能量依赖位势的二阶特征值问题相关的发展方程族及Lax表示.pdf

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1、©aÒ:—?:UDC:?Ò:ÆØ©Uþ•6³AŠ¯Kƒ'uЕ§x9LaxL«_¡A•“6¶:6£Ǥàó’ŒÆžÆ?O:a¬Æ‰!;’¶¡:A^êÆØ©Jžm:2011c09Ø©‰Fžm:2011c12Æǃü:àó’ŒÆ‰F”¬ÌR:µ<:2011c12DissertationSubmittedtoHebeiUniversityofTechnologyforTheMasterDegreeofScienceinAppliedMathematicsTHEEVOLUTION

2、EQUATIONSANDLAXREPRESENTATIONOFTHESECOND-ORDEREIGENVALUEPROBLEMWITHTHEPOTENTIALbyJiaXiaonaSupervisor:Prof.GuZhuquanDecember2011M5(²

3、±²(•ªI²"ÆØ©M5(²{ÆI?d<«ú.ÆØ©Šö¶µFϵ'uÆØ©‡¦^Ç`²<)àó’ŒÆ'uÂ8!•!¦^ÆØ©±e5½µÆkæ^Kf‡.£—ÆØ©3)—·^Ç`²¤ÆØ©Šö¶µFϵ“¶µFϵàó’ŒÆa¬ÆØ©Uþ•6³AŠ¯Kƒ

4、'uЕ§x9LaxL«Á‡©Ì‡?ØUþ•6³AŠ¯KµL'=(@2+u+2v+3p+4q)'=2'xØÓ̯KzXÚkŸ«O.Äk0˜ÄVg,,ÏL9̯K9̃N5^‡,½ÂÜnVHamiltonŽfK;J:¿̯K¤éAuЕ§x.A^•¼F݆Lenard4íS(½Bargmannå.

5、^³¼ê(u,v,p,q)†A¼ê'ƒmå'X,òÙƒAuЕ§xLaxéš‚5z,lAŠ¯KéABargmannXÚ.?ŠâHamiltonåÆ*:

6、ïá˜

7、Ün‹IX,BargmannXÚ3d‹IeHamiltonK•§.•òá‘XÚ=z•k•‘HamiltonŒÈXÚ,¿¼uЕ§x)éÜL«.'…i:AŠ¯K,³,BargmannXÚ,Laxéš‚5z,HamiltonŒÈXÚ,éÜL«.iUþ•6³AŠ¯Kƒ'uЕ§x9LaxL«THEEVOLUTIONEQUATIONSANDLAXOFTHESECOND-ORDEREIGENVALUEPROBLEMWITHTHEPOTENTIALABSTRACTInthispap

8、er,wediscussthesecond-ordereigenvalueproblemwiththeenergydependingonpotential.L'=(@2+u+2v+3p+4q)'=2'xThedierentconstraintoftheauxiliaryproblemisessentialdierence.Firstofall,somebasicconceptsareintroduced.Afterthat,bymean-softheauxiliaryproblemandthe

9、isospectralcompatiblecondition,thereasonablebi-HamiltonoperatorKandJaredened,andtheevolutione-quationsrelatedtothespectralproblemareobtained.UsingthefunctionalgradientandLenardrecursivesequence,theBargmannconstraintisgiven.Bytheconstraintrelationbetweenth

10、epotentialfunction(u,v,p,q)andtheeigenvector,theassociatedLaxpairsarenonlineared,thentheBargmannsystemoftheeigenvalueproblemisfound.AccordingtotheviewpointofHamiltonmechanics,areasonablecoordinatesystemhasbee