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We study the existence of two-dimensional discrete breathers in a two-dimensional face-centred square lattice consisting of alternating light and heavy atoms, with nearest-neighbour coupling containing quartic soft or hard nonlinearity. This study is focused on two-dimensional breathers with frequency in the gap that separates the acoustic and optical bands of the phonon spectrum. We demonstrate the possibility of existence of two-dimensional gap breathers by using the numerical method, the local anharmonicity approximation and the rotating wave approximation. We obtain six types of two-dimensional gap breathers, i.e., symmetric, mirror-symmetric and asymmetric, no matter whether the centre of the breather is on a light or a heavy atom.

Multi-site compact-like discrete breathers in discrete one-dimensional monatomic chains are investigated by discussing a generalized discrete one-dimensional monatomic model. We obtain that the two-site compact-like discrete breathers with codes σ = {0,..., 0,1,1, 0,..., 0} and codes σ = {0,...,0,1, - 1,0,..., 0} can exist in discrete one-dimensional monatomic chain with quartic on-site and inter-site potentials. However, the former can only exist in hard quartic on-site potential and cannot exist in soft quartic on-site potential, whereas the latter is just reversed. All of the two-site Compact-like discrete breathers with codes σ = {0,..., 0,1,1, 0,..., 0} and σ = {0,..., 0,1, - 1, 0,..., 0} cannot exist in a pure K4 chain.

A continuous β-Fermi–Pasta–Ulam (FPU) chain is investigated by using the knowledge of elliptic equation and Jacobian elliptic functions. We obtain the new solutions, two-kink soliton solution, breather solution and breather lattice solution, of the continuous β-FPU chain, besides the kink-soliton solution and chaos solution.

Compact-like discrete breathers in discrete one-dimensional monatomic chains are investigated by discussing a generalized discrete one-dimensional monatomic model. It is proven that compact-like discrete breathers exist not only in soft 4 potential but also in hard 4 potential and K4 chains. The measurements of compact-like discrete breathers' core in soft and hard 4 potential are determined by coupling parameter K4, while those in K4 chains have nothing with coupling parameter K4. The stabilities of compact-like discrete breathers correlate closely to coupling parameter K4 and the boundary condition of lattice.

We study the features of a single q-breather (SQB) in a Fermi-Pasta-Ulam lattice by the numerical method, and obtain that the stability of SQB correlates to coupling constant K and nonlinear parameter β. No matter whether K or β increases, the periodic SQB can be transformed into a quasiperiodic SQB or a chaotic SQB. We also obtain the conditions of excitation of periodic, quasiperiodic and chaotic SQBs.

We study a two-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers can be explicitly constructed by an exact separation of their time and space dependence. DBs can stably exist in the two-dimensional Klein–Gordon lattice with hard on-site potential. When a parametric driving term is introduced in the factor multiplying the harmonic part of the on-site potential of the system, we can obtain the stable quasiperiodic discrete breathers and chaotic discrete breathers by changing the amplitude of the driver.