Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework
The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on...
Gespeichert in:
| Veröffentlicht in: | Physica. D Jg. 172; H. 1; S. 190 - 216 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Elsevier B.V
15.11.2002
|
| Schlagworte: | |
| ISSN: | 0167-2789, 1872-8022 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian
G
2(
C
5)
in choosing the numerical integrator, and the role of the Hodge star operator for relating
⋀
2(
C
5)
and
⋀
3(
C
5)
and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity. |
|---|---|
| AbstractList | The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans function, a complex analytic function whose zeros correspond to eigenvalues. A numerical framework, based on a fast robust shooting algorithm on exterior algebra spaces is introduced. The complete algorithm has several new features, including a rigorous numerical algorithm for choosing starting values, a new method for numerical analytic continuation of starting vectors, the role of the Grassmannian
G
2(
C
5)
in choosing the numerical integrator, and the role of the Hodge star operator for relating
⋀
2(
C
5)
and
⋀
3(
C
5)
and deducing a range of numerically computable forms for the Evans function. The algorithm is illustrated by computing the stability and instability of solitary waves of the fifth-order KdV equation with polynomial nonlinearity. |
| Author | Bridges, Thomas J. Derks, Gianne Gottwald, Georg |
| Author_xml | – sequence: 1 givenname: Thomas J. surname: Bridges fullname: Bridges, Thomas J. – sequence: 2 givenname: Gianne surname: Derks fullname: Derks, Gianne email: g.derks@surrey.ac.uk – sequence: 3 givenname: Georg surname: Gottwald fullname: Gottwald, Georg |
| BookMark | eNqFUMlOwzAUtFCRaAufgOQjHAJekjiBA0IVm6jEocDVcuwX1ZDGYLut-vckBfXApZe36c1oZkZo0LoWEDql5IISml_OuiISJoryjLBzQvIsS_gBGtJCsKQgjA3QcPdyhEYhfBBCqOBiiKpZVJVtbNxg1Rps27DbXY2D6yblN3itVhD6S5wDrm0d54nzBjx-Nu8YvpcqWtdeYYXb5QK81arBtVcLWDv_eYwOa9UEOPnrY_R2f_c6eUymLw9Pk9tporngMam41qUxWcmqlCiRdhPwEjJTZ4ooo2iu05LmFcsLSnLOTVYJoZgWaZGyglI-Rte_vNq7EDzUUnfie2HRK9tISmQfl9zGJfssJGFyG5fkHTr7h_7ydtF534u7-cVBZ21lwcugLbQajPWgozTO7mH4AfLEhVM |
| CitedBy_id | crossref_primary_10_1016_j_jde_2008_07_028 crossref_primary_10_4213_rm9690 crossref_primary_10_1007_s10884_015_9440_3 crossref_primary_10_1137_S0036141004439350 crossref_primary_10_1007_s00220_009_0885_2 crossref_primary_10_1016_j_physd_2010_03_006 crossref_primary_10_1016_j_physd_2006_07_003 crossref_primary_10_1016_S0252_9602_10_60058_6 crossref_primary_10_1137_040605953 crossref_primary_10_1137_10081441X crossref_primary_10_1016_j_physd_2025_134628 crossref_primary_10_1016_j_jde_2022_06_019 crossref_primary_10_1016_j_physd_2006_01_024 crossref_primary_10_1137_140975590 crossref_primary_10_1007_s00211_006_0004_7 crossref_primary_10_1007_s00332_008_9023_0 crossref_primary_10_1140_epjp_s13360_024_04951_4 crossref_primary_10_1007_s00020_013_2117_6 crossref_primary_10_1016_j_physd_2009_05_008 crossref_primary_10_1137_050640849 crossref_primary_10_1137_19M1259705 crossref_primary_10_1016_j_cnsns_2019_105073 crossref_primary_10_1007_s10444_012_9273_0 crossref_primary_10_1088_1402_4896_ab5290 crossref_primary_10_1016_j_physd_2020_132653 crossref_primary_10_1016_j_matcom_2022_06_020 crossref_primary_10_1016_j_physd_2017_12_002 crossref_primary_10_1088_1361_6544_ad8698 crossref_primary_10_1137_040605308 crossref_primary_10_1070_RM9953 crossref_primary_10_1090_S0033_569X_2010_01209_1 crossref_primary_10_1007_s00033_015_0538_6 crossref_primary_10_1016_S0997_7546_03_00036_0 crossref_primary_10_1111_sapm_12136 crossref_primary_10_1007_s13160_020_00428_w crossref_primary_10_1016_j_physd_2015_05_010 crossref_primary_10_1090_S0033_569X_2011_01221_6 crossref_primary_10_1088_0305_4470_39_47_002 crossref_primary_10_1007_s00220_010_1095_7 crossref_primary_10_1016_j_jmaa_2017_08_021 crossref_primary_10_1137_23M1598106 crossref_primary_10_1137_050628271 crossref_primary_10_1088_0305_4470_39_19_S02 crossref_primary_10_1007_s00205_010_0363_1 crossref_primary_10_1007_s00220_008_0487_4 crossref_primary_10_1016_j_mechrescom_2018_05_002 