Generalized solutions of the Cauchy problem for the one-dimensional periodic nonlinear Schrödinger equation, with cubic or quadratic nonlinearities, are not unique. For any s < 0 there exist nonzero generalized solutions varying continuously in the Sobolev space H, with identically vanishing initial data.

The reductions of the Heun equation to the hypergeometric equation by rational changes of its independent variable are classified. Heun-to-hypergeometric transformations are analogous to the classical hypergeometric identities (i.e., hypergeometric-to-hypergeometric transformations) of Goursat. However, a transformation is possible only if the singular point location parameter and normalized ac...

This paper implements the trial equation approach to retrieve cubic–quartic optical solitons in fiber Bragg gratings with aid of methodology. Five forms nonlinear refractive index structures are considered. They Kerr law, parabolic polynomial quadratic–cubic and nonlocal law. Dark singular soliton solutions recovered along Jacobi’s elliptic functions an appropriate modulus ellipticity.

Aoki,T. , (1950) &quot;On stability of the linear transformation in Banach spaces,&quot; Journal of the Mathematical Society of Japan, 2,64-66 Chang, S. and Kim,H. M. , (2002), On the Hyer-Ulam stability of a quadratic functional equations, J. Ineq. Appl. Math. , 33, 1-12. Chang,S. , Lee,E. H. and Kim,H. M. , (2003)On the Hyer-Ulam Rassias stability of a quadratic functional equations, Math. In...

Aoki,T. , (1950) &quot;On stability of the linear transformation in Banach spaces,&quot; Journal of the Mathematical Society of Japan, 2,64-66 Chang, S. and Kim,H. M. , (2002), On the Hyer-Ulam stability of a quadratic functional equations, J. Ineq. Appl. Math. , 33, 1-12. Chang,S. , Lee,E. H. and Kim,H. M. , (2003)On the Hyer-Ulam Rassias stability of a quadratic functional equations, Math. In...

Aoki,T. , (1950) &quot;On stability of the linear transformation in Banach spaces,&quot; Journal of the Mathematical Society of Japan, 2,64-66 Chang, S. and Kim,H. M. , (2002), On the Hyer-Ulam stability of a quadratic functional equations, J. Ineq. Appl. Math. , 33, 1-12. Chang,S. , Lee,E. H. and Kim,H. M. , (2003)On the Hyer-Ulam Rassias stability of a quadratic functional equations, Math. In...

Aoki,T. , (1950) &quot;On stability of the linear transformation in Banach spaces,&quot; Journal of the Mathematical Society of Japan, 2,64-66 Chang, S. and Kim,H. M. , (2002), On the Hyer-Ulam stability of a quadratic functional equations, J. Ineq. Appl. Math. , 33, 1-12. Chang,S. , Lee,E. H. and Kim,H. M. , (2003)On the Hyer-Ulam Rassias stability of a quadratic functional equations, Math. In...