KdV Equation in the Quarter–Plane: Evolution of the Weyl Functions and Unbounded Solutions

A. Sakhnovich

Mathematical Modelling of Natural Phenomena (2012)

  • Volume: 7, Issue: 2, page 131-145
  • ISSN: 0973-5348

Abstract

top
The matrix KdV equation with a negative dispersion term is considered in the right upper quarter–plane. The evolution law is derived for the Weyl function of a corresponding auxiliary linear system. Using the low energy asymptotics of the Weyl functions, the unboundedness of solutions is obtained for some classes of the initial–boundary conditions.

How to cite

top

Sakhnovich, A.. "KdV Equation in the Quarter–Plane: Evolution of the Weyl Functions and Unbounded Solutions." Mathematical Modelling of Natural Phenomena 7.2 (2012): 131-145. <http://eudml.org/doc/222228>.

@article{Sakhnovich2012,
abstract = {The matrix KdV equation with a negative dispersion term is considered in the right upper quarter–plane. The evolution law is derived for the Weyl function of a corresponding auxiliary linear system. Using the low energy asymptotics of the Weyl functions, the unboundedness of solutions is obtained for some classes of the initial–boundary conditions.},
author = {Sakhnovich, A.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {KdV; initial–boundary value problem; Weyl function; evolution; low–energy asymptotics; blow–up solution; initial-boundary value problem; low-energy asymptotics; blow-up solution},
language = {eng},
month = {2},
number = {2},
pages = {131-145},
publisher = {EDP Sciences},
title = {KdV Equation in the Quarter–Plane: Evolution of the Weyl Functions and Unbounded Solutions},
url = {http://eudml.org/doc/222228},
volume = {7},
year = {2012},
}

TY - JOUR
AU - Sakhnovich, A.
TI - KdV Equation in the Quarter–Plane: Evolution of the Weyl Functions and Unbounded Solutions
JO - Mathematical Modelling of Natural Phenomena
DA - 2012/2//
PB - EDP Sciences
VL - 7
IS - 2
SP - 131
EP - 145
AB - The matrix KdV equation with a negative dispersion term is considered in the right upper quarter–plane. The evolution law is derived for the Weyl function of a corresponding auxiliary linear system. Using the low energy asymptotics of the Weyl functions, the unboundedness of solutions is obtained for some classes of the initial–boundary conditions.
LA - eng
KW - KdV; initial–boundary value problem; Weyl function; evolution; low–energy asymptotics; blow–up solution; initial-boundary value problem; low-energy asymptotics; blow-up solution
UR - http://eudml.org/doc/222228
ER -

