×

Construction of multi-soliton solutions for the \(L^2\)-supercritical gKdV and NLS equations. (English) Zbl 1273.35234

The paper proves the existence of multi-soliton solutions of the generalized Korteweg-de Vries (gKdV) and the nonlinear Schrödinger (NLS) equations in the \(L^2\)-supercritical case. The gKdV problem reads \[ u_t +(u_{xx}+u^p)_x=0, \quad (x,t)\in \mathbb{R}^2, p>5, \] and the NLS problem is \[ i u_t +\Delta u +|u|^{p-1}u=0, \quad (x,t)\in \mathbb{R}^{d+1}, p\in \left(1+\tfrac{4}{d},\tfrac{d+2}{d-2}\right), \] where \(\tfrac{d+2}{d-2}\) is replaced by \(\infty\) for \(d=1,2\).
In these regimes soliton solutions of both problems are known to be unstable. Multi-soliton solutions are solutions which at large times behave like the sum of \(N\) given solitons. In more detail, the paper proves that given \(N\) values of speed and spatial shift, a multi-soliton exists which converges to the sum of \(N\) solitons with these speeds and shifts as \(t\to\infty\).
The idea of the proof for the gKdV is the following. In the first step solutions \(u_n\) of gKdV are proved for a sequence of time intervals \(I_n=[T_0,S_n]\) with \(S_n\to\infty\) such that at \(t=S_n\) the solution is a sum of \(N\) solitons plus a correction term given by eigenfunctions of \(L\partial_x\), where \(L\) is the energy Hessian around a soliton. A crucial uniform (in \(n\)) \(H^1\)-estimate of the difference of \(u_n\) and the sum of \(N\) solitons is obtained next. The difference converges to zero exponentially fast as \(t\to\infty\). Finally by a compactness argument \(u_n(t)\) is shown to converge strongly to a gKdV solution \(u^*(t)\in H^{1/2}(\mathbb{R})\) and weakly in \(H^1(\mathbb{R})\). By ruling out the possibility of blowup the solution \(u^*\) exists on \([T_0,\infty)\). The above estimate then shows that \(u^*\) converges to the sum of \(N\) solitons exponentially fast as \(t\to\infty\). The proof for the NLS is analogous.
This work is an extension of [F. Merle, Commun. Math. Phys. 129, No. 2, 223–240 (1990; Zbl 0707.35021)], [Y. Martel, Am. J. Math. 127, No. 5, 1103–1140 (2005; Zbl 1090.35158)] and [Y. Martel and F. Merle, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 23, No. 6, 849–864 (2006; Zbl 1133.35093)].

MSC:

