The category of Set-valued presheaves on a small category B is a topos. Replacing Set by a bicategory S whose objects are sets and morphisms are spans, relations, or partial maps, we consider a category Lax(B,S) of S-valued lax functors on B. When S = Span, the resulting category is equivalent to Cat/B, and hence, is rarely even cartesian closed. Restricting this equivalence gives rise to expon...

In this paper, we obtain the long-time asymptotics of complex mKdV equation via Defit-Zhou method (Non-linear steepest descent method). The Cauchy problem is transformed into corresponding Riemann-Hilbert on basis Lax pair and scattering matrix. After that problems are converted through a decomposition matrix-valued spectral function factorizations jump matrix for problem. Finally, by solving l...

We consider exponentiable objects in lax slices of Top with respect to the specialization order (and its opposite) on a base space B. We begin by showing that the lax slice over B has binary products which are preserved by the forgetful functor to Top if and only if B is a meet (respective, join) semilattice in Top, and go on to characterize exponentiability over a complete Alexandrov space B.

For a small category B and a double category D, let LaxN (B,D) denote the category whose objects are vertical normal lax functors B //D and morphisms are horizontal lax transformations. It is well known that LaxN (B,Cat) ≃ Cat/B, where Cat is the double category of small categories, functors, and profunctors. In [19], we generalized this equivalence to certain double categories, in the case whe...

A behavioral contract in a higher-order language may invoke methods of unknown objects. Although this expressive power allows programmers to formulate sophisticated contracts, it also poses a problem for language designers. Indeed, two distinct semantics have emerged for such method calls, dubbed lax and picky. While lax fails to protect components in certain scenarios, picky may blame an uninv...

We show that a recently introduced Lax pair of the Sawada-Kotera equation is nota new one but is trivially related to the known old Lax pair. Using the so-called trivialcompositions of the old Lax pairs with a differentially constrained arbitrary operators,we give some examples of trivial Lax pairs of KdV and Sawada-Kotera equations.

Abstract In ordinary category theory, limits are known to be equivalent terminal objects in the slice of cones. this paper, we prove that 2-categorical analogues theorem relating 2-limits and 2-terminal various choices 2-categories 2-cones false. Furthermore show that, even when weakening pseudo- or lax-natural transformations, considering bi-type bi-terminal objects, there is still no such cor...

we show that a recently introduced lax pair of the sawada-kotera equation is nota new one but is trivially related to the known old lax pair. using the so-called trivialcompositions of the old lax pairs with a differentially constrained arbitrary operators,we give some examples of trivial lax pairs of kdv and sawada-kotera equations.

We consider the Lax representation of the new two-component coupled integrable system recently discovered by the author. Connection of the hierarchy of infinitely many Lax pairs with each other is presented.