A complete set of conditions describing the transition across the dissipationless undular bore (dispersive shock) is derived for nonlinear weakly dispersive conservative systems that generally are not integrable by the inverse spectral transform method. In the absence of integrable structure, we postulate modulation description of the undular bore with the aid of the averaged Whitham equations ...

In this work we estimate the radius and the mass of a self-gravitating system made of axions. The quantum axion field satisfies the Klein-Gordon equation in a curved space-time and the metric components of this space-time are solutions to the Einstein equations with a source term given by the vacuum expectation value of the energy-momentum operator constructed from the axion field. As a first s...

We show that the following geometric properties of the motion of discrete and continuous curves select integrable dynamics: i) the motion of the curve takes place in the N dimensional sphere of radius R, ii) the curve does not stretch during the motion, iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. Well known examples of integrable evolution equations,...

We study a two-dimensional integrable generalization of the Kaup-Kupershmidt equation, which arises in various problems in mathematical physics. Exact solutions are obtained using the Lie symmetry method in conjunction with the extended tanhmethod and the extended Jacobi elliptic functionmethod. In addition to exact solutions we also present conservation laws which are derived using the multipl...

Abstract In recent years, new properties of space-time duality in the Hamiltonian formalism certain integrable classical field theories have been discovered and led to their reformulation using ideas from covariant theory: this sense, nature r -matrix structure was unravelled. Here, we solve open question extending these results a whole hierarchy. We choose Ablowitz–Kaup–Newell–Segur (AKNS) To ...

We present perturbation theory based on the inverse scattering transform method for solitons described by an equation with linear dispersion law $\ensuremath{\omega}\ensuremath{\sim}1/k$, where $\ensuremath{\omega}$ is frequency and $k$ wave number, cubic nonlinearity. This equation, first suggested Davydova Lashkin describing dynamics of nonlinear short-wavelength ion-cyclotron waves in plasma...