Exact traveling-wave solutions for optical models based on the nonlinear cubic-quintic Schrödinger equation

L. Gagnon

doi:10.1364/JOSAA.6.001477

Journal of the Optical Society of America A
Vol. 6,
Issue 9,
pp. 1477-1483
(1989)
•https://doi.org/10.1364/JOSAA.6.001477

Exact traveling-wave solutions for optical models based on the nonlinear cubic-quintic Schrödinger equation

L. Gagnon

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L. Gagnon, "Exact traveling-wave solutions for optical models based on the nonlinear cubic-quintic Schrödinger equation," J. Opt. Soc. Am. A 6, 1477-1483 (1989)

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Abstract

We present a large set of exact analytic traveling-wavesolutions for the equation i_ut + u_xx = a₁u|u|² + a₂u|u|⁴ that can be of interest in higher-order nonlinear-optical media. This set includes all bright and dark solitary waves as well as periodic ones. Our procedure consists of direct integration; it is then purely algebraic and simple. Solutions are given in simple forms and tabulated.

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Solutions	Parameters and General Behavior	Number
(−2/a₁)^1/2 sech(x)exp(it);	a₁ < 0, <a₂ = 0	1
(−2/a₁)^1/2 sech(x)exp(it);	B, 0 < \|u\|² ≤ −2/a₁	1
${(2 / a_{1})}^{1 / 2} {(b - {sech}^{2} x)}^{1 / 2} exp {i [b {(b - 1)}^{1 / 2} \int \frac{d x}{b - {sech}^{2} x} - (3 b - 1) t]};$	a₁ > 0, a₂ = 0, 1 ≤ b < ∞	2
	D, 2(b − 1)/a₁ < \|u\|² < 2b/a₁	2
${[2 / a_{1} (W_{1} - W_{3})]}^{1 / 2} {[W_{3} c n^{2} (x, k) + W_{2} s n^{2} (x, k)]}^{1 / 2} \times exp i [{(\frac{W_{1} W_{2} W_{3}}{W_{1} - W_{3}})}^{1 / 2} \int \frac{d x}{W_{3} c n^{2} (x, k) + W_{2} s n^{2} (x, k)} - a t],$	a₁ < 0, a₂ = 0, W₁ ≥ 0 ≥ W₂ ≥ W₃,	3
	W₁ ≠ W,
	P, pW₂ < \|u\|²< pW₃
Where $a = \frac{W_{1} + W_{2} + W_{3}}{W_{1} - W_{3}}, k^{2} = \frac{W_{2} - W_{3}}{W_{1} - W_{3}}, p = \frac{2}{a_{1} (W_{1} - W_{3})};$	a₁ > 0, a₂ = 0, 0 ≤ W₃ < W₂ < W₁	4
	P, pW₃ < \|u\|² < pW₂	4
${[\frac{3 W_{4} - {(W_{4} - W_{1})}^{2} W_{1} a_{2} x^{2}}{3 - {(W_{4} - W_{1})}^{2} a_{2} x^{2}}]}^{1 / 2} exp {i [N - (2 a_{2} {W_{1}}^{2} + 3 a_{1} W_{1} / 2) t]},$	a₂ < 0, 0 ≤ W₁ < W₄	5
	B, W₁ < \|u\|² < W₄	5
where $\begin{array}{l} N = S \int \frac{3 - {(W_{4} - W_{1})}^{2} a_{2} x^{2}}{3 W_{4} - {(W_{4} - W_{1})}^{2} W_{1} a_{2} x^{2}} d x, \\ S^{2} = a_{2} {W_{1}}^{3} (W_{1} + a_{1} / 2 a_{2}), W_{4} = - 3 W_{1} - 3 a_{1} / 2 a_{2}; \end{array}$	a₂ < 0, 0 < W₄ < W₁	6
	D, W₄ ≤ \|u\|² < W₁	6
$\begin{array}{l} {(\frac{- 3 a_{1}}{4 a_{2}})}^{1 / 2} {1 + exp [{(\frac{3}{a_{2}})}^{1 / 2} \frac{a_{1}}{2} x]}^{- 1 / 2} exp (i \frac{3 {a_{1}}^{2}}{16 a_{2}} t); \\ {(p W)}^{1 / 2} exp [i (N - a t)], \end{array}$	a₂ > 0, a₁ < 0	7
	K, 0 < \|u\|² < −3a₁/4a₂	7
	a₂ > 0, W₄ ≤ 0 ≤ W₃ < W₁,	8
	W₄ ≠ W₃
	p > 0, ∊ = −1
	D, pW₃ ≤ \|u\|² ≤ pW₁
