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The new methods shown here are quite general but since each particular problem requires its own study, this paper is limited to elasticity and hydrodynamics. In elasticity Hooke’s law gives systems in a nonconservative form; the study of shock waves for these systems gives nonclassical multiplications of distributions of the form Y⋅δ (Y=Heaviside function, δ=Dirac mass at the origin). Using this new mathematical tool new formulas are obtained (more generally new numerical schemes): in a first step ‘‘ambiguous’’ results are obtained; then the ambiguity is removed. In hydrodynamics a formulation is obtained that has a nonconservative form and is at the basis of efficient new numerical schemes. Strictly speaking the reader is not assumed to know anything either on distributions or on elasticity and hydrodynamics, since the basic equations are recalled. 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Phys., 4, 248, 10.1063\u002F1.1703948",{"doi":1175},"10.1063\u002F1.1703948",{"id":24,"text":361,"url":24,"identifiers":1177},{},{"id":24,"text":1179,"url":24,"identifiers":1180},"1964, J. Math. Phys., 5, 75, 10.1063\u002F1.1704066",{"doi":1181},"10.1063\u002F1.1704066",{"id":24,"text":1183,"url":24,"identifiers":1184},"1964, Phys. Rev., 135, A1646, 10.1103\u002FPhysRev.135.A1646",{"doi":1185},"10.1103\u002FPhysRev.135.A1646",{"id":24,"text":361,"url":24,"identifiers":1187},{},{"id":24,"text":361,"url":24,"identifiers":1189},{},{"id":24,"text":1191,"url":24,"identifiers":1192},"1963, Z. Physik, 175, 553, 10.1007\u002FBF01375347",{"doi":1193},"10.1007\u002FBF01375347",{"id":24,"text":1195,"url":24,"identifiers":1196},"1962, Phys. Rev., 126, 2072",{},{"id":24,"text":1198,"url":24,"identifiers":1199},"1964, J. Math. Phys., 5, 127, 10.1063\u002F1.1704057",{"doi":1200},"10.1063\u002F1.1704057",{"id":24,"text":1202,"url":24,"identifiers":1203},"1964, Phys. Rev., 136, B290, 10.1103\u002FPhysRev.136.B290",{"doi":1204},"10.1103\u002FPhysRev.136.B290",{"id":24,"text":1206,"url":24,"identifiers":1207},"1965, J. Math. Phys., 6, 554, 10.1063\u002F1.1704307",{"doi":1208},"10.1063\u002F1.1704307",{"id":24,"text":1210,"url":24,"identifiers":1211},"1963, Helv. Phys. Acta., 36",{},{"id":24,"text":1213,"url":24,"identifiers":1214},"1964, The Free Energy of a Macroscopic System, Arch. Ratl. Mech. and Analysis, 17, 377, 10.1007\u002FBF00250473",{"doi":1215},"10.1007\u002FBF00250473",{"id":24,"text":1217,"url":24,"identifiers":1218},"1963, J. Math. Phys., 4, 1495, 10.1063\u002F1.1703930",{"doi":1219},"10.1063\u002F1.1703930",{"id":24,"text":361,"url":24,"identifiers":1221},{},{"id":24,"text":361,"url":24,"identifiers":1223},{},{"id":24,"text":1225,"url":24,"identifiers":1226},"1914, Proc. Acad. Sci. Amsterdam, 17, 793",{},{"id":24,"text":361,"url":24,"identifiers":1228},{},{"id":1230,"createTime":1231,"updateTime":1231,"relativeEntities":1232,"slug":1233,"properties":1234,"entityType":193,"verifyStatus":194,"verifyTime":1231,"verifyNote":196,"languages":1245,"translateLanguages":24,"viewCount":25,"primaryUrl":1246,"fullTextUrl":24,"authors":1247,"publicationType":235,"publisherRelationship":1282,"citationCount":1335,"citationInfo":1336,"publishDate":1339,"publishYear":1337,"citationAnalyzeStatus":23,"lastCitationAnalyze":24,"indexDatabases":1340,"openAccess":24,"references":1341,"isForceReanalyzing":450},"838b67c4-010c-4758-a643-264621c3c144","2025-02-11T07:08:08.529+00:00",[],"A-restricted-B%C3%A4cklund-transformation",{"openalex":1235,"mag":1237,"abstract":1239,"title":1241,"doi":1243},{"VOID":1236},"W2195568742",{"VOID":1238},"2195568742",{"EN":1240},"\u003Cjats:p>The Bäcklund transformation provides a mathematical tool which displays the interaction of solitons. Here a simple, systematic Bäcklund formalism is introduced which permits the explicit construction of these transformations for a restricted