Stable quadratic solitons consisting of fundamental waves and oscillatory second harmonics subject to boundary confinement

We investigate quadratic solitons consisting of fundamental waves and oscillatory second harmonics, which can be stable when subject to boundary confinement. A family of quadratic solitons whose fundamental waves can be monopole, dipole, or even multipole are found numerically. The existence of solitons, depending on the sample size and degree of nonlocality, are given for strongly, generally, and weakly nonlocal cases, respectively.

Figure 1

(a) Unbounded sine-oscillatory response function Eq. (8). [(b),(c)] Bounded sine-oscillatory response function Eq. (10). Here $\sigma =0.5$.

Figure 2

[(a),(b)] Soliton power $P$ (solid line) and width $w$ (dotted line) for FW vs the degree of nonlocality and sample size, respectively. [(c),(e)] Typical soliton profiles with different $\alpha$ and $l_i$.

Figure 3

Profiles of solitons when $\alpha =0.04$ and $l=66$. Dash-dotted frames represent the ISZ.

Figure 4

[(a),(b)] Profiles of the irreducible dipole solitons, where $\alpha =0.04$ and $l=56$. [(c)–(e)] Profiles of dipole solitons when $\alpha =0.04$ and $l=87$. Dash-dotted frames represent the ISZ.

Figure 5

Soliton power $P$ (solid line) and width $w$ (dotted line) for FW vs the degree of nonlocality $\alpha$ (a) and sample size $l$ (b), respectively.

Figure 6

[(a),(b)] Profiles of quadratic solitons for different $l$ when $\alpha =11$. [(c)–(f)] Simulated propagation of FWs [panels (c) and (d)] and SHs [panels (e) and (f)] corresponding to panels (a) and (b) based on Eqs. (1) and (2). Dash-dotted frames represent the ISZ.

Figure 7

[(a),(b)] Profiles of quadratic solitons for the same $l$ and $\alpha$ as in Fig. 6. (c) Typical dipole quadratic soliton profiles with $l_i=14.95$. Dash-dotted frames represent the ISZ.

Figure 8

[(a), (b)] Soliton power $P$ (solid line) and width $w$ (dotted line) for FW vs the degree of nonlocality $\alpha$ and sample size $l$, respectively. (c) Typical soliton profiles with $l_i=20.08$.

Figure 9

[(a),(b)] Profiles of quadratic solitons for the same $l$ and $\alpha$ as in Fig. 8. (c) Typical dipole quadratic soliton profiles with $l_i=20.08$. Dash-dotted frames represent the ISZ.