## Asymptotic $K$-soliton-like solutions of the Zakharov-Kuznetsov type equations

HTML articles powered by AMS MathViewer

- by Frédéric Valet PDF
- Trans. Amer. Math. Soc.
**374**(2021), 3177-3213 Request permission

## Abstract:

We study here the Zakharov-Kuznetsov equation in dimension $2$, $3$ and $4$ and the modified Zakharov-Kuznetsov equation in dimension $2$. Those equations admit solitons, characterized by their velocity and their shift. Given the parameters of $K$ solitons $R^k$ (with distinct velocities), we prove the existence and uniqueness of a multi-soliton $u$ such that \[ \| u(t) - \sum _{k=1}^K R^k(t) \|_{H^1} \to 0 \quad \text {as} \quad t \to +\infty . \] The convergence takes place in $H^s$ with an exponential rate for all $s \ge 0$. The construction is made by successive approximations of the multi-soliton. We use classical arguments to control of $H^1$-norms of the errors (inspired by Martel [Amer. J. Math. 127 (2005), pp. 1103–1140]), and introduce a new ingredient for the control of the $H^s$-norm in dimension $d\geq 2$, by a technique close to monotonicity.## References

- H. Berestycki and P.-L. Lions,
*Nonlinear scalar field equations. I. Existence of a ground state*, Arch. Rational Mech. Anal.**82**(1983), no. 4, 313–345. MR**695535**, DOI 10.1007/BF00250555 - Debdeep Bhattacharya, Luiz Gustavo Farah, and Svetlana Roudenko,
*Global well-posedness for low regularity data in the 2d modified Zakharov-Kuznetsov equation*, J. Differential Equations**268**(2020), no. 12, 7962–7997. MR**4079024**, DOI 10.1016/j.jde.2019.11.092 - Vianney Combet,
*Multi-soliton solutions for the supercritical gKdV equations*, Comm. Partial Differential Equations**36**(2011), no. 3, 380–419. MR**2763331**, DOI 10.1080/03605302.2010.503770 - Raphaël Côte, Yvan Martel, and Frank Merle,
*Construction of multi-soliton solutions for the $L^2$-supercritical gKdV and NLS equations*, Rev. Mat. Iberoam.**27**(2011), no. 1, 273–302. MR**2815738**, DOI 10.4171/RMI/636 - Raphaël Côte and Claudio Muñoz,
*Multi-solitons for nonlinear Klein-Gordon equations*, Forum Math. Sigma**2**(2014), Paper No. e15, 38. MR**3264254**, DOI 10.1017/fms.2014.13 - Raphaël Côte, Claudio Muñoz, Didier Pilod, and Gideon Simpson,
*Asymptotic stability of high-dimensional Zakharov-Kuznetsov solitons*, Arch. Ration. Mech. Anal.**220**(2016), no. 2, 639–710. MR**3461359**, DOI 10.1007/s00205-015-0939-x - Raphaël Cote and Frédéric Valet,
*Polynomial growth of high Sobolev norms of solutions to the Zakharov-Kuznetsov equation*, 2019. - Raphaël Côte, Yvan Martel, and Xu Yuan,
*Long-time asymptotics of the one-dimensional damped nonlinear Klein-Gordon equation*, 2020. - Anne de Bouard,
*Stability and instability of some nonlinear dispersive solitary waves in higher dimension*, Proc. Roy. Soc. Edinburgh Sect. A**126**(1996), no. 1, 89–112. MR**1378834**, DOI 10.1017/S0308210500030614 - A. V. Faminskiĭ,
*The Cauchy problem for the Zakharov-Kuznetsov equation*, Differentsial′nye Uravneniya**31**(1995), no. 6, 1070–1081, 1103 (Russian, with Russian summary); English transl., Differential Equations**31**(1995), no. 6, 1002–1012. MR**1383936** - Luiz Gustavo Farah, Justin Holmer, and Svetlana Roudenko,
*On instability of solitons in the 2d cubic Zakharov-Kuznetsov equation*, São Paulo J. Math. Sci.**13**(2019), no. 2, 435–446. MR**4025576**, DOI 10.1007/s40863-019-00142-7 - Luiz Gustavo Farah, Justin Holmer, Svetlana Roudenko, and Kai Yang,
