Self-similar blow-up solutions in the generalised Korteweg-de Vries equation: spectral analysis, normal form and asymptotics. (English) Zbl 07900889
Summary: In the present work we revisit the problem of the generalised Korteweg-de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter \(p\), here at \(p=5\). We provide a normal form of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for \(p>5\) in the co-exploding frame.
{© 2024 IOP Publishing Ltd & London Mathematical Society. All rights, including for text and data mining, AI training, and similar technologies, are reserved.}
{© 2024 IOP Publishing Ltd & London Mathematical Society. All rights, including for text and data mining, AI training, and similar technologies, are reserved.}
MSC:
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35C06 | Self-similar solutions to PDEs |
| 34E10 | Perturbations, asymptotics of solutions to ordinary differential equations |
| 34E20 | Singular perturbations, turning point theory, WKB methods for ordinary differential equations |
| 47A75 | Eigenvalue problems for linear operators |
| 37G05 | Normal forms for dynamical systems |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N08 | Finite volume methods for boundary value problems involving PDEs |
References:
| [1] | Ablowitz, M. J.; Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering, 1991, Cambridge University Press · Zbl 0762.35001 |
| [2] | Whitham, G. B., Linear and Nonlinear Waves, 1974, Wiley · Zbl 0373.76001 |
| [3] | Ablowitz, M. J., Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, 2011, Cambridge University Press · Zbl 1232.35002 |
| [4] | Dauxois, T.; Peyrard, M., Physics of Solitons, 2006, Cambridge University Press |
| [5] | Zabusky, N. J.; Kruskal, M. D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states, Phys. Rev. Lett., 15, 240, 1965 · Zbl 1201.35174 · doi:10.1103/PhysRevLett.15.240 |
| [6] | Fermi, EPasta, JUlam, S1955Studies of nonlinear problems. ITechnical Report LA-1940Los Alamos National Laboratory |
| [7] | Gallavotti, G., The Fermi-Pasta-Ulam Problem: A Status Report, 2008, Springer · Zbl 1138.81004 |
| [8] | Zabusky, N. J.; Ames, W. F., A synergetic approach to problems of nonlinear dispersive wave propagation and interaction, Nonlinear Partial Differential Equations: A Symposium on Methods of Solution, 1967, Academic · Zbl 0183.18104 |
| [9] | Korteweg, D. J.; de Vries, G., On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag., 5, 422-43, 1895 · JFM 26.0881.02 · doi:10.1080/14786449508620739 |
| [10] | Olver, P. J., Evolution equations possessing infinitely many symmetries, J. Math. Phys., 18, 1212-5, 1977 · Zbl 0348.35024 · doi:10.1063/1.523393 |
| [11] | Hirota, R., The Direct Method in Soliton Theory, 2004, Cambridge University Press · Zbl 1099.35111 |
| [12] | Remoissenet, M., Waves Called Solitons: Concepts and Experiments, 1999, Springer · Zbl 0933.35003 |
| [13] | Osborne, A. R., Nonlinear Ocean Waves And The Inverse Scattering Transform, 2010, Academic · Zbl 1250.86006 |
| [14] | Nakamura, Y.; Tsukabayashi, I., Modified Korteweg-de Vries ion-acoustic solitons in a plasma, J. Plasma Phys., 34, 402-15, 1985 · doi:10.1017/S0022377800002968 |
| [15] | Bona, J. L.; Souganidis, P. E.; Strauss, W. A., Stability and instability of solitary waves of Korteweg-de Vries type, Proc. R. Soc. A, 411, 395-412, 1987 · Zbl 0648.76005 · doi:10.1098/rspa.1987.0073 |
| [16] | Amodio, P.; Budd, C. J.; Koch, O.; Rottschäfer, V.; Settanni, G.; Weinmüller, E., Near critical, self-similar, blow-up solutions of the generalised Korteweg-de Vries equation: asymptotics and computations, Physica D, 401, 2020 · Zbl 1453.37064 · doi:10.1016/j.physd.2019.132179 |
