnetohydrodynamics. Self-adjointness of the basic equations is investigated. The analysis reveals two exceptional values of the exponent playing a significant role in the model.

A technique to identify new C-integrable and S-integrable systems of nonlinear partial differential equations is reported, with two representative examples displayed and tersely discussed.

In [18] Jacobi introduced a third-order nonlinear ordinary differential equation which links two different moduli of an elliptic integral. In the present paper Lie group analysis is applied to that equation named Jacobi's modular differential equation. A six-dimensional Lie symmetry algebra is obtained...

We study dynamics of the shearless stratified turbulent flows. Using the method of differential constraints we find a class of explicit solutions to the problem under consideration and establish that the differential constraint obtained coincides with the well-known Zeman­Lumley model for stratified...

The purpose of this report is to show the influence of imperfections on creation and evolution of a kink network. Our main finding is a mechanism for reduction of the kinetic energy of kinks which works in both the overdamped and underdamped regimes. This mechanism reduces mobility of kinks and therefore...

1-dimensional polytropic gas dynamics is integrable for trivial reasons, having 2

We discuss the dynamics of an affinely-rigid body in two dimensions. Translational degrees of freedom are neglected. The special stress is laid on completely integrable models solvable in terms of the separation of variables method.

In this note, we present a result to show that the symplectic structures have been naturally encoded into the Painlevé test. In fact, for every principal balance, there is a symplectic change of dependent variables near movable poles.

With the aid of the spectral gradient method of Fuchssteiner, the compatible pair of Hamiltonian operators for the coupled NLS hierarchy is rediscovered. This result enables us to construct a hierarchy, which contains a vector generalization of Fokas-Lenells system. The vector Fokas-Lenells system is...

An algorithm for an asymptotic model of wave propagation in shallow-water is proposed and analyzed. The algorithm is based on the Hamiltonian structure of the equation, and corresponds to a completely integrable particle lattice. Each "particle" in this method travels along a characteristic curve of...

Exact solutions of the multidimensional Liouville equation are constructed.

It is shown that the nonlinear pendulum equation can be transformed into a linear harmonic oscillator in the phase space thanks to Kerner’s method [12]. Moreover, as a mathematical divertissement, the second-order differential equation determining the phase-space trajectories of the nonlinear pendulum...

We study the shape of inflated surfaces introduced in [3] and [12]. More precisely, we analyze profiles of surfaces obtained by inflating a convex polyhedron, or more generally an almost everywhere flat surface, with a symmetry plane. We show that such profiles are in a one-parameter family of curves...

The problem of construction of boundary conditions for nonlinear equations compatible with their higher symmetries is considered. Boundary conditions for the sineGordon, Zhiber­Shabat and KdV equations are discussed. New examples are found for the JS equation.

We show that for a class of boundary value problems, the space of initial functions can be stratified dependently on the limit behavior (as the time variable tends to infinity) of solutions. Using known results on universal phenomena appearing in bifurcations of one parameter families of one-dimensional...

It is shown how pseudoconstants of the Liouville-type equations can be exploited as a tool for construction of the Bäcklund transformations. Several new examples of such transformations are found. In particular we obtained the Bäcklund transformations for a pair of three-component analogs of the dispersive...

We report the recursion operators for a class of symmetry integrable evolution equations of third order which admit fourth-order recursion operators. Under the given assumptions we obtain the complete list of equations, one of which is the well-known Krichever-Novikov equation.

We study non-trivial deformations of the natural action of the Lie algebra Vect(Rn ) on the space of differential forms on Rn . We calculate abstractions for integrability of ifinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation...

As is well-known, any ordinary differential equation in one dimension can be cast as the Euler–Lagrange equation of an appropriate Lagrangian. Additionally, if the initial equation is autonomous, the Lagrangian can always be chosen to be time-independent. In two dimensions, however, the situation is...

We present an extension of a family of second-order integrable mappings to the case where the variables do not commute. In every case we introduce a commutation rule which is consistent with the mapping evolution. Through the proper ordering of variables we ensure the existence of an invariant in the...

This paper shows that several integrable lattices can be transformed into coupled biliear differential-difference equations by introducing auxiliary variables. By testing the Bäcklund transformations for this type of coupled bilinear equations, a new integrable lattice is found. By using the Bäcklund...

Electromagnetic wave propagation through cold collision free plasma is studied using the nonlinear perturbation method. It is found that the equations can be reduced to the modified Kortweg-de Vries equation.

Large classes of Lie solutions of the MHD equations describing the flows of a viscous homogeneous incompressible fluid of finite electrical conductivity are constructed. These classes contain a number of arbitrary functions of time and the general solutions of the heat equation.

