We review recent nonlinear partial differential equation techniques developed to address questions concerning solitons for the quartic generalized Korteweg—de Vries equation (gKdV) and other generalizations of the KdV equation. We draw a comparison between results obtained in this way and some elements of the classical integrability theory for the original KdV equation, which serve as a reference for soliton and multi-soliton problems. First, known results on stability and asymptotic stability of solitons for gKdV equations are reviewed from several different sources. Second, we consider the problem of the interaction of two solitons for the quartic gKdV equation. We focus on recent results and techniques from a previous paper by the present authors concerning the interaction of two almost-equal solitons.

The equation div υ = F has a solution υ in the space of continuous vector fields vanishing at infinity if and only if F acts linearly on BVm/(m−1)(ℝm) (the space of functions in Lm/(m−1)(ℝm) whose distributional gradient is a vector-valued measure) and satisfies the following continuity condition: F(uj) converges to zero for each sequence {uj} such that the measure norms of ∇j are uniformly bounded and uj ⇀ 0 weakly in Lm/(m−1)(ℝm).

We explain the basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by the Peter-Weyl decomposition of L2() for a compact Lie group . After developing a general concept for compact groups and their homogeneous spaces, we give concrete examples for tori, which reflect the situation on ℝn, and for 2 and 3 spheres.

A classification of all four-dimensional power-commutative real division algebras is given. It is shown that every four-dimensional power-commutative real division algebra is an isotope of a particular kind of a quadratic division algebra. The description of such isotopes in dimensions four and eight is reduced to the description of quadratic division algebras. In dimension four, this leads to a complete and irredundant classification. As a special case, the finite-dimensional power-commutative real division algebras that have a unique non-zero idempotent are characterized.

We study Lie algebras endowed with an action by automorphisms of the dicyclic group of degree 3. The close connections of these algebras with Lie algebras graded over the non-reduced root system BC1, with J-ternary algebras and with Freudenthal–Kantor triple systems are explored.

We consider an extension of the notion of sub- and super-solutions for an elliptic boundary-value problem and derive the existence of a classical solution in the presence of pairs of well-ordered or non-well-ordered generalized sub- and super-solutions.

We review some recent results concerning tug-of-war games and their relation to some well-known partial differential equations (PDEs). In particular, we will show that solutions to certain PDEs can be obtained as limits of values of tug-of-war games when the parameter that controls the length of the possible movements goes to zero. Since the equations being studied are nonlinear and are not in divergence form, we will make extensive use of the concept of viscosity solutions.

We study the weak interaction between a pair of well-separated coherent structures in possibly non-local lattice differential equations. In particular, we prove that if a lattice differential equation in one space dimension has asymptotically stable (in the sense of a paper by Chow et al.) travelling-wave solutions whose profiles approach limiting equilibria exponentially fast, then the system admits solutions which are nearly the linear superposition of two such travelling waves moving in opposite directions away from one another. Moreover, such solutions are themselves asymptotically stable. This result is meant to complement analytic or numeric studies into interactions of such pulses over finite times which might result in the scenario treated here. Since the travelling waves are moving in opposite directions, these solutions are not shift-periodic and hence the framework of Chow et al. does not apply. We overcome this difficulty by embedding the original system in a larger one wherein the linear part can be written as a shift-periodic piece plus another piece which, although it is non-autonomous and large, has certain properties which allow us to treat it as if it were a small perturbation.

We use the extended multiple Riccati equations expansion method to construct a series of double-soliton-like solutions, double triangular function solutions and complexiton soliton solutions for nonlinear Jaulent-Miodek PDEs. With the help of symbolic computation using Maple and Mathematica, we obtain many new types of complexiton soliton solutions, i.e. various combinations of trigonometric periodic functions and hyperbolic function solutions, various combinations of trigonometric periodic functions and rational function solutions, and various combinations of hyperbolic functions and rational function solutions.

