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We introduce and study a family of lattice equations which may be viewed either as a strongly nonlinear discrete extension of the Gardner equation, or a non-convex variant of the Lotka-Volterra chain. Their deceptively simple form supports a very rich family of complex solitary patterns. Some of these patterns are also found in the quasi-continuum rendition, but the more intriguing ones, like interlaced pairs of solitary waves, or waves which may reverse their direction either spontaneously or due a collision, are an intrinsic feature of the discrete realm.

In both the Gardner equation and its extensions, the non-convex convection bounds the range of solitons / compactons velocities beyond which they dissolve and kink/anti-kink form. Close to solitons barrier we unfold a narrow strip of velocities where solitons shape undergoes a structural change and rather than grow with velocity, their top flattens and they widen rapidly; $\epsilon ^2 << 1$ change in velocity causes their width to expand ln $(1/ \epsilon)$. To a very good approximation these solitary waves, referred to as flatons, may be viewed as made of a kink and anti-kink placed at an arbitrary distance from each other. Like ordinary solitons, once flatons form they are very robust. A multi-dimensional extension of the Gardner equation reveals that spherical flatons are as prevalent and in many cases every admissible velocity supports an entire sequence of multi-nodal flatons., Comment: 17 pages, 12 figures

In the present paper we study phase waves of self-sustained oscillators with a nearest neighbor dispersive coupling on an infinite lattice. To analyze the underlying dynamics we approximate the lattice with a quasi-continuum, QC. The resulting partial differential model is then further reduced to the Gardner equation which predicts many properties of the underlying solitary structures. Using an iterative procedure on the original lattice equations we determine the shapes of solitary waves, kinks, and the flat-like solitons, that we refer to as flatons. Direct numerical experiments reveal that the interaction of solitons and flatons on the lattice is notably clean. All in all we find that both the QC and the Gardner equation predict remarkably well the discrete patterns and their dynamics.

Using a priori estimates we prove that initially nonnegative, smooth and compactly supported solutions of the $K(m,n)$ equations must lose their smoothness in finite time. Formation of a singularity is a prerequisite for the emergence of compactons.

The standard derivation of Schroedinger's equation from a Lorentz-invariant Feynman path integral consists in taking first the limit of infinite speed of light and then the limit of short time slice. In this order of limits the light cone of the path integral disappears, giving rise to an instantaneous spread of the wave function to the entire space. We ascribe the failure of Schroedinger's equation to retain the light cone of the path integral to the very nature of the limiting process: it is a regular expansion of a singular approximation problem, because the boundary conditions of the path integral on the light cone are lost in this limit. We propose a distinguished limit, which produces an intermediate model between non-relativistic and relativistic quantum mechanics: it produces Schroedinger's equation and preserves the zero boundary conditions on and outside the original light cone of the path integral. These boundary conditions relieve the Schroedinger equation of several annoying, seemingly unrelated unphysical artifacts, including non-analytic wave functions, spontaneous appearance of discontinuities, non-existence of moments when the initial wave function has a jump discontinuity (e.g., a collapsed wave function after a measurement), the EPR paradox, and so on. The practical implications of the present formulation are yet to be seen.

In this article we discuss a new Hamiltonian PDE arising from a class of equations appearing in the study of magma, partially molten rock, in the Earth's interior. Under physically justifiable simplifications, a scalar, nonlinear, degenerate, dispersive wave equation may be derived to describe the evolution of $\phi$, the fraction of molten rock by volume, in the Earth. These equations have two power nonlinearities which specify the constitutive realitions for bulk viscosity and permeability in terms of $\phi$. Previously, they have been shown to admit solitary wave solutions. For a particular relation between exponents, we observe the equation to be Hamiltonian; it can be viewed as a generalization of the Benjamin-Bona-Mahoney equation. We prove that the solitary waves are nonlinearly stable, by showing that they are constrained local minimizers of an appropriate time-invariant Lyapunov functional. A consequence is an extension of the regime of global in time well-posedness for this class of equations to (large) data, which include a neighborhood of a solitary wave. Finally, we observe that these equations have {\it compactons}, solitary traveling waves with compact spatial support at each time., Comment: 27 pages, submitted to DCDS-B