一个能量既依位势又依速度的二阶特征值问题及相关系统的可积性.pdf

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1、©aÒ:—?:UDC:?Ò:ÆØ©˜‡UþQ•³q•„ÝAŠ¯K9ƒ'XÚŒÈ5L•“6¶:6£Ǥàó’ŒÆžÆ?O:a¬Æ‰!;’¶¡:A^êÆØ©Jžm:2011c09Ø©‰Fžm:2011c12Æǃü:àó’ŒÆ‰F”¬ÌR:µ<:2011c12DissertationSubmittedtoHebeiUniversityofTechnologyforTheMasterDegreeofScienceinAppliedMathematicsTHEINTEGRABLESYSTEMONTHESECON

2、D-ORDEREIGENVALUEPROBLEMWITHTHEPOTENTIALANDSPEEDbyYangJunyingSupervisor:Prof.GuZhuquanDecember2011àó’ŒÆa¬ÆØ©˜‡UþQ•³q•„ÝAŠ¯K9ƒ'XÚŒÈ5Á‡©Ì‡?ØUþQ•³q•„ÝAŠ¯KµL'=(@2+u+v)'='x¤éABargmannXÚ.Äk0˜ÄVg,,ÏL9̯K9̃N5^‡½ÂÜnVHamiltonŽfK;J:¿̯K¤éAuЕ§x.A^•¼F݆Lenard4íS

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éÜXy²ÙŒÈ5.ùÒ阇#ŒÈXÚ,¿¼uЕ§x)éÜL«.'…i:AŠ¯K,³,BargmannXÚ,Laxéš‚5z,HamiltonŒÈXÚ,éÜL«.i˜‡UþQ•³q•„ÝAŠ¯K9ƒ'XÚŒÈ5T

7、HEINTEGRABLESYSTEMONTHESECOND-ORDEREIGENVALUEPROBLEMWITHTHEPOTENTIALANDSPEEDABSTRACTInthispaper,theBargmannsystemforthesecond-ordereigenvalueproblemwiththeenergyisdependentonpotentialandspeedL'=(@2+u+v)'='xisdiscussed.Firstofall,somebasicconceptsareintroduced.Afterthat,b

8、ymean-softheauxiliaryproblemandtheisospectralcompatiblecondition,thereasonablebi-HamiltonoperatorKandJaredened,andtheevolutionequationsrelatedtothespectralproblemareobtained.Usingthefunc-tionalgradientandLenardrecursivesequence,theBargmannconstraintisgiven.Bytheconstraint

9、relationbetweenthepotentialfunctionandtheeigenvector,theassociatedLaxpairsarenonlineared,thentheBargmannsystemoftheeigenvalueproblemisfound.AccordingtotheviewpointofHamiltonmechanics,areasonablecoordinatesystemhasbeenfound.TheBargmannsystemistransformedintotheHamiltoncanon

10、icalequationsinthecoordinatesystem.Finally,basedontheLiouvilletheoremandconfo-calinvolutivesystem,theintegrabilityoftheHamiltonsystemisproved.Anewintegrablesystemisfound.Moreover,theinvolutiverepresentationsofthesolutionsfortheevolutionequationsaregenerated.KEYWORDS:eigenv

11、alueproblem,potential,Bargmannsystem,non-linearedLaxpairs,Hamiltonintegrablesystem,involu