Partha P. Banerjee

Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method

The authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations. The physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system. The emphasis is, however, on the mathematical aspects, i.e. the formal procedure necessary to find single solitary wave solutions. By means of examples it is shown how various cases of pulse-type and kink-type solutions are to be obtained by this method. An exhaustive list of equations so treated is presented in tabular form, together with the particular intermediate relations necessary for deriving their solutions. The extension of the technique to construct N-soliton solutions and indicate connections with other existing methods is outlined.

A general physical approach to solitary wave construction from linear solutions

We simplify the physical approach of constructing solitary wave solutions of nondissipative evolution and wave equations from the physical mixing of the real, rather than complex, exponential solutions of the linear equation, in two separate regions. In our new approach, we use mixing in one region only to construct a closed form for the solitary wave solution valid in both regions. Moreover, we extend the approach to deal with equations whose solutions (like tanh2-type) have a constant term in their expansion into real exponentials, and with equations whose linear part allows more than two exponential solutions. Finally, we also demonstrate the application of our technique to a typical dissipative equation, e.g., the Burgers

Derivation and implicit solution of the Harry Dym equation and its connections with the Korteweg-de Vries equation

The Harry Dym equation, which is related to the classical string problem, is derived in three different ways. An implicit cusp solitary-wave solution is constructed via a simple direct method. The existing connections between the Harry Dym and the Korteweg-de Vries equations are uniformised and simplified, and transformations between their respective solutions are carried out explicitly. Whenever possible, physical insights are provided.

Split-Step-Type Angular Plane-Wave Spectrum Method for the Study of Self-Refractive Effects in Nonlinear Wave Propagation

We introduce an efficient algorithm to simulate nonlinear self-induced wave refraction effects. The algorithm is applied to the case of self-focusing, defocusing, and predicted steady-state behavior. The results show reasonably good agreement with available approximate theory and excellent agreement when exact analytic solutions are available for comparison.

Subharmonic generation by resonant three‐wave interaction of deep‐water capillary waves

Subharmonic generation has been observed during the propagation of deep‐water capillary waves. The observations are shown to be in agreement with the theory of degenerate resonant noncollinear three‐wave interaction in a nonlinear, dispersive medium.

Exact Decomposition of Cnoidal Waves into Associated Solitons

Abstract Starting from the Korteweg de Vries equation, a precise model is developed of the decomposition of cnoidal waves into associated solitons.

Assessment of 3-D Angular Movements of Diffuse Objects Using Holographic Interferometry

We have developed a dynamic holographic interferometry (DHI) setup to measure changes in attitudes, distortions and vibrations of objects, using a holocamera. Digital algorithms have been developed to calculate above parameters from DHI images.

Analysis of Beam Propagation in 90-degree Holographic Recording and Readout Using Transfer Functions and Numerical 2-D-Laplace Inversion

Recently, 2-D-Laplace analysis of recording and readout of edge-holograms was reported. Numerical Laplace inversion was examined for simple test cases. Inversion algorithms are applied to examine beam shaping and distortion in photovoltaic and photorefractive materials.