References

top
  1. J.L. Bona, A.S. Fokas. Initial-boundary-value problems for linear and integrable nonlinear dispersive equations. Nonlinearity, 21 (2008), T195-T203.  
  2. J.L. Bona, W.G. Pritchard, L.R. Scott. An evaluation of a model equation for water waves. Philos. Trans. R. Soc. Lond., A302 (1981), 458–510.  
  3. J. Bona, R. Winther. The Korteweg–de Vries equation, posed in a quarter–plane. SIAM J. Math. Anal., 14 (1983), 1056–1106.  
  4. R. Carroll, Q. Bu. Solution of the forced nonlinear Schrödinger (NLS) equation using PDE techniques. Appl. Anal., 41 (1991), 33–51.  
  5. C.K. Chu, L.W. Xiang, Y. Baransky. Solitary waves induced by boundary motion. Commun. Pure Appl. Math., 36 (1983), 495–504.  
  6. S. Clark, F. Gesztesy. Weyl-TitchmarshM-function asymptotics for matrix-valued Schrödinger operators. Proc. Lond. Math. Soc., III. Ser., 82 (2001), 701–724.  
  7. S. Clark, F. Gesztesy, M. Zinchenko. Weyl-Titchmarsh theory and Borg-Marchenko-type uniqueness results for CMV operators with matrix-valued Verblunsky coefficients. Oper. Matrices, 1 (2007), 535–592.  
  8. A.S. Fokas. Integrable nonlinear evolution equations on the half-line. Comm. Math. Phys., 230 (2002), 1–39.  
  9. A.S. Fokas. A unified approach to boundary value problems. CBMS-NSF Regional Conference Ser. in Appl. Math. vol. 78. SIAM, Philadelphia, 2008.  
  10. A.S. Fokas, J. Lenells. Explicit soliton asymptotics for the Korteweg–de Vries equation on the half-line. Nonlinearity, 23 (2010), 937–976.  
  11. G. Freiling, V. Yurko. Inverse Sturm–Liouville Problems and Their Applications. Nova Science Publishers, Huntington, N.Y., 2001.  
  12. F. Gesztesy, B. Simon. On local Borg-Marchenko uniqueness results. Commun. Math. Phys., 211 (2000), 273–287.  
  13. F. Gesztesy, B. Simon. A new approach to inverse spectral theory. II. General real potentials and the connection to the spectral measure. Ann. of Math. (2), 152 (2000), 593–643.  
  14. I. Gohberg, M.A. Kaashoek, A.L. Sakhnovich. Sturm-Liouville systems with rational Weyl functions : explicit formulas and applications. Integr. Equ. Oper. Theory, 30 (1998), 338–377.  
  15. M. Kac, P. van Moerbeke. A complete solution of the periodic Toda problem. Proc. Natl. Acad. Sci. USA, 72 (1975), 2879–2880.  
  16. D.J. Kaup, H. Steudel. Recent results on second harmonic generation. Contemp. Math., 326 (2003), 33–48.  
  17. A. Kostenko, A. Sakhnovich, G. Teschl. Weyl-Titchmarsh theory for Schrödinger operators with strongly singular potentials. Int. Math. Res. Not. 2011, Art. ID rnr065, 49pp.  
  18. P.C. Sabatier. Elbow scattering and inverse scattering applications to LKdV and KdV. J. Math. Phys., 41 (2000), 414–436.  
  19. P.C. Sabatier. Lax equations scattering and KdV. J. Math. Phys., 44 (2003), 3216–3225.  
  20. P.C. Sabatier. Generalized inverse scattering transform applied to linear partial differential equations. Inverse Probl., 22 (2006), 209–228.  
  21. A.L. Sakhnovich. Dirac type and canonical systems : spectral and Weyl-Titchmarsh fuctions, direct and inverse problems. Inverse Probl., 18 (2002), 331–348.  
  22. A.L. Sakhnovich. Second harmonic generation : Goursat problem on the semi-strip, Weyl functions and explicit solutions. Inverse Probl., 21 (2005), 703–716.  
  23. A.L. Sakhnovich. On the compatibility condition for linear systems and a factorization formula for wave functions. J. Differ. Equations, 252 (2012), 3658–3667.  
  24. A.L. Sakhnovich. Sine-Gordon theory in a semi-strip. Nonlinear Analysis, 75 (2012), 964–974.  
  25. L.A. Sakhnovich. Nonlinear equations and inverse problems on the semi-axis (Russian). Preprint 87.30. Mathematical Institute, Kiev, 1987.  
  26. L.A. Sakhnovich. Evolution of spectral data, and nonlinear equations. Ukrain. Math. J., 40 (1988), 459–461.  
  27. L.A. Sakhnovich. Spectral Theory of Canonical Differential Systems. Method of Operator Identities. Operator Theory Adv. Appl. Ser. vol. 107. Birkhäuser, Basel, 1999.  
  28. B.A. Ton. Initial boundary value problems for the Korteweg-de Vries equation. J. Differ. Equations25 (1977), 288–309.  
  29. P.A. Treharne, A.S. Fokas. The generalized Dirichlet to Neumann map for the KdV equation on the half-line. J. Nonlinear Sci., 18 (2008), 191–217.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.