class of nonlinear wave equations. Traditionally a Bäcklund transformation has been viewed as a transformation of a solution surface of a partial differential equation into another surface which may not satisfy the same equation. In the present paper the term ``restricted Bäcklund transformation'' (hereafter abbreviated R-B) is used to refer to the case in which the transformed surface does satisfy the same equation. This formalism clarifies the nature of those transformations which have already been used to study nonlinear interactions in many physical problems. The formalism is introduced through a form of the linear Klein-Gordon equation. For this linear example a complete set of Fourier components is generated by a sequence of R-B transformations. This concrete example also indicates the type of results one can expect in the nonlinear case. For the nonlinear equation φx y = F(φ), a theorem is established which states that R-B transformations exist if and only if the nonlinearity F(·) satisfies F″ = κF, where κ is a constant. For such nonlinearities, the R-B transformations are explicitly constructed and are used to display exact nonlinear interactions. A relationship between the condition F″ = κF, the existence of an infinite number of conservation laws, and the transformation theory is briefly discussed.\u003C\u002Fjats:p>",{"EN":1242},"A restricted Bäcklund transformation",{"VOID":1244},"10.1063\u002F1.1666254",[198],"https:\u002F\u002Fpubs.aip.org\u002Fjmp\u002Farticle\u002F14\u002F12\u002F1817\u002F224509\u002FA-restricted-Backlund-transformation",[1248,1265],{"id":1249,"sortIndex":25,"researcher":24,"roles":1250,"affiliations":1251,"properties":1260,"displayName":1262,"givenName":24,"familyName":24},"e5bf1a67-3932-439e-95dd-4b65c3f84568",[],[1252],{"id":1253,"sortIndex":25,"affiliation":1254,"properties":24},"0bd56fe9-829f-4d99-a20a-69ff08b9b8b8",{"id":1253,"createTime":24,"updateTime":24,"relativeEntities":1255,"slug":24,"properties":1256,"entityType":24,"verifyStatus":24,"verifyTime":24,"verifyNote":24,"languages":24,"translateLanguages":24,"viewCount":24,"url":24,"parentIds":1259,"statistic":24},[],{"title":1257},{"EN":1258},"Department of Mathematics, Iowa State University, Ames, Iowa 50010",[],{"title":1261,"openalex":1263},{"EN":1262},"David W. McLaughlin",{"VOID":1264},"A5108439359",{"id":1266,"sortIndex":94,"researcher":24,"roles":1267,"affiliations":1268,"properties":1277,"displayName":1279,"givenName":24,"familyName":24},"b3c50e5f-f7d7-4ce3-9d87-3444798750d6",[],[1269],{"id":1270,"sortIndex":25,"affiliation":1271,"properties":24},"da1d1393-7182-48e7-bd73-39e651533a95",{"id":1270,"createTime":24,"updateTime":24,"relativeEntities":1272,"slug":24,"properties":1273,"entityType":24,"verifyStatus":24,"verifyTime":24,"verifyNote":24,"languages":24,"translateLanguages":24,"viewCount":24,"url":24,"parentIds":1276,"statistic":24},[],{"title":1274},{"EN":1275},"Department of Electrical and Computer Engineering, The University of Wisconsin, Madison, Wisconsin 53706",[],{"title":1278,"openalex":1280},{"EN":1279},"Alwyn 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J. Fluid Mech., 47, 811, 10.1017\u002FS0022112071001393",{"doi":1345},"10.1017\u002FS0022112071001393",{"id":24,"text":1347,"url":24,"identifiers":1348},"1969, J. Math. Phys., 10, 536, 10.1063\u002F1.1664873",{"doi":1349},"10.1063\u002F1.1664873",{"id":24,"text":1351,"url":24,"identifiers":1352},"1962, Nucl. Phys., 31, 550, 10.1016\u002F0029-5582(62)90774-5",{"doi":1353},"10.1016\u002F0029-5582(62)90774-5",{"id":24,"text":1355,"url":24,"identifiers":1356},"1970, Nuovo Cimento B, 69, 241, 10.1007\u002FBF02710988",{"doi":1357},"10.1007\u002FBF02710988",{"id":24,"text":1359,"url":24,"identifiers":1360},"1973, J. Phys. Soc. Jap., 34, 18, 10.1143\u002FJPSJ.34.18",{"doi":1361},"10.1143\u002FJPSJ.34.18",{"id":24,"text":1363,"url":24,"identifiers":1364},"1971, Rev. Mod. Phys., 