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Abstract
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T h e t r a v e l i n g s o l i t a r y a n d c n o i d a l w a v e s o l u t i o n s o f t h e o n e d i m e n s i o n a l d r i v e n n o n l i n e a r
S c h r € o d i n g e r e q u a t i o n w i t h a g e n e r a l i z e d f o r m o f n o n l i n e a r i t y a r e p r e s e n t e d i n t h i s p a p e r . W e
e x a m i n e t h e m o d u l a t i o n o f n o n l i n e a r s o l i t a r y e x c i t a t i o n s i n t w o k n o w n w e a k l y n o n l i n e a r
m o d e l s o f c l a s s i c o s c i l l a t o r s , n a m e l y , t h e H e l m h o l t z a n d D u f fi n g o s c i l l a t o r s a n d e n v e l o p e
structure formations for different oscillator and driver parameters. It is shown that two distinct
regimes of subcritical and supercritical modulations may occur for nonlinear excitations with
propagation speeds v <
ffiffiffiffiffiffiffiffi
p4F0 and v > pffiffiffiffiffiffiffiffi4F0, respectively, in which F0 is the driver force strength.
The envelope soliton and cnoidal waves in these regimes are observed to be fundamentally
different. The effect of pseudoenergy on the structure of the modulated envelope excitations is
studied in detail for both sub- and supercritical modulation types. The current model for traveling
envelope excitations may be easily extended to pseudopotentials with full nonlinearity relevant to
more realistic gases, fluids, and plasmas.
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