Iterative Structures on Singular Manifolds

2014 ◽  
pp. 173-221
Author(s):  
Bert-Wolfgang Schulze
Keyword(s):  
Singular Manifolds

BILINEAR FORM AND SOLITON SOLUTIONS FOR THE COUPLED NONLINEAR KLEIN–GORDON EQUATIONS

2012 ◽  
Vol 26 (15) ◽  
pp. 1250057
Author(s):  
HE LI ◽  
XIANG-HUA MENG ◽  
BO TIAN
Keyword(s):  
Field Theory ◽  
Bilinear Form ◽  
Gordon Equation ◽  
Relativistic Quantum ◽  
Soliton Solutions ◽  
Relativistic Quantum Mechanics ◽  
Klein Gordon ◽  
Truncated Painlevé Expansion ◽  
Klein Gordon Equation ◽  
Singular Manifolds

With the coupling of a scalar field, a generalization of the nonlinear Klein–Gordon equation which arises in the relativistic quantum mechanics and field theory, i.e., the coupled nonlinear Klein–Gordon equations, is investigated via the Hirota method. With the truncated Painlevé expansion at the constant level term with two singular manifolds, the coupled nonlinear Klein–Gordon equations are transformed to a bilinear form. Starting from the bilinear form, with symbolic computation, we obtain the N-soliton solutions for the coupled nonlinear Klein–Gordon equations.

scholarly journals Singular manifolds, topology change and the dynamics of compactification

2007 ◽  
Vol 2007 (11) ◽  
pp. 062-062 ◽  
Author(s):  
Neil A Butcher ◽  
Paul M Saffin
Keyword(s):  
Topology Change ◽  
Singular Manifolds

An Improved Result Concerning Singular Manifolds of Difference Polynomials

1959 ◽  
Vol 11 ◽  
pp. 222-234 ◽  
Author(s):  
Richard M. Cohn
Keyword(s):  
Low Power ◽  
Necessary Conditions ◽  
Differential Algebra ◽  
Singular Manifold ◽  
Difference Field ◽  
Effective Order ◽  
Difference Polynomial ◽  
Difference Polynomials ◽  
Special Case ◽  
Singular Manifolds

Let be a difference field of characteristic 0, m an irreducible manifold of effective order n over {y}, and F an algebraically irreducible difference polynomial in {y} of effective order n + k, k > 0, which vanishes on 3 m. In an earlier paper (2, p. 447) I gave necessary conditions, restated below as (a), (b), and (c) of the main theorem, for m to be an essential singular manifold of F. These conditions are analogous to the low power criterion of Ritt (1, p. 65) for the corresponding problem of differential algebra. Like that criterion they depend, in the special case that m is the manifold of y, only on which power products appear effectively in F. Unlike the low power criterion, however, conditions (a), (b), and (c) are only necessary, not sufficient.

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