The converse problem for the multipotentialisation of evolution equations and systems

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the {\it converse problem}. Although we mainly study a method for $(1+1)$-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-B\"acklund transformations for the equations/systems. In this respect we define the {\it triangular-auto-B\"acklund transformation} and derive its connections to the converse problem. Several explicit examples are given. In particular we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.Comment: 31 Pages, 7 diagrams, submitted for consideratio

Farming and the city:the changing imaginary of the city and Maputo’s irrigated urban agriculture from 1960 to 2020

Irrigated urban agriculture using various water sources has been consistently present throughout Maputo’s history. Grounded in Infulene Valley, we delve into how urban planning has evolved since 1960 and trace the implications of these policies on urban farming and the livelihoods dependent on it. Documenting the imaginaries of the city over four eras of Maputo’s development, we find that agriculture occupied a prominent place in the post-colonial city, and continues to be significant, despite its vague recognition within urban planning, after the shift to neoliberalism. We advocate for acknowledging urban irrigated agriculture as an intrinsic feature of the city.</p

139La NMR evidence for phase solitons in the ground state of overdoped manganites

Hole doped transition metal oxides are famous due to their extraordinary charge transport properties, such as high temperature superconductivity (cuprates) and colossal magnetoresistance (manganites). Astonishing, the mother system of these compounds is a Mott insulator, whereas important role in the establishment of the metallic or superconducting state is played by the way that holes are self-organized with doping. Experiments have shown that by adding holes the insulating phase breaks into antiferromagnetic (AFM) regions, which are separated by hole rich clumps (stripes) with a rapid change of the phase of the background spins and orbitals. However, recent experiments in overdoped manganites of the La(1-x)Ca(x)MnO(3) (LCMO) family have shown that instead of charge stripes, charge in these systems is organized in a uniform charge density wave (CDW). Besides, recent theoretical works predicted that the ground state is inhomogeneously modulated by orbital and charge solitons, i.e. narrow regions carrying charge (+/-)e/2, where the orbital arrangement varies very rapidly. So far, this has been only a theoretical prediction. Here, by using 139La Nuclear Magnetic Resonance (NMR) we provide direct evidence that the ground state of overdoped LCMO is indeed solitonic. By lowering temperature the narrow NMR spectra observed in the AFM phase are shown to wipe out, while for T<30K a very broad spectrum reappears, characteristic of an incommensurate (IC) charge and spin modulation. Remarkably, by further decreasing temperature, a relatively narrow feature emerges from the broad IC NMR signal, manifesting the formation of a solitonic modulation as T->0.Comment: 5 pages, 4 figure

A note on Verhulst's logistic equation and related logistic maps

We consider the Verhulst logistic equation and a couple of forms of the corresponding logistic maps. For the case of the logistic equation we show that using the general Riccati solution only changes the initial conditions of the equation. Next, we consider two forms of corresponding logistic maps reporting the following results. For the map x_{n+1} = rx_n(1 - x_n) we propose a new way to write the solution for r = -2 which allows better precision of the iterative terms, while for the map x_{n+1}-x_n = rx_n(1 - x_{n+1}) we show that it behaves identically to the logistic equation from the standpoint of the general Riccati solution, which is also provided herein for any value of the parameter r.Comment: 6 pages, 3 figures, 7 references with title