## Analysis & PDE

• Anal. PDE
• Volume 8, Number 3 (2015), 629-674.

### Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers

#### Abstract

We study the long-time dynamics of complex-valued modified Korteweg–de Vries (mKdV) solitons, which are distinguished because they blow up in finite time. We establish stability properties at the $H1$ level of regularity, uniformly away from each blow-up point. These new properties are used to prove that mKdV breathers are $H1$-stable, improving our previous result [Comm. Math. Phys. 324:1 (2013) 233–262], where we only proved $H2$-stability. The main new ingredient of the proof is the use of a Bäcklund transformation which relates the behavior of breathers, complex-valued solitons and small real-valued solutions of the mKdV equation. We also prove that negative energy breathers are asymptotically stable. Since we do not use any method relying on the inverse scattering transform, our proof works even under $L2(ℝ)$ perturbations, provided a corresponding local well-posedness theory is available.

#### Article information

Source
Anal. PDE, Volume 8, Number 3 (2015), 629-674.

Dates
Revised: 4 December 2014
Accepted: 9 February 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.apde/1510843096

Digital Object Identifier
doi:10.2140/apde.2015.8.629

Mathematical Reviews number (MathSciNet)
MR3353827

Zentralblatt MATH identifier
1323.35156

#### Citation

Alejo, Miguel; Muñoz, Claudio. Dynamics of complex-valued modified KdV solitons with applications to the stability of breathers. Anal. PDE 8 (2015), no. 3, 629--674. doi:10.2140/apde.2015.8.629. https://projecteuclid.org/euclid.apde/1510843096

#### References

• M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, London Mathematical Society Lecture Note Series 149, Cambridge University Press, 1991.
• M. A. Alejo, “Geometric breathers of the mKdV equation”, Acta Appl. Math. 121 (2012), 137–155.
• M. A. Alejo and C. Muñoz, “Nonlinear stability of mKdV breathers”, Comm. Math. Phys. 324:1 (2013), 233–262.
• M. A. Alejo, C. Muñoz, and L. Vega, “The Gardner equation and the $L\sp 2$-stability of the $N$-soliton solution of the Korteweg–de Vries equation”, Trans. Amer. Math. Soc. 365:1 (2013), 195–212.
• S. Aubry, “Breathers in nonlinear lattices: existence, linear stability and quantization”, Phys. D 103:1-4 (1997), 201–250.
• T. B. Benjamin, “The stability of solitary waves”, Proc. Roy. Soc. London Ser. A 328 (1972), 153–183.
• B. Birnir, H. P. McKean, and A. Weinstein, “The rigidity of sine-Gordon breathers”, Comm. Pure Appl. Math. 47:8 (1994), 1043–1051.
• J. L. Bona, P. E. Souganidis, and W. A. Strauss, “Stability and instability of solitary waves of Korteweg–de Vries type”, Proc. Roy. Soc. London Ser. A 411:1841 (1987), 395–412.
• J. L. Bona, S. Vento, and F. B. Weissler, “Singularity formation and blowup of complex-valued solutions of the modified KdV equation”, Discrete Contin. Dyn. Syst. 33:11-12 (2013), 4811–4840.
• T. Buckmaster and H. Koch, “The Korteweg–de Vries equation at $H^{-1}$ regularity”, Ann. Inst. H. Poincaré Anal. Non Linéaire (online publication June 2014).
• M. Christ, J. Holmer, and D. Tataru, “Low regularity bounds for mKdV”, preprint, 2012.
• J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, “Sharp global well-posedness for KdV and modified KdV on $\Bbb R$ and $\Bbb T$”, J. Amer. Math. Soc. 16:3 (2003), 705–749.
• P. Deift and X. Zhou, “A steepest descent method for oscillatory Riemann–Hilbert problems: asymptotics for the mKdV equation”, Ann. of Math. $(2)$ 137:2 (1993), 295–368.
• J. Denzler, “Nonpersistence of breather families for the perturbed sine Gordon equation”, Comm. Math. Phys. 158:2 (1993), 397–430.
• C. Gorria, M. A. Alejo, and L. Vega, “Discrete conservation laws and the convergence of long time simulations of the mkdv equation”, J. Comput. Phys. 235 (2013), 274–285.
• M. Grillakis, J. Shatah, and W. Strauss, “Stability theory of solitary waves in the presence of symmetry, I”, J. Funct. Anal. 74:1 (1987), 160–197.
• R. Hirota, “Exact solution of the modified Korteweg–de Vries equation for multiple collisions of solitons”, J. Phys. Soc. Japan 33:5 (1972), 1456–1458.
• A. Hoffman and C. E. Wayne, “Orbital stability of localized structures via Bäcklund transformations”, Differential Integral Equations 26:3-4 (2013), 303–320.
• C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalized Korteweg–de Vries equation via the contraction principle”, Comm. Pure Appl. Math. 46:4 (1993), 527–620.
• C. E. Kenig, G. Ponce, and L. Vega, “On the ill-posedness of some canonical dispersive equations”, Duke Math. J. 106:3 (2001), 617–633.
• S. Kwon and T. Oh, “On unconditional well-posedness of modified KdV”, Int. Math. Res. Not. 2012:15 (2012), 3509–3534.
• G. L. Lamb, Jr., Elements of soliton theory, John Wiley & Sons, New York, 1980.
• P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves”, Comm. Pure Appl. Math. 21 (1968), 467–490.
• J. H. Maddocks and R. L. Sachs, “On the stability of KdV multi-solitons”, Comm. Pure Appl. Math. 46:6 (1993), 867–901.
• Y. Martel and F. Merle, “Blow up in finite time and dynamics of blow up solutions for the $L\sp 2$-critical generalized KdV equation”, J. Amer. Math. Soc. 15:3 (2002), 617–664.
• Y. Martel and F. Merle, “Asymptotic stability of solitons of the subcritical gKdV equations revisited”, Nonlinearity 18:1 (2005), 55–80.
• Y. Martel, F. Merle, and T.-P. Tsai, “Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations”, Comm. Math. Phys. 231:2 (2002), 347–373.
• F. Merle, “On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass”, Comm. Pure Appl. Math. 45:2 (1992), 203–254.
• F. Merle and L. Vega, “$L\sp 2$ stability of solitons for KdV equation”, Int. Math. Res. Not. 2003:13 (2003), 735–753.
• R. M. Miura, C. S. Gardner, and M. D. Kruskal, “Korteweg–de Vries equation and generalizations, II: Existence of conservation laws and constants of motion”, J. Mathematical Phys. 9 (1968), 1204–1209.
• T. Mizumachi and D. Pelinovsky, “Bäcklund transformation and $L^2$-stability of NLS solitons”, Int. Math. Res. Not. 2012:9 (2012), 2034–2067.
• R. L. Pego and M. I. Weinstein, “Asymptotic stability of solitary waves”, Comm. Math. Phys. 164:2 (1994), 305–349.
• P. C. Schuur, Asymptotic analysis of soliton problems: an inverse scattering approach, Lecture Notes in Mathematics 1232, Springer, Berlin, 1986.
• A. Soffer and M. I. Weinstein, “Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations”, Invent. Math. 136:1 (1999), 9–74.
• M. Wadati, “The modified Korteweg–de Vries equation”, J. Phys. Soc. Japan 34 (1973), 1289–1296.
• H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the Korteweg–de Vries equation”, Phys. Rev. Lett. 31 (1973), 1386–1390.
• M. I. Weinstein, “Lyapunov stability of ground states of nonlinear dispersive evolution equations”, Comm. Pure Appl. Math. 39:1 (1986), 51–67.