Stability of bound states in infinite domains

A. T. Il'ichev

Lekcionnye kursy NOC, Stability of bound states in infinite domains (2022), pp. 3-58.

Bibliographie

[1] V. E. Zakharov, S. V. Manakov, S. P. Novikov, L. P. Pitaevskii, Teoriya solitonov, Nauka, M., 1980

[2] A. T. Ilichev, “Ustoichivost solitonov v nelineinykh kompozitnykh sredakh”, ZhETF, 118 (2000), 720–729

[3] A. T. Ilichev, “Teoriya ustoichivosti “petli Eilera” na uprugikh nerastyazhimykh sterzhnyakh”, Trudy Matem. in-ta im. V. A. Steklova, 251 (2005), 154–172

[4] A. T. Ilichev, “Dinamika i spektralnaya ustoichivost solitonopodobnykh struktur v membrannykh trubkakh s zhidkostyu”, UMN, 75:5 (2020), 59–100

[5] A. T. Ilichev, Ustoichivost lokalizovannykh voln v nelineino-uprugikh sterzhnyakh, Fizmatlit, M., 2009

[6] A. T. Ilichev V. Ya. Tomashpolskii, “Neustoichivost solitonov pri izgibe i kruchenii uprugogo sterzhnya”, TMF, 172:3 (2012), 375–386

[7] A. T. Ilichev, A. P. Chugainova, “Teoriya spektralnoi ustoichivosti geteroklinicheskikh reshenii uravneniya Kortevega–de Friza–Byurgersa s proizvolnym potentsialom”, Trudy Matem. in-ta im. V. A. Steklova, 295 (2016), 163–173

[8] A. T. Ilichev, A. P. Chugainova, V. A. Shargatov, “Spektralnaya ustoichivost osobykh razryvov”, Dokl. RAN, 462 (2015), 512–516

[9] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972

[10] R. Kurant, D. Gilbert, Uravneniya matematicheskoi fiziki, Gostekhizdat, M.–L., 1951

[11] A. Lyav, Matematicheskaya teoriya uprugosti, ONTI NKTP SSSR, M., 1935

[12] W. Rudin, Functional analysis, 2nd ed., McGraw-Hill, New York, 1991 | Zbl

[13] B. V. Shabat, Vvvedenie v kompleksnyi analiz. Chast I, Nauka, M., 1985

[14] L. Abdelouhab, J. L. Bona, M. Felland, J.-C. Saut, “Non-local models for nonlinear dispersive waves”, Physica D, 40 (1987), 360–392 | DOI

[15] J. P. Albert, J. L. Bona, D. L. Henry, “Sufficient conditions for stability of solitary wave solutions of model equations for long waves”, Physica D, 24 (1987), 343–366 | DOI | Zbl

[16] J. C. Alexander, R. Sachs, “Linear stability of solitary waves of a Boussinesq-type equation: A computer assisted computation”, Nonlin. World, 2 (1995), 471–507 | Zbl

[17] J. C. Alexander, R. Gardner, C. Jones, “A topological invariant arising in the stability analysis of travelling waves”, J. Reine Angew. Math., 410 (1990), 167–212 | Zbl

[18] J. C. Alexander, M. G. Grillakis, C. K. R. T. Jones, B. Sandstede, “Stability of pulses on optical fibers with phase-sensitive amplifiers”, Z. angew. Math. Phys., 48 (1997), 175–192 | DOI | Zbl

[19] A. L Afendikov, T. J. Bridges, “Instability of the Hocking-Stewartson pulse and its implications for three-dimensional Poiseuille flow”, Proc. Roy. Soc. Lond. A, 457 (2001), 257–272 | DOI | Zbl

[20] T. B. Benjamin, “The stability of solitary waves”, Proc. Roy. Soc. London A, 238 (1972), 153–183

[21] A. Beliaev, A. Il'ichev, “Conditional stability of solitary waves propagating in elastic rods”, Physica D, 90 (1996), 107–118 | DOI | Zbl

[22] D. P. Bennet, R. V. Brown, S. E. Stansfield, J. D. Stroughair, J. L. Bona, “The stability of internal solitary waves”, Math. Proc. Cambridge Philos. Soc., 94 (1983), 351–379 | DOI | Zbl

[23] J. L. Bona, “On the stability of solitary waves”, Proc. Roy. Soc. London A, 344 (1975), 363–374 | DOI | Zbl

