### Abstract

We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε^{2}u″ + V(x)u = |u|^{p-1}u in R, p > 1, when s is a small parameter. Given any finite set of points x_{1} < X_{2} < ⋯ < x_{m} constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n _{i}, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each x_{i} as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

Original language | English |
---|---|

Pages (from-to) | 1653-1671 |

Number of pages | 19 |

Journal | Nonlinearity |

Volume | 15 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2002 Sep |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics
- Mathematical Physics
- Statistical and Nonlinear Physics

### Cite this

*Nonlinearity*,

*15*(5), 1653-1671. https://doi.org/10.1088/0951-7715/15/5/315

**An elementary construction of complex patterns in nonlinear schrödinger equations.** / Del Pino, Manuel; Felmer, Patrido; Tanaka, Kazunaga.

Research output: Contribution to journal › Article

*Nonlinearity*, vol. 15, no. 5, pp. 1653-1671. https://doi.org/10.1088/0951-7715/15/5/315

}

TY - JOUR

T1 - An elementary construction of complex patterns in nonlinear schrödinger equations

AU - Del Pino, Manuel

AU - Felmer, Patrido

AU - Tanaka, Kazunaga

PY - 2002/9

Y1 - 2002/9

N2 - We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

AB - We consider the problem of finding standing waves to a nonlinear Schrödinger equation. This leads to searching for solutions of the equation -ε2u″ + V(x)u = |u|p-1u in R, p > 1, when s is a small parameter. Given any finite set of points x1 < X2 < ⋯ < xm constituted by isolated local maxima or minima of V, and corresponding arbitrary integers n i, i = 1,..., m, we prove that there is a finite energy solution exhibiting a cluster of n/ spikes concentrating around each xi as ε → 0. The clusters consist of spikes with alternating sign if the point is a minimum, and of constant sign if it is a maximum. This construction extends to infinitely many clusters of spikes under appropriate conditions. The proof follows an elementary variational matching approach, which resembles the so-called broken-geodesic method.

UR - http://www.scopus.com/inward/record.url?scp=0041972657&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041972657&partnerID=8YFLogxK

U2 - 10.1088/0951-7715/15/5/315

DO - 10.1088/0951-7715/15/5/315

M3 - Article

AN - SCOPUS:0041972657

VL - 15

SP - 1653

EP - 1671

JO - Nonlinearity

JF - Nonlinearity

SN - 0951-7715

IS - 5

ER -