## Duke Mathematical Journal

### Stability in $H^1$ of the sum of $K$ solitary waves for some nonlinear Schrödinger equations

#### Abstract

In this article we consider nonlinear Schrödinger (NLS) equations in $\mathbb{R}^d$ for $d=1$, $2$, and $3$. We consider nonlinearities satisfying a flatness condition at zero and such that solitary waves are stable. Let $R_k(t,x)$ be $K$ solitary wave solutions of the equation with different speeds $v_1,v_2,\ldots,v_K$. Provided that the relative speeds of the solitary waves $v_k-v_{k-1}$ are large enough and that no interaction of two solitary waves takes place for positive time, we prove that the sum of the $R_k(t)$ is stable for $t\geqslant 0$ in some suitable sense in $H^1$. To prove this result, we use an energy method and a new monotonicity property on quantities related to momentum for solutions of the nonlinear Schrödinger equation. This property is similar to the $L^2$ monotonicity property that has been proved by Martel and Merle for the generalized Korteweg–de Vries (gKdV) equations (see [12, Lem. 16, proof of Prop. 6]) and that was used to prove the stability of the sum of $K$ solitons of the gKdV equations by the authors of the present article (see [15, Th. 1(i)]).

#### Article information

Source
Duke Math. J., Volume 133, Number 3 (2006), 405-466.

Dates
First available in Project Euclid: 13 June 2006

https://projecteuclid.org/euclid.dmj/1150201198

Digital Object Identifier
doi:10.1215/S0012-7094-06-13331-8

Mathematical Reviews number (MathSciNet)
MR2228459

Zentralblatt MATH identifier
1099.35134

#### Citation

Martel, Yvan; Merle, Frank; Tsai, Tai-Peng. 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. doi:10.1215/S0012-7094-06-13331-8. https://projecteuclid.org/euclid.dmj/1150201198

#### References

• H. Berestycki and P.-L. Lions, Nonlinear scalar field equations, I: Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), 313--345.
• V. S. Buslaev and G. S. Perelman, On the stability of solitary waves for nonlinear Schrödinger equations'' in Nonlinear Evolution Equations, Amer. Math. Soc. Transl. Ser. 2 164, Amer. Math. Soc., Providence, 1995, 75--98.
• T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549--561.
• S. Cuccagna, Stabilization of solutions to nonlinear Schrödinger equations, Comm. Pure Appl. Math. 54 (2001), 1110--1145.
• K. El Dika, Asymptotic stability of solitary waves for the Benjamin-Bona-Mahony equation, Discrete Contin. Dyn. Syst. 13 (2005), 583--622.
• K. El Dika and Y. Martel, Stability of $N$-solitary waves for the generalized BBM equations, Dyn. Partial Differ. Equ. 1 (2004), 401--437.
• J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, I: The Cauchy problem, general case, J. Funct. Anal. 32 (1979), 1--32.
• M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry, I, J. Funct. Anal. 74 (1987), 160--197.
• P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109--145.
• J. H. Maddocks and R. L. Sachs, On the stability of KdV multi-solitons, Comm. Pure Appl. Math. 46 (1993), 867--901.
• Y. Martel, Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg --.de Vries equations, Amer. J. Math. 127 (2005), 1103--1140.
• Y. Martel and F. Merle, A Liouville theorem for the critical generalized Korteweg --.de Vries equation, J. Math. Pures Appl. (9) 79 (2000), 339--425.
• —, Asymptotic stability of solitons for subcritical generalized KdV equations, Arch. Ration. Mech. Anal. 157 (2001), 219--254.
• —, Instability of solitons for the critical generalized Korteweg --.de Vries equation, Geom. Funct. Anal. 11 (2001), 74--123.
• 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 (2002), 347--373.
• F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), 555--578.
• R. L. Pego and M. I. Weinstein, Asymptotic stability of solitary waves, Comm. Math. Phys. 164 (1994), 305--349.
• G. Perelman, Some results on the scattering of weakly interacting solitons for nonlinear Schrödinger equations'' in Spectral Theory, Microlocal Analysis, Singular Manifolds, Math. Top. 14, Akademie, Berlin, 1997, 78--137.
• —, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, Comm. Partial Differential Equations 29 (2004), 1051--1095.
• I. Rodnianski, W. Schlag, and A. D. Soffer, Asymptotic stability of $N$-soliton states of NLS, to appear in Comm. Pure Appl. Math.
• M. I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472--491.
• —, Lyapunov stability of ground states of nonlinear dispersive evolution equations, Comm. Pure Appl. Math. 39 (1986), 51--67.
• V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34 (1972), 62--69.