| [17] | Sulem, C.; Sulem, P. L., The Nonlinear Schrödinger Equation, 1999, Springer · Zbl 0928.35157 |
| [18] | Merle, F., Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Am. Math. Soc., 14, 555-78, 2001 · Zbl 0970.35128 · doi:10.1090/S0894-0347-01-00369-1 |
| [19] | Martel, Y.; Merle, F., Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. Math., 155, 235-80, 2002 · Zbl 1005.35081 · doi:10.2307/3062156 |
| [20] | Weinstein, M. I., Nonlinear Schrödinger equations and sharp interpolation estimates, Commun. Math. Phys., 87, 567-76, 1983 · Zbl 0527.35023 · doi:10.1007/BF01208265 |
| [21] | Pego, R. L.; Weinstein, M. I., Eigenvalues and instabilities of solitary waves, Phil. Trans. R. Soc. A, 340, 47-94, 1992 · Zbl 0776.35065 · doi:10.1098/rsta.1992.0055 |
| [22] | Bona, J. L.; Dougalis, V. A.; Karakashian, O. A.; McKinney, W. R., Conservative, high-order numerical schemes for the generalized Korteweg-de Vries equation, Phil. Trans. R. Soc. A, 351, 107-64, 1995 · Zbl 0824.65095 · doi:10.1098/rsta.1995.0027 |
| [23] | Dix, D. B.; McKinney, W. R., Numerical computations of self-similar blow-up solutions of the generalized Korteweg-de Vries equation, Differ. Integral Equ., 11, 679-723, 1998 · Zbl 1007.65061 · doi:10.57262/die/1367329666 |
| [24] | Koch, H., Self-similar solutions to super-critical gKdV, Nonlinearity, 28, 545-75, 2015 · Zbl 1317.35222 · doi:10.1088/0951-7715/28/3/545 |
| [25] | Klein, C.; Peter, R., Numerical study of blow-up and dispersive shocks in solutions to generalized Korteweg-de Vries equations, Physica D, 304-305, 52-78, 2015 · Zbl 1364.65182 · doi:10.1016/j.physd.2015.04.003 |
| [26] | Klein, C., Fourth-order time-stepping for low dispersion Korteweg-de Vries and nonlinear Schrödinger equations, Electron. Trans. Numer. Anal., 29, 116-35, 2008 · Zbl 1186.65134 |
| [27] | Martel, Y.; Merle, F.; Raphaël, P., Blow-up for the critical generalized Korteweg-de Vries equation. I: Dynamics near the soliton, Acta. Math., 212, 59-140, 2014 · Zbl 1301.35137 · doi:10.1007/s11511-014-0109-2 |
| [28] | Martel, Y.; Merle, F.; Raphaël, P., Blow-up for the critical generalized Korteweg-de Vries equation. II: Minimal mass dynamics, J. Eur. Math. Soc., 17, 1855-925, 2015 · Zbl 1326.35320 · doi:10.4171/jems/547 |
| [29] | Kitzhofer, G.; Pulverer, G.; Simon, C.; Koch, O.; Weinmüller, E., The new MATLAB solver bvpsuite for the solution of singular implicit BVPs, JNAIAM J. Numer. Anal. Ind. Appl. Math., 5, 113-34, 2010 · Zbl 1432.65100 |
| [30] | Lan, Y., Stable self-similar blow-up dynamics for slightly L^2-supercritical generalized KDV equations, Commun. Math. Phys., 345, 223-69, 2016 · Zbl 1350.35172 · doi:10.1007/s00220-016-2589-8 |
| [31] | Siettos, C. I.; Kevrekidis, I. G.; Kevrekidis, P. G., Focusing revisited: a renormalization/bifurcation approach, Nonlinearity, 16, 497-506, 2003 · Zbl 1108.35135 · doi:10.1088/0951-7715/16/2/308 |
| [32] | LeMesurier, B JPapanicolaou, G CSulem, CSulem, P L1986Focusing and multi-focusing solutions of the nonlinear Schrödinger equationPhysica D31781078-10; LeMesurier, B JPapanicolaou, G CSulem, CSulem, P L1988Local structure of the self-focusing singularity of the nonlinear Schrödinger equationPhysica D3221026210-26 |
| [33] | McLaughlin, D. W.; Papanicolaou, G. C.; Sulem, C.; Sulem, P. L., Focusing singularity of the cubic Schrödinger equation, Phys. Rev. A, 34, 1200-10, 1986 · doi:10.1103/PhysRevA.34.1200 |
