Solutions to the complex Korteweg-de Vries equation: Blow-up solutions and non-singular solutions

In the paper two kinds of solutions are derived for the complex Korteweg-de Vries equation, including blow-up solutions and non-singular solutions. We derive blow-up solutions from known 1-soliton solution and a double-pole solution. There is a complex Miura transformation between the complex Korteweg-de Vries equation and a modified Korteweg-de Vries equation. Using the transformation, solitons, breathers and rational solutions to the complex Korteweg-de Vries equation are obtained from those of the modified Korteweg-de Vries equation. Dynamics of the obtained solutions are illustrated.Comment: 12 figure

On the origin of the Korteweg-de Vries equation

The Korteweg-de Vries equation has a central place in a model for waves on shallow water and it is an example of the propagation of weakly dispersive and weakly nonlinear waves. Its history spans a period of about sixty years, starting with experiments of Scott Russell in 1834, followed by theoretical investigations of, among others, Lord Rayleigh and Boussinesq in 1871 and, finally, Korteweg and De Vries in 1895. In this essay we compare the work of Boussinesq and Korteweg-de Vries, stressing essential differences and some interesting connections. Although there exist a number of articles, reviewing the origin and birth of the Korteweg-de Vries equations, connections and differences, not generally known, are reported.Comment: minor corrections; 25 pages, 3 figure

De Vries powers: a generalization of Boolean powers for compact Hausdorff spaces

We generalize the Boolean power construction to the setting of compact Hausdorff spaces. This is done by replacing Boolean algebras with de Vries algebras (complete Boolean algebras enriched with proximity) and Stone duality with de Vries duality. For a compact Hausdorff space $X$ and a totally ordered algebra $A$, we introduce the concept of a finitely valued normal function $f:X\to A$. We show that the operations of $A$ lift to the set $FN(X,A)$ of all finitely valued normal functions, and that there is a canonical proximity relation $\prec$ on $FN(X,A)$. This gives rise to the de Vries power construction, which when restricted to Stone spaces, yields the Boolean power construction. We prove that de Vries powers of a totally ordered integral domain $A$ are axiomatized as proximity Baer Specker $A$-algebras, those pairs $(S,\prec)$, where $S$ is a torsion-free $A$-algebra generated by its idempotents that is a Baer ring, and $\prec$ is a proximity relation on $S$. We introduce the category of proximity Baer Specker $A$-algebras and proximity morphisms between them, and prove that this category is dually equivalent to the category of compact Hausdorff spaces and continuous maps. This provides an analogue of de Vries duality for proximity Baer Specker $A$-algebras.Comment: 34 page

Disordering to Order: de Vries behavior from a Landau theory for smectics

We show that Landau theory for the isotropic, nematic, smectic A, and smectic C phases generically, but not ubiquitously, implies de Vries behavior. I.e., a continuous AC transition can occur with little layer contraction; the birefringence decreases as temperature T is lowered above this transition, and increases again below the transition. This de Vries behavior occurs in models with unusually small orientational order, and is preceded by a first order I − A transition. A first order AC transition with elements of de Vries behavior can also occur. These results correspond well with experimental work to date.Comment: 4 pages, 2 page appendi