We obtain weighted norm inequalities for maximal truncated operators of multi-linear singular integrals with non-smooth kernels in the sense of Duong et al. This class of operators extends the class of multi-linear Calderón-Zygmund operators introduced by Coifman and Meyer and includes the higher-order commutators of Calderón. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner et al. The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlar's inequality. As applications of these results, new weighted estimates for the mth order Calderón commutators and their maximal counterparts are deduced.

The harmonic mapping problem asks when there exists a harmonic homeomorphism between two given domains. It arises in the theory of minimal surfaces and in the calculus of variations, specifically in hyperelasticity theory. We investigate this problem for doubly connected domains in the plane, where it already presents a considerable challenge and leads to several interesting open questions.

We use the theory of semiperfect rings to derive decomposition theorems for H- and co-H-spaces, generalizing the results of Wilkerson. The results are then used to prove reducibility of self-homotopy equivalences for arbitrary p-local H- and co-H-spaces.

We continue to study the nonlinear fourth-order problem TΔu – DΔ2u = λ/(L + u)2, –L < u < 0 in Ω, u = 0, Δu = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded smooth domain and λ > 0 is a parameter. When N = 2 and Ω is a convex domain, we know that there is λc > 0 such that for λ ∊ (0, λc) the problem possesses at least two regular solutions. We will see that the convexity assumption on Ω can be removed, i.e. the main results are still true for a general bounded smooth domain Ω. The main technique in the proofs of this paper is the blow-up argument, and the main difficulty is the analysis of touch-down behaviour.

We consider an inverse parabolic problem of reconstruction of the source function, together with the traditional solution. In contrast with older literature, we consider non-homogeneous and time-dependent boundary conditions. We are able to prove a general result of convergence to a stationary state, and of asymptotic expansion as t → ∞.

We establish the local well-posedness for a periodic two-component Camassa–Holm equation. We then present precise blow-up scenarios. Finally, we obtain several blow-up results and the blow-up rate of strong solutions to the equation.

We are interested in the positive doubly periodic solutions, which are even in each variable, of a stationary nonlinear Schrödinger equation in ℝ2, with a small parameter. For any pair of periods (2a, 2b), we construct a branch of solutions that concentrate uniformly to the ground-state solution of the equation.

The three-dimensional compressible magnetohydrodynamic equation in the whole space are studied in this paper. The global classical solution is established when the initial data are small perturbations of some given constant state. Moreover, the optimal decay rate of the solution is also obtained.

We study travelling-front solutions of pyramidal shapes in the Allen–Cahn equation in ℝN with N ≥ 3. It is well known that two-dimensional V-form travelling fronts and three-dimensional pyramidal travelling fronts exist and are stable. The aim of this paper is to show that for N ≥ 4 there exist N-dimensional pyramidal travelling fronts. We construct a supersolution and a subsolution, and find a pyramidal travelling-front solution between them. For the construction of a supersolution we use a multi-scale method.

In numerical computations of tsunamis due to submarine earthquakes, it is frequently assumed that the initial displacement of the water surface is equal to the permanent shift of the seabed and that the initial velocity field is equal to zero and the shallow-water equations are often used to simulate the propagation of tsunamis. We give a mathematically rigorous justification of this tsunami model starting from the full water-wave problem by comparing the solution of the full problem with that of the tsunami model. We also show that, in some cases, we have to impose a non-zero initial velocity field, which arises as a nonlinear effect.

We study the elliptic system −Δu = δ(x)−avp in Ω, −Δv = δ(x)−buq in Ω, subject to homogeneous Dirichlet boundary conditions. Here, Ω ⊂ ℝN, N ≥ 1, is a smooth and bounded domain, δ(x) = dist(x, ∂Ω), a, b ≥ 0 and p, q ∈ ℝ satisfy pq > −1. The existence, non-existence and uniqueness of solutions are investigated in terms of a, b, p and q.