New solutions from nonlocal symmetry of the generalized fifth order KdV equation

The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie’s first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.

EXISTENCE OF PERIODIC SOLUTIONS FOR A DIFFERENTIAL INCLUSION SYSTEMS INVOLVING THE p(t)-LAPLACIAN

We study a nonlinear periodic problem driven by the p（t）-Laplacian and having a nonsmooth potential （hemivariational inequalities）. Using a variational method based on nonsmooth critical point theory for locally Lipschitz functions, we first prove the existence of at least two nontrivial solutions under the generalized subquadratic and then establish the existence of at least one nontrivial solution under the generalized superquadratic.

Explicit solutions from residual symmetry of the Boussinesq equation

The Bcklund transformation related symmetry is nonlocal, which is hard to be applied in constructing solutions for nonlinear equations. In this paper, the residual symmetry of the Boussinesq equation is localized to Lie point symmetry by introducing multiple new variables. By applying the general Lie point method, two main results are obtained： a new type of Backlund transformation is derived, from which new solutions can be generated from old ones; the similarity reduction solutions as well as corresponding reduction equations are found. The localization procedure provides an effective way to investigate interaction solutions between nonlinear waves and solitons.

City level CO_(2) and local air pollutants co-control performance evaluation: A case study of 113 key envir on mental protection cities in China

‘Co-control',or synergistically reducing CO_(2)and local air polluta nt emissions,is an important strategy for cities to achieve'low carb on'and'blue sky'simultaneously.However,there were few studies to evaluate and compare the level of co-control of CO_(2) and local air pollutants in cities year.The present study proposed qualitative and quantitative methods to evaluate the level of co-control of CO_(2)and three local air pollutant(SO_(2).NOX,and particulate matter PM)emissions in key environmental protection cities in China over two periods(2012-2015 and 2015-2018).Statistical analysis found that,though three local air pollutant emissions positively correlated with CO_(2) emission,no significantly positive correlation was found between local air pollutants emission reductions and CO_(2) emission reductions,indicating that co-control effects were poor in general.By using the co-control effect coordinate system,qualitative evaluation showed that less than half of the sample cities could achieve co-control of the total amount of CO_(2) and local air pollutants.By employing the indicator of emission reduction equivalence(EReq),quantitative evaluation showed that the co-control level of the sample cities improved in 2015-2018 compared with 2012-2015.Further regression analysis showed that,reducing coal consumption and economic development significantly enhanced the co-control performance of the observed cities.The present case study proved that the proposed methods for evaluation and comparison of the city co-control performance works well,and can be used in other countries and regions to promote global cities racing to carbon and local air pollutants co-control.