ID-AS ~ DEPT OF MATHEMATICS AND COMPUTER SCIENCE N J ABLOWITZ so APR 96 AFOSR-RI-12AFS-4SS UNCLSSIFIED 98AR9 FS-R8-12ROR9-05 F/G 12/1 NL. Eu'.

Transcription

1 ID-AS ~ DEPT OF MATHEMATICS AND COMPUTER SCIENCE N J ABLOWITZ so APR 96 AFOSR-RI-12AFS-4SS UNCLSSIFIED 98AR9 FS-R8-12ROR9-05 F/G 12/1 NL Eu'."n

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3 -t :uri ry CLASSIF I ~MENTA.TION PAGE UN AD-A l 2*. SCURIT CLA3. DISTRISUTION/AVAI LABILITY OF REPORT 2 b OCSIFIATONOOWNGRAOING SCHEDULE Approved f or public3 release; distibuionunlimited. 4. RGAIZAIONREORTNuMER() ERFRMIG. MOIOIGORGANIZAIO REPWT NUMBER(S) Clarkson University OAAIZTO 6c. ADDRESS tcity. State and ZIP Cc -. 7b. ADDRESS (City, State and ZIP Code) *Department of Mathematics & Cmue cec *Potsdam, New York ' ( ee\q CA C ORGANIZATION Scientific Oft applicable) Air Force Office of Research nl \c4 C) PROGRAM PROJECT TASK 'NO R 10 ' ADORESS fcity. State and ZIP Code) 10. SOURCE OF FUNDING NOS. Boiling AFB, Washington, DC ELEMENT NO. NO. NO, TITLE Include Security Clashification) J \IQ 12. PERSONAL AUTHORISI Mark J. Ablowitz - Principal Investigator 13. O RPOT YP 13b. TIME COVERED (114. DATE OF REPORT (Yr. Mro., Dayl) 115. PAGE COUNT Progress Report jfrom 10/1/85L To48L I 16Api SUPPLEMENTARY NOTATION ciel I I3R0U CODES 18.* SUBJECT TERMS icontinue on revers.e i(rneeisar, and identf( by block num05pri 19 ABSTRACT Continue on rever,, it riecessam- arnd iden tify hy blade number, * The central theme involved in this work is the continuing study of O~certain fundamental features associated with the nonlinear wave propagation C arising in and motivated by physical problems. The usefulness of the * C...~work is attested to by the varied applications, anid wide areas of interest U~ in physics, engineering and mathematics. The work accomplished involves... wave propagation in a number of areas including fluid mechanics, plasma S physics, theoretical physics, statistical mechanics, nonlinear optic", multidimensional solitons, multidimensional inverse problems, Painleve *C,) equations, direct linearizations of certain nonlinear wave equations, * ~ DBAR problems, Riemann-Hilbert boundary value problems, differential S geometry, etc._ 20 DISTRIBUTIONtAVAILABILiTY OF ABSTRACT ~ 21 ABSTRACT SECuOITY (- _;ASSIF CATION j UNCLASSiFIED,UNLIMITED a. SAME AS RPT -- OT!C 5~ERS 22a NAE NCIIOAL F RSPOSILE cr-wk-i 2bTELEPFNCNE lu MBER z21: DrFCE SYVMBCL 22. AME F NDI'IDUA~YA~~ RE~ONSBLE Include l%,,(3 ril-rkja-z-abow'itz, Ptoir.ipa! (3vzti5)~ 96 2f FORM 1473, 83 APR EDITION 'OF I jan 73 IS OBSOLETE 1E u TICL"S FCATI)-

4 AFOSR -TR ANNUAL TECHNICAL REPORT NONLINEAR WAVE PROPAGATION AFOSR Grant AFOSR BY Mark J. Ablowitz Department of Mathematics and K> Computer Science Clarkson University Potsdam, NY October 1, September 30, 1985

5 During the past year significant progress has been made in two related but different areas: (i) Solutions of nonlinear evolution equations (ii) Inverse Scattering. These two fields are bound together via Soliton Theory and the Inverse Scattering Transform. In (i) the recent progress has been twofold. First a new class of solvable nonlinear singular integro-differential equations has been found. The most interesting of which is the sine-hilbert equation: Hut = sin u (1) where Hu = Y W is the Hilbert transform of u. Equation (1) generalizes the sine-gordon equation uxt =sin u (2) to singular integral equations. It should be noted that (1) is the first really interesting solvable singular integral equation since the well known Benjamin-Ono equation (solved by us in 1983): ut + 2uu + Hu 0. (3) It should be noted that the Benjamin-Ono equation may be thought of as the singular integral form of the Korteweg-deVries equation ut + 2uu +u =0. (4) A preprint is almost ready on this work. Secondly, we have recently solved an n-dimensional generalization of the sine-gordon equation which had been studied earlier and derived by a group of differential geometers."-... L. w ". -. #... ' ".. ".....% :. '....,- 2..-,- * ".. * "

6 -2- at Berkeley; Chern, Terng, Tenenblat. These results are very new and indeed demonstrate that interesting multidimensional nonlinear equations can in fact be solved by inverse scattering. We shall write this work up in the near future. Suffice it to say that it can be expected that the results will be of considerable interest in the mathematics and physics community. It should be remarked that earlier work on solvable multidimensional nonlinear partial differential equations had been confined to three independent variables i.e. "2+1" dimensional problems such as the Kadomstev-Petviashvili, Davey Stewartson and three wave interaction equations. We reported on these results in the recent past. With regard to (ii) above: Inverse Scattering, progress continues to be made using the so-called 5 ("DBAR") method. The method allows us to deal naturally and systematically with multidimensional inverse scattering problems in n dimensions. It leads to formulae to reconstruct local potentials as well as characterization conditions which specify what restrictions on the scattering data are necessary in order to allow reconstruction of a local potential. When restricted to the classical multidimensional time independent Schrddinger scattering problem, our results agree with and indeed go further than those of Faddeev. Moreover, our method applies to the time dependent Schrddinger problem as well as elliptic and hyperbolic systems and is new. We expect that reconstruction/ characterization formulae will be able to be found via these methods in other important problems as well e.g. geophysical inverse problems..d. d.... j C. ~.-

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