Higher conservation laws for coherent optical pulse propagation in an inhomogeneously broadened medium.
The description of coherent optical pulse propagation in terms of coupled Maxwell and Schroedinger- Bloch equations in slowly varying envelope approximation possesses conservation laws in addition to those of field energy and field momentum. Comparison with preliminary numerical results has shown that these higher conservation laws can be used to predict the amplitude of each of the 2pi pulses into which an intense coherent pulse may decompose as it propagates through an attenuator. In the present paper a comparison with more extensive numerical results is carried out. It is shown that over a wide range of variation of the parameters involved the conservation laws yield results in agreement with those obtained through numerical computation.
NOETHER'S Theorem and Higher Conservation Laws in Ultrashort Pulse Propagation
LAMB has constructed a number of local conservation laws for the differential equation στ = sin σ describing ultrashort pulse propagation in a resonant medium. On the other hand it is known that from any solution of this equation others may be found by BACKLUND transformation. 
 
 
 
In this paper we introduce extended BACKLUND transformations Ba and compose them to form infinitesimal invariance transformations Ba+e B from which via NOETHER'S theorem a system of conservation laws follows. Some of them are recognized as LAMB'S conservation laws. 
 
 
 
Another conservation law results from a scale invariance called LIE'S transformation. 
 
 
 
LAMB hat eine Anzahl von lokalen Erhaltungssatzen fur die Differentialgleichung στ = sin σ konstruiert, welche die Ausbreitung eines ultrakurzen Lichtimpulses im resonanten Medium beschreibt. Andererseits ist bekannt, das aus einer Losung dieser Gleichung weitere Losungen durch BACKLUND-Transformation gefunden werden konnen. 
 
 
 
In der vorliegenden Arbeit definieren wir erweiterte BACKLUND-Transformationen Ba, setzen diese zu infinitesimalen Invarianztransformationen Ba+e B zusammen und finden nach dem NOETHERschen Satz ein System von Erhaltungssatzen, von denen einige als die von LAMB angegebenen erkannt werden. 
 
 
 
Ein weiterer Erhaltungssatz folgt aus einer Skaleninvarianz, der sog. LIEschen Transformation.
Explicit solutions for N-dimensional Schrödinger equations with position-dependent mass
With the consideration of spherical symmetry for the potential and mass function, one-dimensional solutions of nonrelativistic Schrodinger equations with spatially varying effective mass are successfully extended to arbitrary dimensions within the frame of recently developed elegant nonperturbative technique, where the BenDaniel-Duke effective Hamiltonian in one dimension is assumed like the unperturbed piece, leading to well-known solutions, whereas the modification term due to possible use of other effective Hamiltonians in one dimension and, together with the corrections coming from the treatments in higher dimensions, are considered as an additional term like the perturbation. Application of the model and its generalization for the completeness are discussed.