crossref_primary_10_1007_s00220_010_1175_8 crossref_primary_10_1007_s00332_024_10098_4 crossref_primary_10_1016_j_matpur_2009_10_001 crossref_primary_10_1016_j_mbs_2015_05_009 crossref_primary_10_1007_s00028_020_00630_w crossref_primary_10_1016_j_physd_2013_06_001 crossref_primary_10_1016_j_physd_2021_132872 crossref_primary_10_1137_18M1196121 crossref_primary_10_1016_j_physd_2008_04_009 crossref_primary_10_1007_s00285_010_0379_z crossref_primary_10_1016_j_cnsns_2013_10_027 crossref_primary_10_1007_s42064_022_0144_2 crossref_primary_10_1111_sapm_12755 crossref_primary_10_1016_j_jde_2005_09_001 crossref_primary_10_1016_j_physd_2008_03_008 crossref_primary_10_1137_17M113770X crossref_primary_10_1016_j_jde_2010_07_019 crossref_primary_10_1016_j_physd_2016_05_016 crossref_primary_10_1137_19M1251977 crossref_primary_10_1007_s00205_008_0153_1 crossref_primary_10_1137_080714804 crossref_primary_10_1007_s11232_012_0108_4 crossref_primary_10_1137_140980223 crossref_primary_10_1007_s00205_008_0195_4 crossref_primary_10_4213_rm9953 crossref_primary_10_1007_s00205_008_0112_x crossref_primary_10_1016_j_physd_2022_133610 crossref_primary_10_1016_j_jde_2006_10_006 crossref_primary_10_1016_j_jde_2008_10_006 crossref_primary_10_1007_s40819_020_00907_1 |
| Cites_doi | 10.1007/BF00417931 10.1016/0165-2125(94)90003-5 10.1016/S0375-9601(99)00712-4 10.1007/s002220050303 10.1016/0021-9991(79)90091-3 10.1017/S0022112079000835 10.1007/BF01395882 10.1016/S0375-9601(98)00917-7 10.1103/PhysRevLett.80.5117 10.1088/0951-7715/11/2/009 10.1137/S0036141099361494 10.1080/01630569908816889 10.1137/0523064 10.1090/S0025-5718-00-01237-0 10.1137/1.9780898719543 10.1007/s002110100365 10.1016/S0997-7546(98)80023-X 10.1016/0167-2789(93)90197-9 10.1007/PL00004235 10.1143/JPSJ.33.260 10.1098/rspa.2000.0665 10.1016/0375-9601(95)00752-0 10.1016/0375-9601(90)90307-A 10.1093/imanum/7.1.1 10.1007/978-1-4757-2189-8 10.1098/rspa.1999.0437 10.1512/iumj.1975.24.24096 10.1016/S0167-2789(98)00172-9 10.21711/231766362002/rmc222 10.1016/S0375-9601(96)00772-4 10.1016/S0167-2789(98)00245-0 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1 10.1016/S0010-4655(99)00241-6 10.1017/S0022112097005193 10.1016/S0167-2789(97)00209-1 10.1137/S0036141099361834 10.1016/S0167-2789(01)00283-4 10.1007/978-1-4757-3946-6 10.1007/s002110050020 10.1016/0025-5564(77)90076-1 |
| ContentType | Journal Article |
| Copyright | 2002 Elsevier Science B.V. |
| Copyright_xml | – notice: 2002 Elsevier Science B.V. |
| DBID | AAYXX CITATION |
| DOI | 10.1016/S0167-2789(02)00655-3 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Engineering Physics |
| EISSN | 1872-8022 |
| EndPage | 216 |
| ExternalDocumentID | 10_1016_S0167_2789_02_00655_3 S0167278902006553 |
| GroupedDBID | --K --M -~X .~1 0R~ 1B1 1RT 1~. 1~5 29O 4.4 457 4G. 5VS 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO ABAOU ABFNM ABMAC ABNEU ABXDB ABYKQ ACAZW ACDAQ ACFVG ACGFS ACNCT ACNNM ACRLP ADBBV ADEZE ADGUI ADIYS ADMUD AEBSH AEKER AFFNX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AIEXJ AIGVJ AIKHN AITUG AIVDX AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BBWZM BKOJK BLXMC EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-Q GBLVA HMV HVGLF HZ~ H~9 IHE J1W K-O KOM M38 M41 MHUIS MO0 MVM N9A NDZJH O-L O9- OAUVE OGIMB OZT P-8 P-9 P2P PC. Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SPD SPG SSQ SSW SSZ T5K TN5 TWZ WUQ XJT XPP YNT YYP ~02 ~G- 9DU AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ADNMO ADVLN AEIPS AFJKZ AGQPQ AIIUN ANKPU APXCP CITATION EFKBS ~HD |
| ID | FETCH-LOGICAL-c373t-b3cc9dd592b40a74d59e39e5df5a0ada16c4916b26810633d5b77a2c748428113 |
| ISICitedReferencesCount | 96 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000179357700011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0167-2789 |
| IngestDate | Tue Nov 18 21:10:33 EST 2025 Sat Nov 29 03:57:19 EST 2025 Fri Feb 23 02:16:03 EST 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 1 |
| Keywords | Evans function Fifth-order KdV Linear stability Numerical exterior algebra |
| Language | English |