43, 99, 10.1103\u002FRevModPhys.43.99",{"doi":1365},"10.1103\u002FRevModPhys.43.99",{"id":24,"text":1367,"url":24,"identifiers":1368},"1965, Phys. Rev. Lett., 15, 240",{},{"id":24,"text":361,"url":24,"identifiers":1370},{},{"id":24,"text":361,"url":24,"identifiers":1372},{},{"id":24,"text":1374,"url":24,"identifiers":1375},"1971, Riv. Nuovo Cimento, 1, 227, 10.1007\u002FBF02820622",{"doi":1376},"10.1007\u002FBF02820622",{"id":24,"text":361,"url":24,"identifiers":1378},{},{"id":24,"text":1380,"url":24,"identifiers":1381},"1970, J. Math. Phys., 11, 258, 10.1063\u002F1.1665057",{"doi":1382},"10.1063\u002F1.1665057",{"id":24,"text":361,"url":24,"identifiers":1384},{},{"id":24,"text":1386,"url":24,"identifiers":1387},"1953, Z. Phys., 34, 173",{},{"id":24,"text":445,"url":24,"identifiers":1389},{"doi":447},{"id":24,"text":1391,"url":24,"identifiers":1392},"1968, Commun. Pure Appl. Math., 21, 467, 10.1002\u002Fcpa.3160210503",{"doi":1393},"10.1002\u002Fcpa.3160210503",{"id":24,"text":1395,"url":24,"identifiers":1396},"1968, J. Math. Phys., 9, 1204, 10.1063\u002F1.1664701",{"doi":1397},"10.1063\u002F1.1664701",{"id":24,"text":304,"url":24,"identifiers":1399},{},{"id":24,"text":1401,"url":24,"identifiers":1402},"1972, Sov. Phys.-JETP, 34, 62",{},{"id":24,"text":361,"url":24,"identifiers":1404},{},{"id":24,"text":361,"url":24,"identifiers":1406},{},{"id":24,"text":1408,"url":24,"identifiers":1409},"1953, J. Analyse Math., 2, 219",{},{"id":24,"text":361,"url":24,"identifiers":1411},{},{"id":24,"text":1413,"url":24,"identifiers":1414},"1970, Acta Phys. Austriaca, 31, 80",{},{"id":24,"text":1416,"url":24,"identifiers":1417},"1972, Ann. Soc. Sci. Brux. 1, 86, 211",{},{"id":24,"text":1419,"url":24,"identifiers":1420},"1969, Z. Angew. Math. Mech., 49, 333, 10.1002\u002Fzamm.19690490603",{"doi":1421},"10.1002\u002Fzamm.19690490603",{"id":24,"text":1423,"url":24,"identifiers":1424},"1876, Math. Ann., 9, 297",{},{"id":24,"text":1426,"url":24,"identifiers":1427},"1882, Math. Ann., 19, 387",{},{"id":24,"text":1429,"url":24,"identifiers":1430},"1903, Ann. Fac. Sci. Univ. Toulouse, 5, 437, 10.5802\u002Fafst.210",{"doi":1431},"10.5802\u002Fafst.210",{"id":24,"text":361,"url":24,"identifiers":1433},{},{"id":24,"text":361,"url":24,"identifiers":1435},{},{"id":24,"text":361,"url":24,"identifiers":1437},{},{"id":24,"text":361,"url":24,"identifiers":1439},{},{"id":24,"text":1441,"url":24,"identifiers":1442},"1973, Bull. Am. Phys. Soc., 18, 64",{},{"id":24,"text":1444,"url":24,"identifiers":1445},"1973, Proc. IEEE, 61",{},{"id":24,"text":361,"url":24,"identifiers":1447},{},{"id":24,"text":1449,"url":24,"identifiers":1450},"1970, Trans. Am. Math. Soc., 150, 327",{},{"id":24,"text":1452,"url":24,"identifiers":1453},"1972, Trans. Am. Math. Soc., 166, 371",{},{"id":1455,"createTime":1456,"updateTime":1456,"relativeEntities":1457,"slug":1458,"properties":1459,"entityType":193,"verifyStatus":194,"verifyTime":1456,"verifyNote":196,"languages":1470,"translateLanguages":24,"viewCount":25,"primaryUrl":1471,"fullTextUrl":24,"authors":1472,"publicationType":235,"publisherRelationship":1507,"citationCount":1559,"citationInfo":1560,"publishDate":1563,"publishYear":1561,"citationAnalyzeStatus":23,"lastCitationAnalyze":24,"indexDatabases":1564,"openAccess":24,"references":1565,"isForceReanalyzing":450},"bda54ae2-8701-44b1-97fe-eae773872200","2025-02-11T07:08:06.872+00:00",[],"Multicomponent-integrable-reductions-in-the-Kadomtsev-Petviashvilli-hierarchy",{"openalex":1460,"mag":1462,"abstract":1464,"title":1466,"doi":1468},{"VOID":1461},"W2058712380",{"VOID":1463},"2058712380",{"EN":1465},"\u003Cjats:p>New types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy are considered on the basis of Sato’s approach. Within this approach the KP hierarchy is represented by infinite sets of equations for potentials u2,u3,..., of pseudodifferential operators and their eigenfunctions Ψ and adjoint eigenfunctions Ψ*. The KP hierarchy was studied under constraints of the following type (∑ni=1 ΨiΨ*i)x = Sκ,x where Sκ,x are symmetries for the KP equation and Ψi(λi), Ψ*i(λi) are eigenfunctions with eigenvalue λi. It is shown that for the first three cases κ=2,3,4 these constraints give rise to hierarchies of 1+1-dimensional commuting flows for the variables u2, Ψ1,...,Ψn, Ψ*1,...,Ψ*n. Bi-Hamiltonian structures for the new hierarchies are presented.