Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.     

Interaction between nonlinear diffusion and geometry of domain

Let Ω be a domain in ${\mathbb{R}}^N$, where $N?2$ and ?Ω is not necessarily bounded. We consider nonlinear diffusion equations of the form ${?}_{t}u=\mathrm{\Delta }?(u)$. Let $u=u(x,t)$ be the solution of either the initial-boundary value problem over Ω, where the initial value equals zero and the boundary value equals 1, or the Cauchy problem where the initial data is the characteristic function of the set ${\mathbb{R}}^N?\Omega$.We consider an open ball B in Ω whose closure intersects ?Ω only at one point, and we derive asymptotic estimates for the content of substance in B for short times in terms of geometry of Ω. Also, we obtain a characterization of the hyperplane involving a stationary level surface of u by using the sliding method due to Berestycki, Caffarelli, and Nirenberg. These results tell us about interactions between nonlinear diffusion and geometry of domain.     

On weakly nonlinear evolution of convective flow in a passive mushy layer

The problem of weakly nonlinear convective flow in a mushy layer, with a permeable mush–liquid interface and constant permeability, is studied under operating conditions for an experiment. A Landau type nonlinear evolution equation for the amplitude of the secondary solutions, which is based on the Landau theory and formulation for the Rayleigh, $R$, number close to its critical value, $R_c$, is developed. Using numerical and analytical methods, the solutions to the evolution equation are calculated for both supercritical and subcritical conditions. We found, in particular, that for $R<R_c$, the amplitude of the secondary solutions decays with time. For $R>R_c$, the tendency for chimney formation in the mushy layer increases with time. In addition, in such a supercritical regime, the basic flow is linearly unstable and we see the presence of steady flow for large values of time. These results suggest a possible slow transition to turbulence in such a flow system.     

A quasilinearization method for a class of second order singular nonlinear differential equations with nonlinear boundary conditions

We consider the differential equation $-(1/w)({\mathit{pu}}^{\prime }{)}^{\prime }+\mu u=\mathit{Fu}$, where F is a nonlinear operator, with nonlinear boundary conditions. Under appropriate assumptions on $p,w,F$ and the boundary conditions, the existence of solutions is established. If the problem has a lower solution and an upper solution, then we use a quasilinearization method to obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation.     

Upper and lower solutions and quasilinearization for a class of second order singular nonlinear differential equations with nonlinear boundary conditions

We consider the differential equation $-(1/w)({\mathit{pu}}^{\prime }{)}^{\prime }=f(·,u)$, where $f$ is a nonlinear function, with nonlinear boundary conditions. Under appropriate assumptions on $p,w,f$ and the boundary conditions, the existence of solutions is established. If the problem has a lower solution and an upper solution, then we use a quasilinearization method to obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation.     

Viscosity solutions of obstacle problems for fully nonlinear path-dependent PDEs

In this article, we adapt the definition of viscosity solutions to the obstacle problem for fully nonlinear path-dependent PDEs with data uniformly continuous in $(t,\omega )$, and generator Lipschitz continuous in $(y,z,\gamma )$. We prove that our definition of viscosity solutions is consistent with the classical solutions, and satisfy a stability result. We show that the value functional defined via the second order reflected backward stochastic differential equation is the unique viscosity solution of the variational inequalities.