35Q51 Soliton equations
35Q53 KdV equations (Korteweg-de Vries equations)
35Q55 NLS equations (nonlinear Schrödinger equations)
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Berestycki, H. and Lions, P.-L.: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), no. 4, 313-345. · Zbl 0533.35029
[2] Bona, J.L., Souganidis, P.E. and Strauss, W.A.: Stability and instability of solitary waves of Korteweg-de Vries type. Proc. Roy. Soc. London Ser. A 411 (1987), no. 1841, 395-412. · Zbl 0648.76005
[3] Cazenave, T. and Lions, P.-L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Comm. Math. Phys. 85 (1982), no. 4, 549-561. · Zbl 0513.35007
[4] Cazenave, T. and Weissler, F.: The Cauchy problem for the critical nonlinear Schrödinger equation in \(H^s\). Nonlinear Anal. 14 (1990), 807-836. · Zbl 0706.35127
[5] Duyckaerts, T. and Merle, F.: Dynamics of threshold solutions for energy-critical wave equation. Int. Math. Res. Pap. IMRP 2007 , no. 4, Art. ID rpn002, 67 pp. (2008). · Zbl 1159.35043
[6] Duyckaerts, T. and Roudenko, S.: Threshold solutions for the focusing 3D cubic Schrödinger equation. Rev. Mat. Iberoam. 26 (2010), no. 1, 1-56. · Zbl 1195.35276
[7] El Dika, K.: Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation. Discrete Contin. Dyn. Syst. 13 (2005), 583-622. · Zbl 1083.35019
[8] El Dika, K. and Martel, Y.: Stability of \(N\) solitary waves for the generalized BBM equations. Dyn. Partial Differ. Equ. 1 (2004), 401-437. · Zbl 1080.35116
[9] Gidas, B., Ni, W.M. and Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbbR^n\). In Mathematical analysis and applications, Part A , 369-402. Adv. in Math. Suppl. Stud. 7 . Academic Press, New York, 1981. · Zbl 0469.35052
[10] Ginibre, J. and Velo, G.: On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case. J. Funct. Anal. 32 (1979), no. 1, 1-32. · Zbl 1159.35043
[11] Grillakis, M.: Analysis of the linearization around a critical point of an infinite dimensional hamiltonian system. Comm. Pure Appl. Math. 43 (1990), no. 3, 299-333. · Zbl 0731.35010
[12] Grillakis, M., Shatah, J. and Strauss, W.: Stability theory of solitary waves in the presence of symmetry. I. J. Funct. Anal. 74 (1987), no. 1, 160-197. · Zbl 0656.35122
[13] Kato, T.: On the Cauchy problem for the (generalized) Korteweg-de Vries equation. In Studies in applied mathematics , 93-128. Adv. Math. Suppl. Stud. 8 . Academic Press, New York, 1983. · Zbl 0549.34001
[14] Kenig, C.E., Ponce, G. and Vega, L.: Well-posedness and scattering result for the generalized Korteweg-De Vries equation via the contraction principle. Comm. Pure Appl. Math. 46 (1993), 527-620. · Zbl 0808.35128
[15] Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u-u+u^ p=0\) in \(\mathbbR^n\). Arch. Rational Mech. Anal. 105 (1989), no. 3, 243-266. · Zbl 0676.35032
[16] Martel, Y.: Asymptotic \(N\)-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations. Amer. J. Math. 127 (2005), no. 5, 1103-1140. · Zbl 1090.35158
[17] Martel, Y. and Merle, F.: Asymptotic stability of solitons for subcritical generalized KdV equations. Arch. Ration. Mech. Anal. 157 (2001), no. 3, 219-254. · Zbl 0981.35073
[18] Martel, Y. and Merle, F.: Instability of solitons for the critical generalized Korteweg-De Vries equation. Geom. Funct. Anal. 11 (2001), no. 1, 74-123. · Zbl 0985.35071
[19] Martel, Y. and Merle, F.: 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. JSTOR: · Zbl 1005.35081
[20] Martel, Y. and Merle, F.: Multi solitary waves for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), 849-864. · Zbl 1133.35093
[21] Martel, Y., Merle, F. and Tsai, T.-P.: Stability and asymptotic stability in the energy space of the sum of \(n\) solitons for subcritical gKdV equations. Comm. Math. Phys. 231 (2002), 347-373. · Zbl 1017.35098
[22] Martel, Y., Merle, F. and Tsai, T.-P.: Stability in \(H^1\) of the sum of \(K\) solitary waves for some nonlinear Schrödinger equations. Duke Math. J. 133 (2006), no. 3, 405-466. · Zbl 1099.35134
[23] Merle, F.: 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. · Zbl 0707.35021
[24] Miura, R.M.: The Korteweg-de Vries equation: a survey of results. SIAM Rev. 18 (1976), 412-459. JSTOR: · Zbl 0333.35021
[25] Mizumachi, T.: Asymptotic stability of solitary wave solutions to the regularized long-wave equation. J. Differential Equations 200 (2004), 312-341. · Zbl 1053.35119
[26] Pego, R.L. and Weinstein, M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340 (1992), no. 1656, 47-94. JSTOR: · Zbl 0776.35065
[27] Pego, R.L. and Weinstein, M.I.: Asymptotic stability of solitary waves. Comm. Math. Phys. 164 (1994), 305-349. · Zbl 0805.35117
[28] Schlag, W.: Spectral theory and nonlinear partial differential equations: a survey. Discrete Contin. Dyn. Syst. 15 (2006), no. 3, 703-723. · Zbl 1121.35121
[29] Weinstein, M.I.: Modulational stability of ground states of nonlinear Schrödinger equations. SIAM J. Math. Anal. 16 (1985), 472-491. · Zbl 0583.35028
[30] Weinstein, M.I.: Lyapunov stability of ground states of nonlinear dispersive evolution equations. Comm. Pure Appl. Math. 39 (1986), 51-68. · Zbl 0594.35005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.