The symmetry classification and reduction of a non–stationary spherically symmetric energy–transport model for semiconductors was investigated by Molati and Wafo Soh (2005). In this work the exact solutions of the reduced model in the stationary case are constructed.

In this paper, we study and classify the conservation laws of the Jaulent–Miodek equations and other systems of KdV type equations which arises in, inter alia, shallow water equations. The main focus of the paper is the construction of the conservation laws as a consequence of the interplay between symmetry...

We provide an alternative method for obtaining of compatible Poisson structures on Lie groups by means of the adjoint representations of Lie algebras. In this way we calculate some compatible Poisson structures on four dimensional and nilpotent six dimensional symplectic real Lie groups. Then using Magri-Morosi’s...

We introduce the notion of a random symmetry. It consists of taking the action given by a deterministic flow that maintains the solutions of a given differential equation invariant and replacing it with a stochastic flow. This generates a random action, which we call a random symmetry.

This paper is a continuation of [1] where the classical model was analyzed. Discussed are some quantization problems of two-dimensional affinely rigid body with the double dynamical isotropy. Considered are highly symmetric models for which the variables can be separated. Some explicit solutions are...

The peakons are peaked traveling wave solutions of an integrable shallow water eqution. We present a variational proof of their stability.

A general structure of commutator representations for the hierarchy of nonlinear evolution equations (NLEEs) is proposed. As two concrete examples, the Harry-Dym and Kaup-Newell cases are discused.

The analytical solutions of a nonlinear fin problem with variable thermal conductivity and heat transfer coefficients are investigated by considering theory of Lie groups and its relations with λ-symmetries and Prelle-Singer procedure. Additionally, the classification problem with respect to different...

A reciprocal transformation is introduced for a three-component Camassa-Holm type equation and it is showed that the transformed system is a reduction of the first negative flow in a generalized MKdV hierarchy.

It is known that many integrable systems can be reduced from self-dual Yang-Mills equations. The formal solution space to the self-dual Yang-Mills equations is given by the so called ADHM construction, in which the solution space are graded by vector spaces with dimensionality concerning topological...

The problem of studying the maximal Lie symmetry of some nonlinear generalization of the vector subsystem of the Maxwell equations is completely solved.

The singular manifold expansion of Weiss, Tabor and Carnevale [1] has been successfully applied to integrable ordinary and partial differential equations. They yield information such as Lax pairs, Bäcklund transformations, symmetries, recursion operators, pole dynamics, and special solutions. On the...

We classify the Noether point symmetries of a generalized Lane-Emden equation. We obtain first integrals of the various cases which admit Noether point symmetry and find reduction to quadratures for these cases. Three new cases are found for the function f (y). One of them is f (y) = ?y r, where r

This paper studies structures of the 3-Lie algebra M realized by the general linear Lie algebra gl(n, ℂ). We show that M has only one nonzero proper ideal. We then give explicit expressions of both derivations and inner derivations of M. Finally, we investigate substructures of the (inner) derivation...

The flow of a superfluid film adsorbed on a porous medium can be modeled by a meromorphic differential on a Riemann surface of high genus. In this paper, we define the mixed Hodge metric of meromorphic differentials on a Riemann surface and justify using this metric to approximate the kinetic energy...

We derive discrete systems which result from a second, not studied up to now, form of the q-PVI equation. The derivation is based on two different procedures: “limits” and “degeneracies”. We obtain several new discrete Painlevé equations along with some linearisable systems. The parallel between the...

A family of real Hamiltonian forms (RHF) for the special class of affine 1 + dimensional Toda field theories is constructed. Thus the method, proposed in [1] for systems with finite number of degrees of freedom is generalized to infinite-dimensional Hamiltonian systems. We show that each of these RHF...

Quantum Schrödinger equation, describing dynamical spin-interaction of two electrons with external magnetic field, is considered as an object for cybernetic research. Indeed, because of having a possibility to change external magnetic field, we can influence the interaction of particles. The algorithm...

In this paper we prove an extension of the usual freezing trick argument which can be applied to a number of quasi-exactly solvable spin models of Calogero­Sutherland type. In order to illustrate the application of this method we analyze a partially solvable spin chain presenting near-neighbors interactions...

The Cauchy problem for the Liouville equation with a small perturbation is considered. We are interested in the asymptotics of the perturbed solution under the assumption that one has singularity. The main goal is to study both the asymptotic approximation of the singular lines and the asymptotic approximation...