[24] J. L. Bona, R. L. Sachs, “Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation”, Comm. Math. Phys., 118 (1988), 15–29 | DOI | Zbl

[25] J. L. Bona, P. E. Souganidis, W. A. Strauss, “Stability and instability of solitary waves of Korteweg–de Vries type”, Proc. Roy. Soc. London A, 411 (1987), 395–412 | DOI | Zbl

[26] T. J. Bridges, G. Derks, G. Gottwald, “Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework”, Physica D, 172 (2002), 190–216 | DOI | Zbl

[27] A. P. Chugainova, A. T. Il'ichev, A. G. Kulikovskii, V. A. Shargatov, “Problem of arbitrary discontinuity disintegration for the generalized Hopf equation: selection conditions for a unique solution”, IMA J. Appl. Math., 82 (2017), 496–525 | Zbl

[28] A. P. Chugainova, A. T. Ilichev, V. A. Shargatov, “Stability of shock wave structures in nonlinear elastic media”, Math. Mech. Solids, 24 (2019), 3456–3471 | DOI | Zbl

[29] A. Clebsch, Theorie der Elasticität Fester Körper, B. G. Teubner, Leipzig, 1862

[30] E. A. Coddington, N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955 | Zbl

[31] B. D. Coleman, E. H. Dill, “Flexure waves in elastic rods”, J. Acoustical Soc. America, 91 (1992), 2663–2673 | DOI

[32] B. D. Coleman, J. M. Xu, “On the interaction of solitary waves of flexure in elastic rods”, Acta Mech., 110 (1995), 173–182 | DOI | Zbl

[33] B. D. Coleman, E. H. Dill, D. Swigon, “On the dynamics of flexure and stretch in the theory of elastic rods”, Arch. Ration. Mech. Anal., 129 (1995), 147–174 | DOI | Zbl

[34] B. D. Coleman, E. H. Dill, M. Lembo, Z. Lu, I. Tobias, “On the dynamics of rods in the theory of Kirchoff and Clebsch”, Arch. Ration. Mech. Anal., 121 (1993), 339–359 | DOI

[35] W. A. Coppel, Stability and asymptotic bahaviour of differential equations, D. C. Heath and Company, Boston, 1965

[36] D. J. Dichmann, J. H. Maddocks, R. L. Pego, “Hamiltonian dynamics of an elastica and the stability of solitary waves”, Arch. Ration. Mech. Anal., 135 (1996), 347–396 | DOI

[37] E. H. Dill, “Kirchoff's theory of rods”, Arch. Hist. Exact Sci., 44 (1992), 1–23 | DOI | Zbl

[38] J. V. Evans, “Nerve axon equations, III: Stability of the nerve impulse”, Indiana Univ. Math. J., 22 (1972), 577–594 | DOI

[39] R. A. Gardner, K. Zumburin, “The gap lemma and geometric criteria for instability of viscous shock profiles”, Comm. Pure Appl. Math., 41 (1998), 797–855 | 3.0.CO;2-1 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[40] M. Grillakis, J. Shatah, W. Strauss, “Stability theory of solitary waves in the presence of symmetry. I”, J. Funct. Anal., 74 (1987), 160–197 | DOI | Zbl

[41] M. Grillakis, J. Shatah, W. Strauss, “Stability theory of solitary waves in the presence of symmetry. II”, J. Funct. Anal., 94 (1990), 308–348 | DOI | Zbl

[42] V. V. Gubernov, G. N. Mercer, H. S. Sidhu, R. O. Weber, “Evans function stability of combustion waves”, SIAM Journ. Appl. Math., 63 (2003), 1259–1275 | DOI | Zbl

[43] Y. B. Fu, A. T. Il'ichev, “Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow”, Math. Mech. Solids, 20 (2015), 1198–2014 | DOI

[44] D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin, Heidelberg and New York, 1981 | Zbl

[45] Z. Huang, H. S. Sidhu, I. N. Towers, Z. Jovanoski, V. V. Gubernov, “Stability analysis of combustion waves for competitive exothermic reactions using Evans function”, Appl. Math. Modelling, 54 (2018), 347–360 | DOI | Zbl

[46] J. Humpreus, B. Sandstede, K. Zumburin, “Efficient computation of analytic bases in Evans function analysis of large systems”, Numerishe Math., 103 (2006), 631–642 | DOI

[47] A. Il'ichev, “Stability of solitary waves in nonlinear composite media”, Physica D, 150 (2001), 261–277