where $\begin{array}{l} W = \frac{μ + ν cosh θ}{ρ + σ cosh θ}, θ = {[4 a_{2} (W_{1} - W_{3}) (W_{1} - W_{4}) / 3]}^{1 / 2} p x, \\ N = \frac{S {(3 / 4 a_{2})}^{1 / 2}}{ν p^{2} {[(W_{1} - W_{3}) (W_{1} - W_{4})]}^{1 / 2}} [σ θ - (μ σ - ν ρ) \int \frac{d θ}{μ + ν cosh θ}], \\ μ = - 2 W_{3} W_{4} + W_{1} (W_{3} + W_{4}), ν = ∊ W_{1} (W_{4} - W_{3}), \\ ρ = 2 W_{1} - (W_{3} + W_{4}), σ = ∊ (W_{4} - W_{3}), \\ S^{2} = - a_{2} p^{4} {W_{1}}^{2} W_{3} W_{4} / 3, W_{4} = - 2 W_{1} - W_{3} - 3 a_{1} / 2 a_{2} p, \\ a = - a_{2} p^{2} [{W_{1}}^{2} + 2 W_{1} (W_{3} + W_{4}) + W_{3} W_{4}] / 3; \end{array}$	a₂ > 0, 0 = W₁ < W₃ < W₄	9
	p > 0, ∊ = −1
	B, 0 < \|u\|² ≤ pW₃
	a₂ < 0, W₄ < W₃ ≤ 0 ≤ W₁, W₃ ≠ W₁	10
	p < 0, ∊ = 1
	P, pW₃ ≤ \|u\|² ≤ pW₄
	a₂< 0, 0 ≤ W₃ < W₁ < W₄	11
	p > 0, ∊ = 1	12
	B, pW₁ < \|u\|² ≤ pW₄
	p > 0, ∊ = −1
	D, pW₁ < \|u\|² ≤ pW₃
	a₂ < 0, W₄ < W₁ = 0 < W₃	13
	p > 0, ∊ = −1
	B, 0 < \|u\|² ≤ pW₃
	p < 0, ∊ = 1	14
	B, 0 < \|u\|² ≤ pW₄	14
(pW)^1/2exp[i(N − at)],	a₂ > 0, W₄ ≤ 0 ≤ 5 W_s < W₂ < W₁	15
where $\begin{array}{l} W = \frac{W_{3} (W_{2} - W_{4}) - W_{4} (W_{2} - W_{3}) s n^{2} (θ, k)}{W_{2} - W_{4} - (W_{2} - W_{3}) s n^{2} (θ, k)}, \\ θ = {[a_{2} (W_{1} - W_{3}) (W_{2} - W_{4}) / 3]}^{1 / 2} p x, k^{2} = \frac{(W_{2} - W_{3}) (W_{1} - W_{4})}{(W_{1} - W_{3}) (W_{2} - W_{4})}, \\ N = \frac{S {(3 / a_{2})}^{1 / 2}}{p^{2} {[(W_{1} - W_{3}) (W_{2} - W_{4})]}^{1 / 2}} \int d θ / W, \\ W_{4} = - W_{1} - W_{2} - W_{3} - 3 a_{1} / 2 a_{2} p, S^{2} = a_{2} p^{4} W_{1} W_{2} W_{3} W_{4} / 3, \\ a = - a_{2} p^{2} [W_{1} W_{2} + (W_{1} + W_{2}) (W_{3} + W_{4}) + W_{3} W_{4}] / 3; \end{array}$	W₃ ≠ W₄, p > 0
	P, pW₃ ≤ \|u\| ≤ pW₂
(pW)^1/2exp[i(N − at)],	a₂ < 0, 0 ≤ W₄ < W₃ < W₂ < W₁	16
where $\begin{array}{l} W = \frac{W_{2} (W_{1} - W_{3}) - W_{3} (W_{1} - W_{2}) s n^{2} (θ, k)}{W_{1} - W_{3} - (W_{1} - W_{2}) s n^{2} (θ, k)}, \\ k^{2} = \frac{(W_{1} - W_{2}) (W_{3} - W_{4})}{(W_{1} - W_{3}) (W_{2} - W_{4})}, W_{4}, S, and a as in solution (15), \end{array}$	P > 0
	P, pW₂ ≤ \|u\|² ≤ pW₁
	A₂< 0, W₄ < W₃ < 0 < W₂ < W₁	17
	W₃ ≠ W₂, p > 0
	P, pW₂ ≤ \|u\|² ≤ pW₁
N and θ as in solution (15) with the change a₂ → −a₂;
(pW)^1/2 exp[i(N − at)],	A₂ < 0, 0 < W₄ < W₃ < W₂ < W₁	18
Where	p > 0
Where	P, pW₁ ≤ \|u\| ≤ pW₃
$W = \frac{W_{4} (W_{3} - W_{1}) - W_{1} (W_{3} - W_{4}) s n^{2} (θ, k)}{W_{3} - W_{1} - (W_{3} - W_{4}) s n^{2} (θ, k)},$	a₂ < 0, W₄ < W₃ < 0 < W₂ < W₁	19
	W₃ ≠ W₂, p < 0
	P, pW₃ ≤ \|u\|² ≤ pW₄
N, θ, k², S, and W₄ as in solution (16) or (17);
(pW)^1/2 exp[i(N − at)],	a₂ < 0, 0 ≤ W₂ < W₁, q > 0	20
where $\begin{array}{l} W = \frac{W_{2} A + W_{1} B - (W_{1} - W_{2} A) c n (θ, k)}{A + B + (A - B) c n (θ, k)}, \\ θ = 2 {(A B a_{2} / 3)}^{1 / 2} p x, A^{2} = {(W_{1} - r)}^{2} + q^{2}, B^{2} = {(W_{2} - r)}^{2} + q^{2}, \\ k^{2} = \frac{{(W_{1} - W_{2})}^{2} - {(A - B)}^{2}}{4 A B}, N = {[\frac{W_{1} W_{2} (r^{2} + q^{2})}{4 A B}]}^{1 / 2} \int d θ / W, \\ - 2 r = W_{1} + W_{2} + 3 a_{1} / 2 a_{2} p, a = \frac{- a_{2} p^{2}}{3} [W_{1} W_{2} + (W_{1} + W_{2}) 2 r + r^{2} + q^{2}] . \end{array}$	p > 0
	P, pW₂ ≤ \|u\|² ≤ pW₁

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Journal of the Optical Society of America A