两类非线性偏微分方程组正解的正则性,对称性和单调性的开题报告题目：两类非线性偏微分方程组正解的正则性、对称性和单调性摘要：本文研究了两类非线性偏微分方程组正解的正则性、对称性和单调性。第一类方程组为二维Schrodinger方程，它在量子力学、数学物理和非线性分析等领域有广泛的应用。我们研究了具有能量凸函数和带有奇异项的初始数据的正解的局部存在性和全局存在性，并对正解的稳定性和单调性进行了分析。第二类方程组为Korteveg-deVries方程组，它在水波、声波、气体动力学等领域有广泛应用。我们研究了具有有界变差解和带有奇异项的初始数据的正解的全局存在性和渐近行为，并证明了其存在唯一性和单调性。本文的研究对相关领域的理论和应用具有一定的参考价值。关键词：非线性偏微分方程组；正解；正则性；对称性；单调性；Schrodinger方程；Korteveg-deVries方程组Abstract:Thispaperstudiestheregularity,symmetry,andmonotonicityofthepositivesolutionsoftwotypesofnonlinearpartialdifferentialequationsystems.Thefirsttypeofequationsystemisthetwo-dimensionalSchrodingerequation,whichhaswideapplicationsinquantummechanics,mathematicalphysics,andnonlinearanalysis.Westudythelocalandglobalexistenceofpositivesolutionswithenergy-convexfunctionsandsingulartermsintheinitialdata,andanalyzethestabilityandmonotonicityofpositivesolutions.ThesecondtypeofequationsystemistheKorteveg-deVriesequationsystem,whichhaswideapplicationsinwaterwaves,soundwaves,gasdynamics,andotherfields.Westudytheglobalexistenceandasymptoticbehaviorofpositivesolutionswithboundedvariationandsingulartermsintheinitialdata,andprovetheiruniquenessandmonotonicity.Theresearchinthispaperhassomereferencevalueforthetheoryandapplicationofrelatedfields.Keywords:nonlinearpartialdifferentialequationsystem;positivesolution;regularity;symmetry;monotonicity;Schrodingerequation;Korteveg-deVriesequationsystem