| License | https://www.elsevier.com/tdm/userlicense/1.0 |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c373t-b3cc9dd592b40a74d59e39e5df5a0ada16c4916b26810633d5b77a2c748428113 |
| PageCount | 27 |
| ParticipantIDs | crossref_citationtrail_10_1016_S0167_2789_02_00655_3 crossref_primary_10_1016_S0167_2789_02_00655_3 elsevier_sciencedirect_doi_10_1016_S0167_2789_02_00655_3 |
| PublicationCentury | 2000 |
| PublicationDate | 2002-11-15 |
| PublicationDateYYYYMMDD | 2002-11-15 |
| PublicationDate_xml | – month: 11 year: 2002 text: 2002-11-15 day: 15 |
| PublicationDecade | 2000 |
| PublicationTitle | Physica. D |
| PublicationYear | 2002 |
| Publisher | Elsevier B.V |
| Publisher_xml | – name: Elsevier B.V |
| References | Bridges, Reich (BIB15) 2001; 156 Champneys (BIB18) 1998; 112 Kichenassamy, Olver (BIB39) 1992; 23 Bridges (BIB11) 1999; 455 Haragus-Courcelle, Ill’ichev (BIB31) 1998; 17 Barashenkov, Zemlyanaya (BIB8) 2000; 126 Champneys, Groves (BIB19) 1997; 342 P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981. Gardner, Zumbrun (BIB28) 1998; 51 Kawahara (BIB37) 1972; 33 Groves (BIB30) 1998; 11 Soffer, Weinstein (BIB46) 1999; 136 Afendikov, Bridges (BIB2) 2001; 457 Pego, Smereka, Weinstein (BIB45) 1993; 67 Ascher, Chin, Reich (BIB6) 1994; 67 Evans (BIB26) 1975; 24 T.J. Bridges, G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal. 33 (2002) 1356–1378. M. Liefvendahl, G. Kreiss, Numerical investigation of examples of unstable viscous shock waves, in: Proceedings of Eight International Conference on Hyper. Problems Hyp’2000, Universität of Magdeburg, 2000. Bridges, Derks (BIB12) 1999; 251 Alexander, Sachs (BIB4) 1995; 2 M. Marcus, Finite Dimensional Multilinear Algebra, Part II, Marcel Dekker, New York, 1975. Cooper (BIB21) 1987; 7 Alexander, Gardner, Jones (BIB3) 1990; 410 Levandosky (BIB40) 1999; 125 J. Harris, Algebraic Geometry, Springer, Berlin, 1992. Ying, Katz (BIB48) 1988; 53 Benzoni-Gavage, Serre, Zumbrun (BIB9) 2001; 32 Ng, Reid (BIB44) 1979; 30 Ill’ichev, Semenov (BIB33) 1992; 3 W.J.F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM Publications, Philadelphia, 2000. Ablowitz, Segur (BIB1) 1979; 92 Evans, Feroe (BIB27) 1977; 37 T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, Heidelberg, 1984. Swinton, Elgin (BIB47) 1990; 145 H.B. Keller, Numerical Solution of Two-Point Boundary-Value Problems, CBMS-NSF Conference Series, vol. 24, SIAM, Philadelphia, 1974. R.O. Wells, Differential Analysis on Complex Manifolds, Springer, Berlin, 1980. Bridges, Derks (BIB13) 2001; 156 Beyn, Lorenz (BIB10) 1999; 20 Karpman (BIB35) 1996; 210 Barashenkov, Pelinovsky, Zemlyanaya (BIB7) 1998; 80 E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. L. Allen, T.J. Bridges, Numerical exterior algebra and the compound matrix method, Numer. Math. 92 (2002) 197–232. Brin (BIB16) 2001; 70 Kapitula, Sandstede (BIB34) 1998; 124 J.R. Magnus, H. Neudecker, Matrix Differential Calculus, Wiley, New York, 1988. Dey, Khare, Kumar (BIB23) 1996; 223 L.Q. Brin, K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves, Preprint, Indiana University, 2002. Dias, Kuznetsov (BIB24) 1999; 263 Craig, Groves (BIB22) 1994; 19 10.1016/S0167-2789(02)00655-3_BIB20 Haragus-Courcelle (10.1016/S0167-2789(02)00655-3_BIB31) 1998; 17 Craig (10.1016/S0167-2789(02)00655-3_BIB22) 1994; 19 Bridges (10.1016/S0167-2789(02)00655-3_BIB12) 1999; 251 Champneys (10.1016/S0167-2789(02)00655-3_BIB18) 1998; 112 Groves (10.1016/S0167-2789(02)00655-3_BIB30) 1998; 11 Ng (10.1016/S0167-2789(02)00655-3_BIB44) 1979; 30 Kapitula (10.1016/S0167-2789(02)00655-3_BIB34) 1998; 124 Afendikov (10.1016/S0167-2789(02)00655-3_BIB2) 2001; 457 10.1016/S0167-2789(02)00655-3_BIB17 10.1016/S0167-2789(02)00655-3_BIB14 Barashenkov (10.1016/S0167-2789(02)00655-3_BIB7) 1998; 80 Champneys (10.1016/S0167-2789(02)00655-3_BIB19) 1997; 342 10.1016/S0167-2789(02)00655-3_BIB32 Bridges (10.1016/S0167-2789(02)00655-3_BIB11) 1999; 455 Bridges (10.1016/S0167-2789(02)00655-3_BIB15) 2001; 156 Kichenassamy (10.1016/S0167-2789(02)00655-3_BIB39) 1992; 23 10.1016/S0167-2789(02)00655-3_BIB29 10.1016/S0167-2789(02)00655-3_BIB5 Beyn (10.1016/S0167-2789(02)00655-3_BIB10) 1999; 20 10.1016/S0167-2789(02)00655-3_BIB25 