[48] A. Il'ichev, “Instability of solitary waves on Euler's elastica”, Z. angew. Math. Phys., 57 (2006), 547–566 | DOI

[49] A. T. Il'ichev, Y. B. Fu, “Stability of aneurysm solutions in a fluid-filled elastic membrane tube”, Acta Mechanica Sinica, 28 (2012), 1209–1218 | DOI

[50] A. T. Il'ichev, Y. B. Fu, “Stability of an inflated hyperelastic membrane tube with localized wall thinning”, Internat. J. Engrg. Sci., 80 (2014), 53–61 | DOI

[51] A. T. Il'ichev, V. A. Shargatov, Y. B. Fu, “Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membrane tube”, Acta Mech., 231 (2020), 4095–4110 | DOI

[52] A. T. Il'ichev, A. Yu. Semenov, “Stability of solitary waves in dispersive media described by a fifth-order evolution equation”, Theor. Comput. Fluid Dynamics, 3 (1992), 307–326 | DOI

[53] C. K. R. T. Jones, “Stability of the travelling wave solution of the FitzHugh–Nagumo system”, Trans. Amer. Math. Soc., 286 (1984), 431–469 | DOI | Zbl

[54] T. Kapitula, “The Evans function and generalized Melnikov integrals”, SIAM J. Math. Anal., 30 (1998), 273–297 | DOI

[55] T. Kapitula, “Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation”, Physica D, 116 (1998), 95–120 | DOI | Zbl

[56] T. Kapitula, B. Sandstede, “Stability for bright solitary wave solutions to perturbed nonlinear Schrödinger equations”, Physica D, 124 (1998), 58–103 | DOI | Zbl

[57] T. Kapitula, B. Sandstede, “Edge bifurcations for near integrable systems via Evans function techniques”, SIAM Journ. Vath. Anal., 33 (2002), 1117–1143 | DOI | Zbl

[58] T. Kapitula, N. Kutz, B. Sandstede, “The Evans function for nonlocal equations”, Indiana Univ. Math. Journ., 53 (2004), 1095–1126 | DOI | Zbl

[59] G. Kirchhoff, Vorlezungen über mathematische Physic, Mechanik, Vorl. 28, D. G. Teubner, Leipzig, 1876

[60] M. Oh, B. Sandstede, “Evans function for periodic waves on infinite cylindrical domains”, Journ. Diff. Equations, 248 (2010), 544–555 | DOI | Zbl

[61] S. P. Pearce, Y. B. Fu, “Characterisation and stability of localised bulging/necking in inflated membrane tubes”, IMA J. Appl. Math., 75 (2010), 581–602 | DOI | Zbl

[62] R. L. Pego, M. I. Weinstein, “Eigenvalues and instabilities of solitary waves”, Philos. Trans. Roy. Soc. Lond. A, 340 (1992), 47–94 | DOI | Zbl

[63] R. L. Pego, P. Smereka, M. I. Weinstein, “Oscillatory instability of travelling waves for KdV–Burgers equation”, Physica D, 67 (1993), 45–65 | DOI | Zbl

[64] B. Sandstede, “Evans function and nonlinear stability of traveling waves in nonlocal network models”, Int. Journ. Bifur. Chaos, 17 (2007), 2693–2709 | DOI

[65] V. A. Shargatov, S. V. Gorkunov, A. T. Il'ichev, “Dynamics of front-like water evaporation phase transition interfaces”, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 223–236 | DOI | Zbl

[66] J. Shatah, W. Strauss, “Instability of nonlinear bound states”, Commun. Math. Phys., 100 (1985), 173–190 | DOI | Zbl

[67] W. A. Strauss, Nonlinear wave equations, Conference board of the mathematical sciences. Regional conference series in mathematics, 73, Amer. Math. Soc., 1989 | Zbl

[68] J. Swinton, J. Eglin, “Stability of travelling pulse to a laser equation”, Phys. Lett. A, 145 (1990), 428–433 | DOI

[69] M. I. Weinstein, “Liapunov stability of ground states of nonlinear dispersive evolution equations”, Comm. Pure Appl. Math., 39 (1986), 51–68 | DOI

[70] M. I. Weinstein, “Existence and dynamic stability of solitary wave solutions of equations, arising in long wave propagation”, Comm. Partial Diff. Eqn., 12 (1987), 1133–1173 | DOI | Zbl