\u003C\u002Fjats:p>",{"EN":1467},"Multicomponent integrable reductions in the Kadomtsev–Petviashvilli hierarchy",{"VOID":1469},"10.1063\u002F1.530416",[198],"https:\u002F\u002Fpubs.aip.org\u002Fjmp\u002Farticle\u002F34\u002F4\u002F1429\u002F229143\u002FMulticomponent-integrable-reductions-in-the",[1473,1490],{"id":1474,"sortIndex":25,"researcher":24,"roles":1475,"affiliations":1476,"properties":1485,"displayName":1487,"givenName":24,"familyName":24},"c4ad63b9-0525-4fdd-bbb3-eed29e4c663b",[],[1477],{"id":1478,"sortIndex":25,"affiliation":1479,"properties":24},"a27fabd0-7887-4520-81bc-7be2b7edb75f",{"id":1478,"createTime":24,"updateTime":24,"relativeEntities":1480,"slug":24,"properties":1481,"entityType":24,"verifyStatus":24,"verifyTime":24,"verifyNote":24,"languages":24,"translateLanguages":24,"viewCount":24,"url":24,"parentIds":1484,"statistic":24},[],{"title":1482},{"EN":1483},"University of Lwow, Faculty of Mathematics 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A, 89, 332, 10.1016\u002F0375-9601(82)90186-4",{"doi":1711},"10.1016\u002F0375-9601(82)90186-4",{"id":24,"text":361,"url":24,"identifiers":1713},{},{"id":1715,"createTime":1716,"updateTime":1716,"relativeEntities":1717,"slug":1718,"properties":1719,"entityType":193,"verifyStatus":194,"verifyTime":1730,"verifyNote":196,"languages":1731,"translateLanguages":24,"viewCount":25,"primaryUrl":1732,"fullTextUrl":24,"authors":1733,"publicationType":235,"publisherRelationship":1768,"citationCount":1820,"citationInfo":1821,"publishDate":1823,"publishYear":290,"citationAnalyzeStatus":23,"lastCitationAnalyze":24,"indexDatabases":1824,"openAccess":24,"references":1825,"isForceReanalyzing":450},"dfa6b7cd-cd55-461d-a699-8ad90d1bf1e2","2025-02-11T07:08:06.420+00:00",[],"New-reductions-of-the-Kadomtsev-Petviashvili-and-two-dimensional-Toda-lattice-hierarchies-via-symmetry-constraints",{"openalex":1720,"mag":1722,"abstract":1724,"title":1726,"doi":1728},{"VOID":1721},"W2075115641",{"VOID":1723},"2075115641",{"EN":1725},"\u003Cjats:p>New types of reductions of the Kadomtsev–Petviashvili (KP) hierarchy and the two-dimensional Toda lattice (2DTL) hierarchy are considered on the basis of Sato’s approach. Within this approach these hierarchies are represented by infinite sets of equations for potentials u1,u2,u3,..., of pseudodifferential operators and their eigenfunctions ψ and adjoint eigenfunctions ψ*. The KP and the 2DTL hierarchies are studied under constraints of the following type: ∑n=1N αnSn(u1,u2,u3,...)=Ωx, where Sn are symmetries for these hierarchies, αn are arbitrary constants, and Ω is an arbitrary linear functional of the quantity ψ(λ)ψ*(μ). It is shown that for the KP hierarchy these constraints give rise to hierarchies of 1+1-dimensional commuting flows for the variables u2,u3,...,uN,ψ,ψ*. Many known systems and several new ones are among them. Symmetry reductions for the 2DTL hierarchy give rise both to finite-dimensional dynamical systems and 1+1-dimensional discrete systems. 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A proof, which is independent of the Kinnersley–Chitre formalism, establishes that the HHP transforms the potential (for certain closed self-dual 2 forms) F0(x, t) of any given member of V into the potential F (x, t) of another member of V. Two illustrative examples involving the Minkowski space F0(x, t) are given. 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