10.1016/S0167-2789(02)00655-3_BIB42 Karpman (10.1016/S0167-2789(02)00655-3_BIB35) 1996; 210 10.1016/S0167-2789(02)00655-3_BIB41 10.1016/S0167-2789(02)00655-3_BIB43 Brin (10.1016/S0167-2789(02)00655-3_BIB16) 2001; 70 Pego (10.1016/S0167-2789(02)00655-3_BIB45) 1993; 67 Bridges (10.1016/S0167-2789(02)00655-3_BIB13) 2001; 156 Levandosky (10.1016/S0167-2789(02)00655-3_BIB40) 1999; 125 Ill’ichev (10.1016/S0167-2789(02)00655-3_BIB33) 1992; 3 Benzoni-Gavage (10.1016/S0167-2789(02)00655-3_BIB9) 2001; 32 10.1016/S0167-2789(02)00655-3_BIB38 Evans (10.1016/S0167-2789(02)00655-3_BIB27) 1977; 37 Gardner (10.1016/S0167-2789(02)00655-3_BIB28) 1998; 51 10.1016/S0167-2789(02)00655-3_BIB36 Dias (10.1016/S0167-2789(02)00655-3_BIB24) 1999; 263 Dey (10.1016/S0167-2789(02)00655-3_BIB23) 1996; 223 Ablowitz (10.1016/S0167-2789(02)00655-3_BIB1) 1979; 92 Soffer (10.1016/S0167-2789(02)00655-3_BIB46) 1999; 136 Ascher (10.1016/S0167-2789(02)00655-3_BIB6) 1994; 67 Kawahara (10.1016/S0167-2789(02)00655-3_BIB37) 1972; 33 Alexander (10.1016/S0167-2789(02)00655-3_BIB3) 1990; 410 Barashenkov (10.1016/S0167-2789(02)00655-3_BIB8) 2000; 126 10.1016/S0167-2789(02)00655-3_BIB49 Alexander (10.1016/S0167-2789(02)00655-3_BIB4) 1995; 2 Ying (10.1016/S0167-2789(02)00655-3_BIB48) 1988; 53 Evans (10.1016/S0167-2789(02)00655-3_BIB26) 1975; 24 Cooper (10.1016/S0167-2789(02)00655-3_BIB21) 1987; 7 Swinton (10.1016/S0167-2789(02)00655-3_BIB47) 1990; 145 |
| References_xml | – reference: W.J.F. Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria, SIAM Publications, Philadelphia, 2000. – volume: 112 start-page: 158 year: 1998 end-page: 186 ident: BIB18 article-title: Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics publication-title: Physica D – volume: 33 start-page: 260 year: 1972 end-page: 264 ident: BIB37 article-title: Oscillatory solitary waves in dispersive media publication-title: J. Phys. Soc. Jpn. – volume: 145 start-page: 428 year: 1990 end-page: 433 ident: BIB47 article-title: Stability of travelling pulse to a laser equation publication-title: Phys. Lett. A – volume: 156 start-page: 1 year: 2001 end-page: 87 ident: BIB13 article-title: The symplectic Evans matrix, and the linearization about solitary waves and fronts publication-title: Arch. Rat. Mech. Anal. – reference: J.R. Magnus, H. Neudecker, Matrix Differential Calculus, Wiley, New York, 1988. – reference: J. Harris, Algebraic Geometry, Springer, Berlin, 1992. – volume: 156 start-page: 219 year: 2001 end-page: 238 ident: BIB15 article-title: Computing Lyapunov exponents on a Stiefel manifold publication-title: Physica D – volume: 23 start-page: 1141 year: 1992 end-page: 1166 ident: BIB39 article-title: Existence and nonexistence of solitary wave solutions to higher order model evolution equations publication-title: SIAM J. Math. Anal. – volume: 136 start-page: 9 year: 1999 end-page: 74 ident: BIB46 article-title: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations publication-title: Inven. Math. – volume: 19 start-page: 367 year: 1994 end-page: 389 ident: BIB22 article-title: Hamiltonian long-wave approximations to the water-wave problem publication-title: Wave Motion – volume: 51 start-page: 797 year: 1998 end-page: 855 ident: BIB28 article-title: The gap lemma and geometric criteria for instability of shock profiles publication-title: Commun. Pure Appl. Math. – volume: 32 start-page: 929 year: 2001 end-page: 962 ident: BIB9 article-title: Alternate Evans functions and viscous shock waves publication-title: SIAM J. Math. Anal. – reference: R.O. Wells, Differential Analysis on Complex Manifolds, Springer, Berlin, 1980. – reference: T.J. Bridges, G. Derks, Linear instability of solitary wave solutions of the Kawahara equation and its generalizations, SIAM J. Math. Anal. 33 (2002) 1356–1378. – volume: 455 start-page: 3019 year: 1999 end-page: 3040 ident: BIB11 article-title: The Orr–Sommerfeld equation on a manifold publication-title: Proc. Roy. Soc. London A – volume: 3 start-page: 307 year: 1992 end-page: 326 ident: BIB33 article-title: Stability of solitary waves in dispersive media described by a fifth-order evolution equation publication-title: Theor. Comp. Fluid Dyn. – volume: 7 start-page: 1 year: 1987 end-page: 13 ident: BIB21 article-title: Stability of Runge–Kutta methods for trajectory problems publication-title: IMA J. Numer. Anal. – volume: 67 start-page: 131 year: 1994 end-page: 149 ident: BIB6 article-title: Stabilization of DAEs and invariant manifolds publication-title: Numer. Math. – volume: 24 start-page: 1169 year: 1975 end-page: 1190 ident: BIB26 article-title: Nerve axon equations IV. The stable and unstable impulse publication-title: Indiana Univ. Math. J. – volume: 124 start-page: 58 year: 1998 end-page: 103 ident: BIB34 article-title: Stability for bright solitary wave solutions to perturbed nonlinear Schrödinger equations publication-title: Physica D – volume: 70 start-page: 1071 year: 2001 end-page: 1088 ident: BIB16 article-title: Numerical testing of the stability of viscous shock waves publication-title: Math. Comp. – volume: 30 start-page: 125 year: 1979 end-page: 136 ident: BIB44 article-title: An initial-value method for eigenvalue problems using compound matrices publication-title: J. Comp. Phys. – volume: 223 start-page: 449 year: 1996 end-page: 452 ident: BIB23 article-title: Stationary solutions of the fifth-order KdV-type equations and their stabilization publication-title: Phys. Lett. A – reference: H.B. Keller, Numerical Solution of Two-Point Boundary-Value Problems, CBMS-NSF Conference Series, vol. 24, SIAM, Philadelphia, 1974. – volume: 11 start-page: 341 year: 1998 end-page: 353 ident: BIB30 article-title: Solitary-wave solutions to a class of fifth-order model equations publication-title: Nonlinearity – volume: 342 start-page: 199 year: 1997 end-page: 229 ident: BIB19 article-title: A global investigation of solitary-wave solutions to a two-parameter model for water waves publication-title: J. Fluid Mech. – reference: M. Liefvendahl, G. Kreiss, Numerical investigation of examples of unstable viscous shock waves, in: Proceedings of Eight International Conference on Hyper. Problems Hyp’2000, Universität of Magdeburg, 2000. – volume: 410 start-page: 167 year: 1990 end-page: 212 ident: BIB3 article-title: A topological invariant arising in the stability analysis of traveling waves publication-title: J. Reine Angew. Math. – volume: 251 start-page: 363 year: 1999 end-page: 372 ident: BIB12 article-title: Hodge duality and the Evans function publication-title: Phys. Lett. A – volume: 92 start-page: 691 year: 1979 end-page: 715 ident: BIB1 article-title: On the evolution of packets of water waves publication-title: J. Fluid Mech. – reference: P.G. Drazin, W.H. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1981. – reference: L.Q. Brin, K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves, Preprint, Indiana University, 2002. – volume: 37 start-page: 23 year: 1977 end-page: 50 ident: BIB27 article-title: Local stability theory of the nerve impulse publication-title: Math. Biosci. – volume: 126 start-page: 22 year: 2000 end-page: 27 ident: BIB8 article-title: Oscillatory instabilities of gap solitons: a numerical study publication-title: Comp. Phys. Commun. – volume: 263 start-page: 98 year: 1999 end-page: 104 ident: BIB24 article-title: Nonlinear stability of solitons in the fifth-order Korteweg–de Vries equation publication-title: Phys. Lett. A – reference: L. Allen, T.J. Bridges, Numerical exterior algebra and the compound matrix method, Numer. Math. 92 (2002) 197–232. – volume: 67 start-page: 45 year: 1993 end-page: 65 ident: BIB45 article-title: Oscillatory instability of traveling waves for a KdV–Burgers equation publication-title: Physica D – volume: 20 start-page: 201 year: 1999 end-page: 244 ident: BIB10 article-title: Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals publication-title: Numer. Func. Anal. Optim. – reference: T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, Heidelberg, 1984. – volume: 457 start-page: 257 year: 2001 end-page: 272 ident: BIB2 article-title: Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow publication-title: Proc. R. Soc. Lond. A – volume: 2 start-page: 471 year: 1995 end-page: 507 ident: BIB4 