The different second-order nonlinear partial equations are found that are invariant under the representation D(1 2, 0) D(0, 1 2) of the Poincaré group P(1, 3) and also under conformal group C(1, 3). The some exact solutions are constructed for the one of these equations.

The inverse scattering problem for a first order system of three equations on the half-line with nonsingular diagonal matrix multiplying the derivative and general boundary conditions is considered. It is focused the case of two repeated diagonal elements of diagonal matrix. The scattering matrix on...

In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peaked solitons (peakons). The 3CH model is proven to be integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system admits peakons and multi-peakon solutions....

The classical and quantum mechanics of systems on Lie groups and their homogeneous spaces are described. The special stress is laid on the dynamics of deformable bodies and the mutual coupling between rotations and deformations. Deformative modes are discretized, i.e., it is assumed that the relevant...

Various solutions are displayed and analyzed (both analytically and numerically) of a recently-introduced many-body problem in the plane which includes both integrable and nonintegrable cases (depending on the values of the coupling constants); in paticular the origin of certain periodic behaviors is...

In this paper we continue studies of the functional representation of the Ablowitz­ Ladik hierarchy (ALH). Using formal series solutions of the zero-curvature condition we rederive the functional equations for the tau-functions of the ALH and obtain some new equations which provide more straightforward...

We discuss some special classes of canonical transformations of the extended phase space, which relate integrable systems with a common Lagrangian submanifold. Vrious parametric forms of trajectories are associated with different integrals of motion, Lax equations, separated variables and action-angles...

In the framework of the theory of approximate transformation groups proposed by Baikov, Gaziziv and Ibragimov [1], the first-order approximate symmetry operator is calculated for the Navier-Stokes equations. The symmetries of the coupled system obtained by expanding the dependent variables of the Navier-Stokes...

We introduce a differential geometry description of the path lines, stream lines and particles contours in hydrodynamics. We present a generalized form of a Korteweg-de Vries type of equation for the exterior of a circle. Nonlinearities from the boundary conditions, surface tension and the Euler equations...

Hierarchies of evolution partial differential equations have become well-established in the literature over the last thirty years. More recently sequences of ordinary differential equations have been introduced. Of these perhaps the most notable is the Riccati Sequence which has beautiful singularity,...

The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on n vertices and 2n − 2 edges, induce – under the orientation mapping – infinitesimal symmetries of classical Poisson...

We clarify and extend some remarks raised in [5] [Constantin A, J. Math. Phys. 46 (2005), 023506] about the evolution of compactly supported initial data under the Camassa-Holm flow.

In this paper we discuss a theoretical model for both the free-surface and interfacial profiles of progressive nonlinear waves which result from introducing an obstacle of finite height, in the form of a ramp of gentle slope, attached to the bottom below the flow of a stratified, ideal, two-layer fluid....

This article is a direct continuation of our paper which was published in the Journal of Nonlinear Mathematical Physics 1994, V.1, N 1, 75­113.

In this paper, the (2+1)-dimensional extended quantum Zakharov-Kuznetsov equation is further explored. The equation is shown to be self-adjoint and conserved vector is constructed according to the related theorem. Then the corresponding optimal system of one-dimensional subgroups is determined. Similarity...

A new definition for the electromagnetic field velocity is proposed. The velocity depends on the physical fields. The question posed by the title of this paper is, surprisingly, not yet answered uniquely today; not even by way of definition. According to modern assumptions the light is the electromagnetic...

We study the integrability properties of the hierarchy of a class of nonlinear ordinary differential equations and point out some of the properties of these equations and their connection to the Ermakov-Pinney equation.

Recently a solvable N-body problem featuring several free parameters has been investigated, and conditions on these parameters have been identified which guarantee that this system is isochronous (all its solutions are periodic with a fixed period) and that it possesses equilibria. The N coordinates...

In this paper we investigate compatible overdetermined systems of PDEs on the plane with one common characteristic. Lie's theorem states that its integration is equivalent to a system of ODEs, and we give a new proof by relating it to the geometry of rank 2 distributions. We find a criterion for...

Novikov super-algebras are related to quadratic conformal super-algebras which correspond to Hamiltonian pairs and play fundamental role in completely integrable systems. In this paper, we focus on quadratic Novikov super-algebras, which are Novikov super-algebras with associative non-degenerate super-symmetric...

We introduce a nonlinear and noncanonical gauge transformation which allows the rduction of a complex nonlinearity, contained in a Schrödinger equation, into a real one. This Schrödinger equation describes a canonical system, whose kinetics is governed by a generalized Exclusion-Inclusion Principle....