*Blow-up in finite or infinite time of the 2D cubic Zakharov-Kuznetsov equation*, 2018. - Eduard Feireisl,
*Finite energy travelling waves for nonlinear damped wave equations*, Quart. Appl. Math.**56**(1998), no. 1, 55–70. MR**1604876**, DOI 10.1090/qam/1604876 - Axel Grünrock and Sebastian Herr,
*The Fourier restriction norm method for the Zakharov-Kuznetsov equation*, Discrete Contin. Dyn. Syst.**34**(2014), no. 5, 2061–2068. MR**3124726**, DOI 10.3934/dcds.2014.34.2061 - Daniel Han-Kwan,
*From Vlasov-Poisson to Korteweg–de Vries and Zakharov-Kuznetsov*, Comm. Math. Phys.**324**(2013), no. 3, 961–993. MR**3123542**, DOI 10.1007/s00220-013-1825-8 - Sebastian Herr and Shinya Kinoshita,
*The Zakharov-Kuznetsov equation in high dimensions: Small initial data of critical regularity*, 2020. - Jacek Jendrej, Michał Kowalczyk, and Andrew Lawrie,
*Dynamics of strongly interacting kink-antikink pairs for scalar fields on a line*, 2020. - Shinya Kinoshita,
*Well-posedness for the Cauchy problem of the Klein-Gordon-Zakharov system in 2D*, Discrete Contin. Dyn. Syst.**38**(2018), no. 3, 1479–1504. MR**3809003**, DOI 10.3934/dcds.2018061 - Man Kam Kwong,
*Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $\textbf {R}^n$*, Arch. Rational Mech. Anal.**105**(1989), no. 3, 243–266. MR**969899**, DOI 10.1007/BF00251502 - David Lannes, Felipe Linares, and Jean-Claude Saut,
*The Cauchy problem for the Euler-Poisson system and derivation of the Zakharov-Kuznetsov equation*, Studies in phase space analysis with applications to PDEs, Progr. Nonlinear Differential Equations Appl., vol. 84, Birkhäuser/Springer, New York, 2013, pp. 181–213. MR**3185896**, DOI 10.1007/978-1-4614-6348-1_{1}0 - Felipe Linares and Ademir Pastor,
*Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation*, SIAM J. Math. Anal.**41**(2009), no. 4, 1323–1339. MR**2540268**, DOI 10.1137/080739173 - Felipe Linares and Ademir Pastor,
*Local and global well-posedness for the 2D generalized Zakharov-Kuznetsov equation*, J. Funct. Anal.**260**(2011), no. 4, 1060–1085. MR**2747014**, DOI 10.1016/j.jfa.2010.11.005 - Y. Martel and F. Merle,
*Instability of solitons for the critical generalized Korteweg-de Vries equation*, Geom. Funct. Anal.**11**(2001), no. 1, 74–123. MR**1829643**, DOI 10.1007/PL00001673 - Yvan Martel,
*Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations*, Amer. J. Math.**127**(2005), no. 5, 1103–1140. MR**2170139**, DOI 10.1353/ajm.2005.0033 - Yvan Martel and Frank Merle,
*Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation*, Ann. of Math. (2)**155**(2002), no. 1, 235–280. MR**1888800**, DOI 10.2307/3062156 - Yvan Martel and Frank Merle,
*Multi solitary waves for nonlinear Schrödinger equations*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**23**(2006), no. 6, 849–864 (English, with English and French summaries). MR**2271697**, DOI 10.1016/j.anihpc.2006.01.001 - Yvan Martel and Frank Merle,
*Note on coupled linear systems related to two soliton collision for the quartic gKdV equation*, Rev. Mat. Complut.**21**(2008), no. 2, 327–349. MR**2441957**, DOI 10.5209/rev_{R}EMA.2008.v21.n2.16378 - Yvan Martel, Frank Merle, and Tai-Peng Tsai,
*Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations*, Comm. Math. Phys.**231**(2002), no. 2, 347–373. MR**1946336**, DOI 10.1007/s00220-002-0723-2 - Yvan Martel and Pierre Raphaël,
