Two-component Generalizations Research Articles

Blow-up criteria for a two-component nonlinear dispersive wave system

We investigate the blow-up phenomena for the two-component generalizations of nonlinear dispersive wave equation on the real line that includes the two-component Camassa-Holm system as well as the system for the hyperelastic-rod wave equation as special cases. We establish a blow-up criterion for system of coupled equations under certain natural initial profiles. The obtained criterion involves only the properties of the data u0 in a neighbourhood of a single point and consequently, it is local-in-space with respect to first component of the system under consideration.

Two-component generalizations of the Novikov equation

Some two-component generalizations of the Novikov equation, except the Geng-Xue equation, are presented, as well as their Lax pairs and bi-Hamiltonian structures. Furthermore, we study the Hamiltonians of the Geng-Xue equation and discuss the homogeneous and local properties of them.

Local-in-space blow-up criteria for two-component nonlinear dispersive wave system

We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial profiles. Presented result extends and specifies the earlier blow-up criteria for such type systems.

Two-component generalizations of the Camassa–Holm equation

A classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa–Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators of specific forms is carried out, in order to obtain bi-Hamiltonian structures for the same systems of equations. Using reciprocal transformations, some exact solutions and Lax pairs are also constructed for the systems considered.

Coupled solitons of intense high-frequency and low-frequency waves in Zakharov-type systems.

One-parameter families of exact two-component solitary-wave solutions for interacting high-frequency (HF) and low-frequency (LF) waves are found in the framework of Zakharov-type models, which couple the nonlinear Schrödinger equation for intense HF waves to the Boussinesq (Bq) or Korteweg-de Vries (KdV) equation for the LF component through quadratic terms. The systems apply, in particular, to the interaction of surface (HF) and internal (LF) waves in stratified fluids. These solutions are two-component generalizations of the single-component Bq and KdV solitons. Perturbed dynamics and stability of the solitary waves are studied in detail by means of analytical and numerical methods. Essentially, they are stable against separation of the HF and LF components if the latter one is shaped as a potential well acting on the HF field, and unstable, against splitting of the two components, with a barrier-shaped LF one. Collisions between the solitary waves are studied by means of direct simulations, demonstrating a trend to merger of in-phase solitons, and elastic interactions of out-of-phase ones.

Two-component integrable generalizations of Burgers equations with nondiagonal linearity

Two-component second- and third-order Burgers type systems with nondiagonal constant matrix of leading order terms are classified for higher symmetries. New integrable systems are obtained. Master symmetries of the obtained symmetry integrable systems, and bi-Poisson structures of those that also possess conservation laws, are given.

Non-metric two-component Euler equations on the circle

A geometric approach to the study of natural two-component generalizations of the periodic Hunter–Saxton is presented. We give rigorous evidence of the fact that these systems can be realized as geodesic equations with respect to symmetric linear connections on the semidirect product of a suitable subgroup of the diffeomorphism group of the circle \({{\text{\sc Diff}}(\mathbb{S})}\) with the space of smooth functions on the circle. An immediate consequence of this approach is a well-posedness result of the corresponding Cauchy problems in the smooth category.

Soliton Solutions of the Kaup-Kupershmidt and Sawada-Kotera Equations

In this paper I seek soliton solutions of two-component generalizations of the Kaup- Kupershmidt and Sawada-Kotera equations, for this purpose I will apply the extended tanh method. The extended tanh method with a computerized symbolic computation, is used for constructing the travelling wave solutions of coupled nonlinear equations arising in physics. The obtained solutions include soliton, kink and plane periodic solutions. KeyWords: Soliton Solutions; Kaup-Kupershmidt Equation; Sawada-Kotera Equation

The geometry of the two-component Camassa–Holm and Degasperis–Procesi equations

We use geometric methods to study two natural two-component generalizations of the periodic Camassa–Holm and Degasperis–Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff ( S 1 ) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa–Holm equation, giving explicit examples of large subspaces of positive curvature.

Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy

We consider two-component integrable generalizations of the dispersionless two-dimensional Toda lattice (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 equations and Lax–Sato equations are introduced, and a dressing scheme based on the vector nonlinear Riemann problem is formulated. The simplest two-component generalization of the dispersionless 2DTL equation is derived, and its differential reduction analogous to the Dunajski interpolating system is presented. A symmetric two-component generalization of the dispersionless elliptic 2DTL equation is also constructed.

Two-component generalizations of the Camassa-Holm equation

Two-component generalizations of the Camassa-Holm equation

On negative flows of the AKNS hierarchy and a class of deformations of a bihamiltonian structure of hydrodynamic type

A deformation parameter of a bihamiltonian structure of hydrodynamic type is shown to parameterize different extensions of the AKNS hierarchy to include negative flows. This construction establishes a purely algebraic link between, on the one hand, two realizations of the first negative flow of the AKNS model and, on the other, two-component generalizations of Camassa-Holm and Dym type equations. The two-component generalizations of Camassa-Holm and Dym type equations can be obtained from the negative order Hamiltonians constructed from the Lenard relations recursively applied on the Casimir of the first Poisson bracket of hydrodynamic type. The positive order Hamiltonians, which follow from Lenard scheme applied on the Casimir of the second Poisson bracket of hydrodynamic type, are shown to coincide with the Hamiltonians of the AKNS model. The AKNS Hamiltonians give rise to charges conserved with respect to equations of motion of two-component Camassa-Holm and two-component Dym type equations.

Multicomponent binary spreading process.

I investigate numerically the phase transitions of two-component generalizations of binary spreading processes in one dimension. In these models pair annihilation AA --> emptyset, BB --> emptyset, explicit particle diffusion, and binary pair production processes compete with each other. Several versions with spatially different production are explored, and it is shown that for the cases 2A --> 3A, 2B--> 3B and 2A --> 2AB, 2B--> 2BA a phase transition occurs at zero production rate (sigma=0), which belongs to the class of N-component, asymmetric branching and annihilating random walks, characterized by the order parameter exponent beta=2. In the model with particle production AB --> ABA, BA --> BAB a phase transition point can be located at sigma(c)=0.3253 which belongs to the class of one-component binary spreading processes.