Inthispaper,weinvestigateaclassofHamiltoniansystemsarisinginnonlinearcompositemedia.Bydetailedanalysisandcomputationweobtainadecayingestimatesonthesemigroupandprovetheorbitalinstabilityoftwofamiliesofexplicitsolitarywavesolutions(slowfamilyinanisotropiccaseandsolitarywavesinisotropiccase),whichtheoreticallyverifytherelatedguessandnumericalresults.