Interaction of Dirac δ-waves in the nonlinear Klein-Gordon equation☆
Section snippets
Introduction and contents
The Klein-Gordon equation has over 90 years of history. Proposed in 1926 and 1927 by Klein, Fock, and Gordon [1], [2], [3], [4], it was considered the relativistic counterpart of the Schrödinger wave equation. Nowadays, it is accepted that the (nonlinear) Klein-Gordon equation describes the evolution of a spin-zero scalar field in the presence of a potential (see [5] pp. 20, 21). This equation plays a prominent role in a wide range of fields such as physics, engineering, chemistry, and biology
Multiplication of distributions
For readers' convenience and paper's self-containedness, this section summarizes Sarrico's distributional products theory to provide all that is essential to understand the present work. To get an up to date overview of his distributional products, see [72] Sec. 2. For a thorough presentation, see [41], [73].
Let be the space of indefinitely differentiable complex-valued functions defined on with compact support, and let be the space of Schwartz distributions. In his theory, each function
Powers of distributions
This section defines α-powers for some distributions with the aim of later composing entire functions with certain distributions relevant for this work.
The α-product (6) allows to define α-powers with and by the recurrence relation See [51] for details. Naturally, for all . As an example, if , by (7) it follows that , , and for , . Since Sarrico's distributional products are
Composition of an entire function with a distribution
This section defines the composition of entire functions with certain distributions. Such definition affords a rigorous meaning to the right-hand side of the Klein-Gordon equation (1) for the distributions u studied in this work.
Let and be an entire function given by the MacLaurin series with coefficients on and the entire function given by the series The sum of the above series can easily be computed by the following formula
The α-solution concept
This section presents the α-solution concept and relates it with the classical solution concept.
Let be the space of continuously differentiable maps in the sense of the usual topology of . For , the notation is sometimes used to stress that the distribution acts on functions depending on x.
Let be the space of functions such that:
- (a)
for each , ;
- (b)
, defined by is in .
Interaction of δ-waves
This section solves the Cauchy problem (1), (2), (3) in solution space (4) by considering ϕ as an entire function taking real values on the real axis.
Having in mind the identification , equation (1) must be substituted by equation (33) and the solution (4) by where are -functions. For short, will denote the set of functions of the form (35) and Q denotes the space of entire functions taking real values on the real axis.
Let us
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