| [34] | Kopell, N.; Landman, M. J., Spatial structure of the focusing singularity of the nonlinear Schrödinger equation: a geometrical analysis, SIAM J. Appl. Math., 55, 1297-323, 1995 · Zbl 0836.34041 · doi:10.1137/S0036139994262386 |
| [35] | Chapman, S. J.; Kavousanakis, M.; Kevrekidis, I. G.; Kevrekidis, P. G., Normal form for the onset of collapse: the prototypical example of the nonlinear Schrödinger equation, Phys. Rev. E, 104, 2021 · doi:10.1103/PhysRevE.104.044202 |
| [36] | Jon Chapman, S.; Kavousanakis, M.; Charalampidis, E. G.; Kevrekidis, I. G.; Kevrekidis, P. G., A spectral analysis of the nonlinear Schrödinger equation in the co-exploding frame, Physica D, 439, 2022 · Zbl 1497.35428 · doi:10.1016/j.physd.2022.133396 |
| [37] | Bernoff, A. J.; Witelski, T. P., Stability and dynamics of self-similarity in evolution equations, J. Eng. Math., 66, 11-31, 2010 · Zbl 1194.35091 · doi:10.1007/s10665-009-9309-8 |
| [38] | Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations I, 1993, Springer · Zbl 0789.65048 |
| [39] | Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, 2023, Springer · Zbl 1511.37003 |
| [40] | Bender, C. M.; Orszag, S. A., Advanced Mathematical Methods for Scientists and Engineers, Asymptotic Methods and Perturbation Theory, 1999, Springer · Zbl 0938.34001 |
| [41] | Meyer, R. E., A simple explanation of the Stokes phenomenon, SIAM Rev., 31, 435-45, 1989 · Zbl 0679.34068 · doi:10.1137/1031090 |
| [42] | Paris, R. B.; Wood, A. D., Stokes phenomenon demystified, IMA Bull., 31, 21-28, 1995 · Zbl 0827.34002 |
| [43] | Olde Daalhuis, A. B.; Chapman, S. J.; King, J. R.; Ockendon, J. R.; Tew, R. H., Stokes phenomenon and matched asymptotic expansions, SIAM J. Appl. Math., 55, 1469-83, 1995 · Zbl 0834.41026 · doi:10.1137/S0036139994261769 |
| [44] | Chapman, S. J., Subcritical transition in channel flows, J. Fluid Mech., 451, 35-97, 2002 · Zbl 1037.76023 · doi:10.1017/S0022112001006255 |
| [45] | Note that the work of [] does not include the first term on the RHS of Equation . However, omitting this term does not affect the solvability condition. Essentially the analysis in [] perturbs only the p which appears as a coefficient, not that which appears as an exponent, that is, they solve \(\frac{ \partial w}{ \partial \tau}=-\frac{ \partial^3 w}{ \partial \xi^3}-\frac{ \partial w^5}{ \partial \xi}+G(\frac{2}{ p - 1}w+\xi\frac{ \partial w}{ \partial \xi})+\frac{ \partial w}{ \partial \xi}.\) |
| [46] | We note that there is a missing factor of 2 in the corresponding equations in []. |
| [47] | If we followed [] and missed out all the terms associated with perturbing the power, then we would find \(p_1=-c\frac{{\displaystyle \int_{- \operatorname{\infty}}^{\operatorname{\infty}} ( - 20 \frac{ \partial w_0}{ \partial \xi} w_0^3 \hat{w}_{\operatorname{odd}} + \frac{w_0}{2} + \xi \frac{ \partial w_0}{ \partial \xi} ) \hat{w}_{\operatorname{even}} \operatorname{d} \xi}}{{\displaystyle \int_{- \operatorname{\infty}}^{\operatorname{\infty}} \frac{1}{8} w_0^2 \operatorname{d} \xi}}\approx-5.07c.\) It seems like the extra terms integrate to zero, i.e. \({\displaystyle \int_{- \operatorname{\infty}}^{\operatorname{\infty}} \frac{ \partial w_0}{ \partial \xi} ( - 20 w_0^3 W_1 - w_0^4 - 5 w_0^4 \log w_0 ) \hat{w}_{\operatorname{odd}} + ( \frac{w_0}{2} + \xi \frac{ \partial w_0}{ \partial \xi} ) W_1 \operatorname{d} \xi = 0}\) though we have been unable to show this. |
| [48] | Budd, C. J.; Rottschäfer, V.; Williams, J. F., Multibump, blow-up, self-similar solutions of the complex Ginzburg-Landau eqaution, SIAM J. Appl. Dyn. Sys., 4, 649-78, 2005 · Zbl 1170.35530 · doi:10.1137/040610866 |
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