Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations

COMPACTION IN A CLASS OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

We inspect the compaction structure in a class of nonlinear dispersive conditions in this article. The compaction sort of lone waves free of exponential tails and width self-sufficient of abundance is formally created. We further set up particular examples of answers for the defocusing parts of these models.

Long-range transport of ozone across the eastern China seas: A case study in coastal cities in southeastern China

Tropospheric ozone (O₃) can be transported influenced by large-scale circulation. In this study, an ozone pollution episode in 6 cities of southeastern coastal area of China (SCA) in autumn 2017 was investigated. Compared with the typical local ozone pollution, there was no significant diurnal variations in this pollution episode, the O₃ concentrations maintained a stable level of about 47 ppb continuously. The WRF-CMAQ model as well as the coupled process analysis (PA) and source apportionment modules were used to simulate the formation and transport and quantify the contributions to O₃. Besides, the HYSPLIT model was used to calculate the backward trajectories arriving in the cities. We find that this pollution was mainly caused by O₃ transport from the eastern China seas (ECS). Under the movement of the Mongolian high-pressure, the O₃ precursors emitted from Beijing-Tianjin-Hebei (BTH), Northeast China (NEC) and Japan-Korea (JK) were transported to ECS then generated O₃ through photochemical reactions. Due to the weak nitrogen oxide titration and the extremely weak ozone deposition on the water surface, O₃ concentrations maintained high during the movement of air masses over ECS and finally affected SCA after long-range transport. The contributions of horizontal advections were significant basically all the day with hourly contribution about 10 ppb hr^-1 and extended from surface to 500 m above the ground level. JK contributed the most, with multi-days averaged contribution about 5 ppb and peak up to 30 ppb. The contributions of BTH and NEC were comparable, with average about 2 ppb and hourly peak of 19 and 10 ppb, respectively. For the first time, this study clearly shows that the O₃ precursors emitted from northern China and Japan-Korea contribute to the O₃ pollution in SCA under certain weather conditions, which will help to better understand and predict the O₃ pollution in that area.

Exact Solutions to Time-Fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry

We study a fifth order time-fractional KdV equation (FKdV) under meaning of the conformal fractional derivative. By trial equation method based on symmetry, we construct the abundant exact traveling wave solutions to the FKdV equation. These solutions show rich evolution patterns including solitons, rational singular solutions, periodic and double periodic solutions and so forth. In particular, under the concrete parameters, we give the representations of all these solutions.

Compact-envelope bright solitary wave in a DNA double strand

We study the nonlinear dynamics of a homogeneous DNA chain that is based on site-dependent finite stacking and pairing enthalpies. We introduce an extended nonlinear Schrödinger equation describing the dynamics of modulated wave in DNA model. We obtain envelope bright solitary waves with compact support as a solution. Analytical criteria of existence of this solution are derived. The stability of bright compactons is confirmed by numerical simulations of the exact equations of the lattice. The impact of the finite stacking energy is investigated, and we show that some of these compact bright solitary waves are robust, while others decompose very quickly depending on the finite stacking parameters.

Properties of compacton-anticompacton collisions

We study the properties of compacton-anticompacton collision processes. We compare and contrast results for the case of compacton-anticompacton solutions of the K(l,p) Rosenau-Hyman (RH) equation for l = p = 2, with compacton-anticompacton solutions of the L(l,p) Cooper-Shepard-Sodano (CSS) equation for p = 1 and l = 3. This study is performed using a Padé discretization of the RH and CSS equations. We find a significant difference in the behavior of compacton-anticompacton scattering. For the CSS equation, the scattering can be interpreted as "annihilation" as the wake left behind dissolves over time. In the RH equation, the numerical evidence is that multiple shocks form after the collision, which eventually lead to "blowup" of the resulting wave form.

Stability and dynamical properties of Cooper-Shepard-Sodano compactons

Extending a Padé approximant method used for studying compactons in the Rosenau-Hyman (RH) equation, we study the numerical stability of single compactons of the Cooper-Shepard-Sodano (CSS) equation and their pairwise interactions. The CSS equation has a conserved Hamiltonian which has allowed various approaches for studying analytically the nonlinear stability of the solutions. We study three different compacton solutions and find they are numerically stable. Similar to the collisions between RH compactons, the CSS compactons re-emerge with same coherent shape when scattered. The time evolution of the small-amplitude ripple resulting after scattering depends on the values of the parameters l and p characterizing the corresponding CSS equation. The simulation of the CSS compacton scattering requires a much smaller artificial viscosity to obtain numerical stability than in the case of RH compacton propagation.

Stability and dynamical properties of Rosenau-Hyman compactons using Padé approximants

We present a systematic approach for calculating higher-order derivatives of smooth functions on a uniform grid using Padé approximants. We illustrate our findings by deriving higher-order approximations using traditional second-order finite-difference formulas as our starting point. We employ these schemes to study the stability and dynamical properties of K(2,2) Rosenau-Hyman compactons including the collision of two compactons and resultant shock formation. Our approach uses a differencing scheme involving only nearest and next-to-nearest neighbors on a uniform spatial grid. The partial differential equation for the compactons involves first, second, and third partial derivatives in the spatial coordinate and we concentrate on four different fourth-order methods which differ in the possibility of increasing the degree of accuracy (or not) of one of the spatial derivatives to sixth order. A method designed to reduce round-off errors was found to be the most accurate approximation in stability studies of single solitary waves even though all derivates are accurate only to fourth order. Simulating compacton scattering requires the addition of fourth derivatives related to artificial viscosity. For those problems the different choices lead to different amounts of "spurious" radiation and we compare the virtues of the different choices.

Bright compact breathers

In this communication we will consider the potential of some general classes of nonlinear lattice models to support bright discrete compact breather solutions (compactlets). We analyze the conditions for which such solutions are possible and classify the models as belonging in three general categories: a class with no compact breather solutions, one with one-parameter families of solutions, and a class with "isolated" solutions (i.e., no free parameters). In the latter two cases we construct the solutions and analyze their linear stability. The drastically different stability features of these solutions in comparison with their smoothly decaying counterparts are discussed. Stable breather solutions with compact support are identified in the one-parameter families of solutions, while the corresponding solutions found in the zero-parameter families are always found to be unstable.