article-title: Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation publication-title: Nonlin. World – volume: 210 start-page: 77 year: 1996 end-page: 84 ident: BIB35 article-title: Stabilization of soliton instabilities by higher-order dispersion: KdV-type equations publication-title: Phys. Lett. A – reference: M. Marcus, Finite Dimensional Multilinear Algebra, Part II, Marcel Dekker, New York, 1975. – volume: 53 start-page: 143 year: 1988 end-page: 163 ident: BIB48 article-title: A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain publication-title: Numer. Math. – volume: 125 start-page: 222 year: 1999 end-page: 240 ident: BIB40 article-title: A stability analysis for fifth-order water-wave models publication-title: Physica D – reference: E.A. Coddington, N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. – volume: 17 start-page: 739 year: 1998 end-page: 768 ident: BIB31 article-title: Three-dimensional solitary waves in the presence of additional surface effects publication-title: Euro. J. Mech. B – volume: 80 start-page: 5117 year: 1998 end-page: 5120 ident: BIB7 article-title: Vibrations and oscillatory instabilities of gap solitons publication-title: Phys. Rev. Lett. – ident: 10.1016/S0167-2789(02)00655-3_BIB20 – volume: 3 start-page: 307 year: 1992 ident: 10.1016/S0167-2789(02)00655-3_BIB33 article-title: Stability of solitary waves in dispersive media described by a fifth-order evolution equation publication-title: Theor. Comp. Fluid Dyn. doi: 10.1007/BF00417931 – volume: 19 start-page: 367 year: 1994 ident: 10.1016/S0167-2789(02)00655-3_BIB22 article-title: Hamiltonian long-wave approximations to the water-wave problem publication-title: Wave Motion doi: 10.1016/0165-2125(94)90003-5 – volume: 263 start-page: 98 year: 1999 ident: 10.1016/S0167-2789(02)00655-3_BIB24 article-title: Nonlinear stability of solitons in the fifth-order Korteweg–de Vries equation publication-title: Phys. Lett. A doi: 10.1016/S0375-9601(99)00712-4 – ident: 10.1016/S0167-2789(02)00655-3_BIB43 – volume: 136 start-page: 9 year: 1999 ident: 10.1016/S0167-2789(02)00655-3_BIB46 article-title: Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations publication-title: Inven. Math. doi: 10.1007/s002220050303 – volume: 30 start-page: 125 year: 1979 ident: 10.1016/S0167-2789(02)00655-3_BIB44 article-title: An initial-value method for eigenvalue problems using compound matrices publication-title: J. Comp. Phys. doi: 10.1016/0021-9991(79)90091-3 – volume: 92 start-page: 691 year: 1979 ident: 10.1016/S0167-2789(02)00655-3_BIB1 article-title: On the evolution of packets of water waves publication-title: J. Fluid Mech. doi: 10.1017/S0022112079000835 – volume: 53 start-page: 143 year: 1988 ident: 10.1016/S0167-2789(02)00655-3_BIB48 article-title: A reliable argument principle algorithm to find the number of zeros of an analytic function in a bounded domain publication-title: Numer. Math. doi: 10.1007/BF01395882 – volume: 251 start-page: 363 year: 1999 ident: 10.1016/S0167-2789(02)00655-3_BIB12 article-title: Hodge duality and the Evans function publication-title: Phys. Lett. A doi: 10.1016/S0375-9601(98)00917-7 – volume: 80 start-page: 5117 year: 1998 ident: 10.1016/S0167-2789(02)00655-3_BIB7 article-title: Vibrations and oscillatory instabilities of gap solitons publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.80.5117 – volume: 11 start-page: 341 year: 1998 ident: 10.1016/S0167-2789(02)00655-3_BIB30 article-title: Solitary-wave solutions to a class of fifth-order model equations publication-title: Nonlinearity doi: 10.1088/0951-7715/11/2/009 – ident: 10.1016/S0167-2789(02)00655-3_BIB14 doi: 10.1137/S0036141099361494 – volume: 20 start-page: 201 year: 1999 ident: 10.1016/S0167-2789(02)00655-3_BIB10 article-title: Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals publication-title: Numer. Func. Anal. Optim. doi: 10.1080/01630569908816889 – volume: 23 start-page: 1141 year: 1992 ident: 10.1016/S0167-2789(02)00655-3_BIB39 article-title: Existence and nonexistence of solitary wave solutions to higher order model evolution equations publication-title: SIAM J. Math. Anal. doi: 10.1137/0523064 – volume: 