Research Article
Dynamics of Generalized Unstable Nonlinear Schrödinger Equation: Instabilities, Solitons, and Rogue Waves
Issue:
Volume 11, Issue 1, March 2025
Pages:
1-18
Received:
30 October 2024
Accepted:
10 December 2024
Published:
20 January 2025
DOI:
10.11648/j.ijamtp.20251101.11
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Abstract: This study delves into the dynamics of the unstable Schrödinger equation, employing three distinct analytical methods: the complex envelope function ansatz, the generalized Tanh method, and the Bernoulli sub-ODE method. By leveraging the complex envelope function technique, we uncover solutions for various optical soliton types, including dark optical solitons, bright optical solitons, and bright-dark optical solitons. Notably, this method facilitates an in-depth examination of individual soliton intensity profiles, providing valuable insights into their behavior. Furthermore, we utilize the generalized Tanh method and the Bernoulli sub-ODE method to derive solutions involving hyperbolic and trigonometric functions. These solutions shed light on the intricate dynamics of nonlinear optical phenomena within the framework of the Schrödinger equation. The obtained solutions are graphically illustrated, showcasing dark, bright, dark-bright, and singular solitons. Our research contributes significantly to the understanding of unstable Schrödinger equation dynamics, offering a comprehensive analysis of optical soliton behavior. The conservation laws of the model equation are also constructed, providing a deeper understanding of the underlying physical principles. This study’s findings have important implications for the development of advanced optical communication systems and the study of nonlinear optical phenomen.
Abstract: This study delves into the dynamics of the unstable Schrödinger equation, employing three distinct analytical methods: the complex envelope function ansatz, the generalized Tanh method, and the Bernoulli sub-ODE method. By leveraging the complex envelope function technique, we uncover solutions for various optical soliton types, including dark opti...
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Research Article
Injective Envelopes of Real C*- and AW*-Algebras
Abdugafur Rakhimov*
,
Laylo Ramazonova
Issue:
Volume 11, Issue 1, March 2025
Pages:
19-23
Received:
26 March 2025
Accepted:
8 April 2025
Published:
29 April 2025
Abstract: Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of type II1. Therefore, it seems interesting to study the so called maximal injective W* and C*-subalgebras or what is equivalent, the smallest injective C*-algebra containing a given algebra, which is called an injective envelope of C*- algebra. It is shown that every outer *-automorphism of a real C*-algebra can be uniquely extended to an injective envelope of real C*-algebra. It is proven that if a real C*-algebra is a simple, then its injective envelope is also simple, and it is a real AW*-factor. The example of a real C*-algebra that is not real AW*-algebra and the injective envelope is a real AW*-factor of type III, which is not a real W*-algebra is constructed. This leads to the interesting result that up to isomorphism, the class of injective real (resp. complex) AW*-factors of type III is at least one larger than the class injective real (resp. complex) W*-factors of type III.
Abstract: Injective (complex and real) W*- and C*- algebras, in particular, factors have been studied quite well. On the other hand, in an arbitrary case, i.e., in the non-injective case, it is quite difficult to study (up to isomorphism) the W*-algebras, in particular, it is known that there is a continuum of pairwise non-isomorphic non-injective factors of...
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