Communications in Mathematical Sciences

Volume 13 (2015)

Number 5

Multi-hump solitary waves of a nonlinear Dirac equation

Pages: 1219 – 1242

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n5.a7

Authors

Jian Xu (LMAM and School of Mathematical Sciences, Peking University, Beijing, China)

Sihong Shao (LMAM and School of Mathematical Sciences, Peking University, Beijing, China)

Huazhong Tang (HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing, China)

Dongyi Wei (LMAM and School of Mathematical Sciences, Peking University, Beijing, China)

Abstract

This paper concentrates on a $(1+1)$-dimensional nonlinear Dirac (NLD) equation with a general self-interaction, being a linear combination of the scalar, pseudoscalar, vector and axial vector self-interactions to the power of the integer $k+1$. The solitary wave solutions to the NLD equation are analytically derived, and the upper bounds of the hump number in the charge, energy and momentum densities for the solitary waves are proved analytically in theory. The results show that: (1) for a given integer $k$, the hump number in the charge density is not bigger than $4$, while that in the energy density is not bigger than $3$; (2) those upper bounds can only be achieved in the situation of higher nonlinearity, namely, $k \in \lbrace 5,6,7, \cdots \rbrace$ for the charge density and $k \in \lbrace 3,5,7, \cdots \rbrace$ for the energy density; (3) the momentum density has the same multi-hump structure as the energy density; (4) more than two humps (resp. one hump) in the charge (resp. energy) density can only happen under the linear combination of the pseudoscalar self-interaction and at least one of the scalar and vector (or axial vector) self-interactions. Our results on the multi-hump structure will be interesting in the interaction dynamics for the NLD solitary waves.

Keywords

nonlinear Dirac equation, solitary wave, multi-hump, self-interaction

2010 Mathematics Subject Classification

35L60, 35Q51, 37K40, 81Q05

Full Text (PDF format)