*Strongly interacting blow up bubbles for the mass critical nonlinear Schrödinger equation*, Ann. Sci. Éc. Norm. Supér. (4)**51**(2018), no. 3, 701–737 (English, with English and French summaries). MR**3831035**, DOI 10.24033/asens.2364 - Frank Merle,
*Construction of solutions with exactly $k$ blow-up points for the Schrödinger equation with critical nonlinearity*, Comm. Math. Phys.**129**(1990), no. 2, 223–240. MR**1048692**, DOI 10.1007/BF02096981 - Mei Ming, Frederic Rousset, and Nikolay Tzvetkov,
*Multi-solitons and related solutions for the water-waves system*, SIAM J. Math. Anal.**47**(2015), no. 1, 897–954. MR**3315224**, DOI 10.1137/140960220 - Robert M. Miura,
*The Korteweg-de Vries equation: a survey of results*, SIAM Rev.**18**(1976), no. 3, 412–459. MR**404890**, DOI 10.1137/1018076 - Luc Molinet and Didier Pilod,
*Bilinear Strichartz estimates for the Zakharov-Kuznetsov equation and applications*, Ann. Inst. H. Poincaré C Anal. Non Linéaire**32**(2015), no. 2, 347–371. MR**3325241**, DOI 10.1016/j.anihpc.2013.12.003 - Nguyễn Tiến Vinh,
*Strongly interacting multi-solitons with logarithmic relative distance for the gKdV equation*, Nonlinearity**30**(2017), no. 12, 4614–4648. MR**3734149**, DOI 10.1088/1361-6544/aa8cab - Tiến Vinh Nguyễn,
*Existence of multi-solitary waves with logarithmic relative distances for the NLS equation*, C. R. Math. Acad. Sci. Paris**357**(2019), no. 1, 13–58 (English, with English and French summaries). MR**3907597**, DOI 10.1016/j.crma.2018.11.012 - Francis Ribaud and Stéphane Vento,
*A note on the Cauchy problem for the 2D generalized Zakharov-Kuznetsov equations*, C. R. Math. Acad. Sci. Paris**350**(2012), no. 9-10, 499–503 (English, with English and French summaries). MR**2929056**, DOI 10.1016/j.crma.2012.05.007 - Francis Ribaud and Stéphane Vento,
*Well-posedness results for the three-dimensional Zakharov-Kuznetsov equation*, SIAM J. Math. Anal.**44**(2012), no. 4, 2289–2304. MR**3023376**, DOI 10.1137/110850566 - R. Sipcic and D. J. Benney,
*Lump interactions and collapse in the modified Zakharov-Kuznetsov equation*, Stud. Appl. Math.**105**(2000), no. 4, 385–403. MR**1793348**, DOI 10.1111/1467-9590.00157 - Michael I. Weinstein,
*Nonlinear Schrödinger equations and sharp interpolation estimates*, Comm. Math. Phys.**87**(1982/83), no. 4, 567–576. MR**691044**, DOI 10.1007/BF01208265 - Michael I. Weinstein,
*Modulational stability of ground states of nonlinear Schrödinger equations*, SIAM J. Math. Anal.**16**(1985), no. 3, 472–491. MR**783974**, DOI 10.1137/0516034 - Michael I. Weinstein,
*Lyapunov stability of ground states of nonlinear dispersive evolution equations*, Comm. Pure Appl. Math.**39**(1986), no. 1, 51–67. MR**820338**, DOI 10.1002/cpa.3160390103 - V. E. Zakharov and E. A. Kuznetsov,
*On three dimensional solitons*, Zh. Eksp. Teoret. Fiz.**66**(1974), 594–597.

## Additional Information

**Frédéric Valet**- Affiliation: IRMA UMR 7501, Université de Strasbourg, CNRS, F-67000 Strasbourg, France
- Address at time of publication: Postboks 7803, 5020 Bergen, Norway
- Email: frederic.valet@uib.no
- Received by editor(s): June 2, 2020
- Published electronically: March 8, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 3177-3213 - MSC (2020): Primary 35Q53, 35Q35, 35B40, 37K40
- DOI: https://doi.org/10.1090/tran/8331
- MathSciNet review: 4237946