Dan Grecu

Solitary Waves in a Madelung Fluid Description of Derivative NLS Equations

Abstract 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 with stationary profile current velocity the fluid density satisfies generalized stationary Gardner equation, and solitary wave solutions are found. for the completely integrable cases these are compared with existing solutions in literature.

Cylindrical nonlinear Schrödinger equation versus cylindrical Korteweg‐de Vries equation

A correspondence between the family of cylindrical nonlinear Schrodinger (cNLS) equations and the one of cylindrical Korteweg‐de Vries (cKdV) equations is constructed. It associates non stationary solutions of the first family with the ones of the second family. This is done by using a correspondence, recently found, between the families of generalized NLS equation and generalized KdV equation, and their solutions in the form of travelling waves, respectively. In particular, non‐stationary soliton‐like solutions of the cNLS equation can be associated with non‐stationary soliton‐like solutions of cKdV equation.

Modulational Instability of Some Nonlinear Continuum and Discrete Models

Modulational instability (also known as the Benjamin‐Feir instability) of quasi‐monochromatic waves propagating in dispersive and weakly nonlinear media is a general phenomenon encountered in hydrodynamics, plasma physics, condensed matter and is responsible for the generation of robust solitary waves (sometime solitons). The statistical approach is reviewed for several nonlinear systems: the nonlinear Schrodinger equation, the discrete self‐trapping equation and Ablowitz‐Ladik equation. An integral stability equation is deduced from a linearized kinetic equation for the two‐point correlation function. This is solved for several choices of the unperturbed initial spectral function.

On the mapping connecting the cylindrical nonlinear von Neumann equation with the standard von Neumann equation

The Wigner transformation is used to define the quasidistribution (Wigner function) associated with the wave function of the cylindrical nonlinear Schroedinger equation (CNLSE) in a way similar to that of the standard nonlinear Schroedinger equation (NLSE

Beyond Nonlinear Schrödinger Equation Approximation for an Anharmonic Chain with Harmonic Long Range Interactions

Abstract Multi-scales method is used to analyze a nonlinear differential-difference equation. In the order ε3 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 terms. The zero dispersion point case is also analyzed and the relevant equation is a modified NLS eq. with a third order derivative term included.

Exact solutions of a mixed KdV + mKdV and Benjamin-Ono equation

Abstract The effect of an “inverse power”-like harmonic long range interaction potential in an anharmonic chain with a cubic + quartic nearest neighbour interaction is considered. By the reductive perturbative method a mixed KdV + mKdV + Benjamin-Ono equation is found. Exact nonsingular rational solutions are determined using Nakamuras transformation and the bilinearization method of Hirota.

Modulational Instability of Cylindrical and Spherical NLS Equations. Statistical Approach

The modulational (Benjamin‐Feir) instability for cylindrical and spherical NLS equations (c/s NLS equations) is studied using a statistical approach (SAMI). A kinetic equation for a two‐point correlation function is written and analyzed using the Wigner‐Moyal transform. The linear stability of the Fourier transform of the two‐point correlation function is studied and an implicit integral form for the dispersion relation is found. This is solved for different expressions of the initial spectrum (δ‐spectrum, Lorentzian, Gaussian), and in the case of a Lorentzian spectrum the total growth of the instability is calculated. The similarities and differences with the usual one‐dimensional NLS equation are emphasized.

On the lattice corrections to the free energy of kink-bearing nonlinear one-dimensional scalar systems

The effective potential obtained by Trullinger and Sasaki is discused.Using asymptotic methods from the theory of differential equations depending on a large parameter, the lattice corrections to the kink and kink-kink contributions to the free energy are calculated. The results are in complete agreement with a first order correction to the energy of the static kink

Statistical approach of modulational instability beyond Gaussian approximation

A statistical description of the modulation instability is discussed for the NLS equation. The kinetic equation for the two-point correlation function is obtained by using a decoupling procedure beyond the usual Gaussian approximation for the averaged values of products of four and six field variables. The induced evolution generated by a plane wave perturbation are calculated. They comprise both an evolution of the background and of the second harmonics.

Periodic and Solitary Wave Solutions of Two Component Zakharov-Yajima-Oikawa System, Using Madelung's Approach ⋆

Using the multiple scales method, the interaction between two bright and one dark solitons is studied. Provided that a long wave-short wave resonance condition is sa- tisfied, the two-component Zakharov{Yajima{Oikawa (ZYO) completely integrable system is obtained. By using a Madelung fluid description, the one-soliton solutions of the corre- sponding ZYO system are determined. Furthermore, a discussion on the interaction between one bright and two dark solitons is presented. In particular, this problem is reduced to solve a one-component ZYO system in the resonance conditions.