70 start-page: 1071 year: 2001 ident: 10.1016/S0167-2789(02)00655-3_BIB16 article-title: Numerical testing of the stability of viscous shock waves publication-title: Math. Comp. doi: 10.1090/S0025-5718-00-01237-0 – ident: 10.1016/S0167-2789(02)00655-3_BIB25 – ident: 10.1016/S0167-2789(02)00655-3_BIB42 – ident: 10.1016/S0167-2789(02)00655-3_BIB29 doi: 10.1137/1.9780898719543 – ident: 10.1016/S0167-2789(02)00655-3_BIB36 – ident: 10.1016/S0167-2789(02)00655-3_BIB5 doi: 10.1007/s002110100365 – volume: 17 start-page: 739 year: 1998 ident: 10.1016/S0167-2789(02)00655-3_BIB31 article-title: Three-dimensional solitary waves in the presence of additional surface effects publication-title: Euro. J. Mech. B doi: 10.1016/S0997-7546(98)80023-X – volume: 67 start-page: 45 year: 1993 ident: 10.1016/S0167-2789(02)00655-3_BIB45 article-title: Oscillatory instability of traveling waves for a KdV–Burgers equation publication-title: Physica D doi: 10.1016/0167-2789(93)90197-9 – volume: 156 start-page: 1 year: 2001 ident: 10.1016/S0167-2789(02)00655-3_BIB13 article-title: The symplectic Evans matrix, and the linearization about solitary waves and fronts publication-title: Arch. Rat. Mech. Anal. doi: 10.1007/PL00004235 – volume: 33 start-page: 260 year: 1972 ident: 10.1016/S0167-2789(02)00655-3_BIB37 article-title: Oscillatory solitary waves in dispersive media publication-title: J. Phys. Soc. Jpn. doi: 10.1143/JPSJ.33.260 – volume: 457 start-page: 257 year: 2001 ident: 10.1016/S0167-2789(02)00655-3_BIB2 article-title: Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow publication-title: Proc. R. Soc. Lond. A doi: 10.1098/rspa.2000.0665 – volume: 210 start-page: 77 year: 1996 ident: 10.1016/S0167-2789(02)00655-3_BIB35 article-title: Stabilization of soliton instabilities by higher-order dispersion: KdV-type equations publication-title: Phys. Lett. A doi: 10.1016/0375-9601(95)00752-0 – ident: 10.1016/S0167-2789(02)00655-3_BIB41 – volume: 145 start-page: 428 year: 1990 ident: 10.1016/S0167-2789(02)00655-3_BIB47 article-title: Stability of travelling pulse to a laser equation publication-title: Phys. Lett. A doi: 10.1016/0375-9601(90)90307-A – volume: 7 start-page: 1 year: 1987 ident: 10.1016/S0167-2789(02)00655-3_BIB21 article-title: Stability of Runge–Kutta methods for trajectory problems publication-title: IMA J. Numer. Anal. doi: 10.1093/imanum/7.1.1 – ident: 10.1016/S0167-2789(02)00655-3_BIB32 doi: 10.1007/978-1-4757-2189-8 – volume: 455 start-page: 3019 year: 1999 ident: 10.1016/S0167-2789(02)00655-3_BIB11 article-title: The Orr–Sommerfeld equation on a manifold publication-title: Proc. Roy. Soc. London A doi: 10.1098/rspa.1999.0437 – volume: 24 start-page: 1169 year: 1975 ident: 10.1016/S0167-2789(02)00655-3_BIB26 article-title: Nerve axon equations IV. The stable and unstable impulse publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1975.24.24096 – volume: 124 start-page: 58 year: 1998 ident: 10.1016/S0167-2789(02)00655-3_BIB34 article-title: Stability for bright solitary wave solutions to perturbed nonlinear Schrödinger equations publication-title: Physica D doi: 10.1016/S0167-2789(98)00172-9 – ident: 10.1016/S0167-2789(02)00655-3_BIB17 doi: 10.21711/231766362002/rmc222 – volume: 223 start-page: 449 year: 1996 ident: 10.1016/S0167-2789(02)00655-3_BIB23 article-title: Stationary solutions of the fifth-order KdV-type equations and their stabilization publication-title: Phys. Lett. A doi: 10.1016/S0375-9601(96)00772-4 – volume: 125 start-page: 222 year: 1999 ident: 10.1016/S0167-2789(02)00655-3_BIB40 article-title: A stability analysis for fifth-order water-wave models publication-title: Physica D doi: 10.1016/S0167-2789(98)00245-0 – volume: 51 start-page: 797 year: 1998 ident: 10.1016/S0167-2789(02)00655-3_BIB28 article-title: The gap lemma and geometric criteria for instability of shock profiles publication-title: Commun. Pure Appl. Math. doi: 10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1 – volume: 2 start-page: 471 year: 1995 ident: 10.1016/S0167-2789(02)00655-3_BIB4 article-title: Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation publication-title: Nonlin. World – volume: 126 start-page: 22 year: 2000 ident: 10.1016/S0167-2789(02)00655-3_BIB8 article-title: Oscillatory instabilities of gap solitons: a numerical study publication-title: Comp. Phys. Commun. doi: 10.1016/S0010-4655(99)00241-6 – volume: 342 start-page: 199 year: 1997 ident: 10.1016/S0167-2789(02)00655-3_BIB19 article-title: A global investigation of solitary-wave solutions to a two-parameter model for water waves publication-title: J. Fluid Mech. doi: 10.1017/S0022112097005193 – volume: 112 start-page: 158 year: 1998 ident: 10.1016/S0167-2789(02)00655-3_BIB18 article-title: Homoclinic orbits in reversible systems and their applications in mechanics, fluids and optics publication-title: Physica D doi: 10.1016/S0167-2789(97)00209-1 – volume: 410 start-page: 167 year: 1990 ident: 10.1016/S0167-2789(02)00655-3_BIB3 article-title: A topological invariant arising in the stability analysis of traveling waves publication-title: J. Reine Angew. Math. – volume: 32 start-page: 929 year: 2001 ident: 10.1016/S0167-2789(02)00655-3_BIB9 article-title: Alternate Evans functions and viscous shock waves publication-title: SIAM J. Math. Anal. doi: 10.1137/S0036141099361834 – volume: 156 start-page: 219 year: 2001 ident: 10.1016/S0167-2789(02)00655-3_BIB15 article-title: Computing Lyapunov exponents on a Stiefel manifold publication-title: Physica D doi: 10.1016/S0167-2789(01)00283-4 – ident: 10.1016/S0167-2789(02)00655-3_BIB49 doi: 10.1007/978-1-4757-3946-6 – ident: 10.1016/S0167-2789(02)00655-3_BIB38 – volume: 67 start-page: 131 year: 1994 ident: 10.1016/S0167-2789(02)00655-3_BIB6 article-title: Stabilization of DAEs and invariant manifolds publication-title: Numer. Math. doi: 10.1007/s002110050020 – volume: 37 start-page: 23 year: 1977 ident: 10.1016/S0167-2789(02)00655-3_BIB27 article-title: Local stability theory of the nerve impulse publication-title: Math. Biosci. doi: 10.1016/0025-5564(77)90076-1 |
| SSID | ssj0001737 |
| Score | 2.0366378 |
| Snippet | The spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation is formulated in terms of the Evans... |
| SourceID | crossref elsevier |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 190 |
| SubjectTerms | Evans function Fifth-order KdV Linear stability Numerical exterior algebra |
| Title | Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework |
| URI | https://dx.doi.org/10.1016/S0167-2789(02)00655-3 |
| Volume | 172 |
| WOSCitedRecordID | wos000179357700011&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1872-8022 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0001737 issn: 0167-2789 databaseCode: AIEXJ dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1bb9MwFLZKB9J4QDBAjJv8wAMocknspE54G2hchjRNYqC-RY7tiIkq3dp021_gX3N8iZOpiJvES5RacSz5fD3-fHLOZ4SexZoxWTFFpjqtSUoFJ7AFY0QlUkgRK1hipD1sgh8e5rNZcTQafe9qYc7nvGnyy8vi9L-aGtrA2KZ09i_MHV4KDXAPRocrmB2uf2R4oI824bUTVlqF38ALVybdzSTKXQgvN2uIZ31St1-JVeGMPqovkT5bh6QPETVr91VnHtVdJteQ0h45S0_63OHXTjqizz-KDiaBMOvlNxeMB1z2n_TfLdr2wgepbZj-SjSCmrI8V4_pQmQbZTIuagne2JTcukXHedqcgyuO6VVXzOkG5pxjTdyhon6Npq4-c8P9u0jEpzAekHQjMFsYopUR1q95IRPRPmurgal9iF1DW5RnRT5GW3sf9mcHYVlPuBNg7d7dl4O97Ad8HtMXfrCfE50BeTm-jW75XQfec2i5g0a62UE3B1qUO-iGs-PqLqoCgjAgCA8QhBc17hCELYJMCyAIDxCEAUG4Q9ArLHDADw74uYc-v90_fvOe-KM4iGSctaRiUhZKZQWt0ljwFO40K3Sm6kzEQolkKlPYaFTUyNtNGVNZxbmg0ijV0jxJ2H00bhaNfoAw54oxluVKCaCWPKm45hUsLDJh0tDzXZR281ZKr1NvjkuZl4OExCkvzXSXMS3tdJdsF01Ct1Mn1PK7DnlnlNKzTcciS0DTr7s-_Peuj9B2_7d5jMbtcq2foOvyvD1ZLZ96zP0Az3SdSA |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Stability+and+instability+of+solitary+waves+of+the+fifth-order+KdV+equation%3A+a+numerical+framework&rft.jtitle=Physica.+D&rft.au=Bridges%2C+Thomas+J.&rft.au=Derks%2C+Gianne&rft.au=Gottwald%2C+Georg&rft.date=2002-11-15&rft.pub=Elsevier+B.V&rft.issn=0167-2789&rft.eissn=1872-8022&rft.volume=172&rft.issue=1&rft.spage=190&rft.epage=216&rft_id=info:doi/10.1016%2FS0167-2789%2802%2900655-3&rft.externalDocID=S0167278902006553 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0167-2789&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0167-2789&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0167-2789&client=summon |