Lax pair, rogue-wave and soliton solutions for a variable-coefficient generalized nonlinear Schrödinger equation in an optical fiber, fluid or plasma

Abstract

In this paper, a variable-coefficient generalized nonlinear Schrödinger equation, which can be used to describe the nonlinear phenomena in the optical fiber, fluid or plasma, is investigated. Lax pair, higher-order rogue-wave and multi-soliton solutions, Darboux transformation and generalized Darboux transformation are obtained. Wave propagation and interaction are analyzed: (1) The Hirota and Lakshmanan–Porsezian–Daniel coefficients affect the propagation velocity and path of each one soliton; three types of soliton interaction have been attained: the bound state, one bell-shape soliton’s catching up with the other and two bell-shape soliton head-on interaction. Multi-soliton interaction is elastic. (2) The Hirota and Lakshmanan–Porsezian–Daniel coefficients affect the propagation direction of the first-step rogue waves and interaction range of the higher-order rogue waves.

Appendix

The second-order rogue-wave solutions of Eq. (1):

$$\begin{aligned} \psi ^{[2]}= & {} \big (e^{\frac{7 i x}{\gamma }} \big (384 x^5 (6 t \alpha -13 i \gamma ) \gamma \big (36 \alpha ^2+169 \gamma \big )^2+64 x^6 \big (36 \alpha ^2+169 \gamma \big )^3+24 x \gamma ^5 \\ &\,\big (96 t^5 \alpha -208 i t^4 \gamma -208 t^3 \alpha \gamma +696 i t^2 \gamma ^2-234 t \alpha \gamma ^2+291 i \gamma ^3\big )+\gamma ^6 \\ &\,\big (64 t^6-144 t^4 \gamma -180 t^2 \gamma ^2+45 \gamma ^3\big )-12 x^2 \gamma ^4 \big (9984 i t^3 \alpha \gamma -6720 i t \alpha \gamma ^2 \\ &-\,16 t^4 \big (180 \alpha ^2+169 \gamma \big )+15 \gamma ^2 \big (100 \alpha ^2+401 \gamma \big )+24 t^2 \gamma \big (204 \alpha ^2+1261 \gamma \big )\big ) \\ &+\,192 x^3 \gamma ^3 \big ({-}52 i t^2 \gamma \big (108 \alpha ^2+169 \gamma \big )-6 t \alpha \gamma \big (252 \alpha ^2+3107 \gamma \big )-i \gamma ^2 \\ &\, \big (612 \alpha ^2+10309 \gamma \big )+24 t^3 \big (60 \alpha ^3+169 \alpha \gamma \big )\big )-48 x^4 \gamma ^2 \big (2496 i t \alpha \gamma \big (36 \alpha ^2+169 \gamma \big ) \\ &-\,4 t^2 \big (6480 \alpha ^4+36504 \alpha ^2 \gamma +28561 \gamma ^2\big )+\gamma \big (10800 \alpha ^4+175032 \alpha ^2 \gamma \\ &+\,173563 \gamma ^2\big )\big )\big )\big )/\big (\sqrt{\gamma } \big (2304 t x^5 \alpha \gamma \big (36 \alpha ^2+169 \gamma \big )^2+64 x^6 \big (36 \alpha ^2+169 \gamma \big )^3 \\ &+\,48 t x \alpha \gamma ^5 \big (48 t^4-8 t^2 \gamma +51 \gamma ^2\big )+\gamma ^6 \big (64 t^6+48 t^4 \gamma +108 t^2 \gamma ^2+9 \gamma ^3\big ) \\ &+\,1152 t x^3 \alpha \gamma ^3 \big (4 t^2 \big (60 \alpha ^2+169 \gamma \big )-\gamma \big (108 \alpha ^2+1079 \gamma \big )\big )\\&+12 x^2 \gamma ^4 \big ({-}360 t^2 \gamma \big (4 \alpha ^2+39 \gamma \big )+16 t^4 \big (180 \alpha ^2+169 \gamma \big )+3 \gamma ^2 \big (556 \alpha ^2+5379 \gamma \big )\big ) \\ &+\,48 x^4 \gamma ^2 \big (4 t^2 \big (6480 \alpha ^4+36504 \alpha ^2 \gamma +28561 \gamma ^2\big )+13 \gamma \big ({-}432 \alpha ^4-2232 \alpha ^2 \gamma \\ &+\,30589 \gamma ^2\big )\big )\big )\big ), \end{aligned}$$

The third-order rogue-wave solutions of Eq. (1):

$$\begin{aligned} \psi ^{[3]}= & {} -\big (e^{\frac{7 i x}{\gamma }} \big (49152 x^{11} (6 t \alpha -13 i \gamma ) \gamma \big (36 \alpha ^2+169 \gamma \big )^5+4096 x^{12} \big (36 \alpha ^2+169 \gamma \big )^6 \\ &+\,\gamma ^{12} \big (4096 t^{12}-18432 t^{10} \gamma -57600 t^8 \gamma ^2-172800 t^6 \gamma ^3+226800 t^4 \gamma ^4 \\ &+\,113400 t^2 \gamma ^5-14175 \gamma ^6\big )+48 x \gamma ^{11} \big (6144 t^{11} \alpha -13312 i t^{10} \gamma -33280 t^9 \alpha \gamma \\ &+\,111360 i t^8 \gamma ^2-57600 t^7 \alpha \gamma ^2+1920 i t^6 \gamma ^3-187200 t^5 \alpha \gamma ^3+165600 i t^4 \gamma ^4 \\ &+\,286200 t^3 \alpha \gamma ^4-936900 i t^2 \gamma ^5+141750 t \alpha \gamma ^5-137025 i \gamma ^6\big )+24 x^2 \gamma ^{10} \\ &\, \big ({-}1597440 i t^9 \alpha \gamma +10690560 i t^7 \alpha \gamma ^2-5760 t^6 \big (356 \alpha ^2-4545 \gamma \big ) \gamma ^2 \\ &+\,10828800 i t^5 \alpha \gamma ^3-7200 t^4 \big (1596 \alpha ^2-1993 \gamma \big ) \gamma ^3+34675200 i t^3 \alpha \gamma ^4 \\ &-\,14536800 i t \alpha \gamma ^5+675 \big (1116 \alpha ^2-8837 \gamma \big ) \gamma ^5+1024 t^{10} \big (396 \alpha ^2+169 \gamma \big ) \\ &-\,3840 t^8 \gamma \big (612 \alpha ^2+1261 \gamma \big )+2700 t^2 \gamma ^4 \big (3700 \alpha ^2+33247 \gamma \big )\big )+960 x^3 \\ &\,\gamma ^9 \big ({-}576 t^5 \alpha \big (684 \alpha ^2-30817 \gamma \big ) \gamma ^2+720 i t^2 \big (13620 \alpha ^2-6893 \gamma \big ) \gamma ^4 \\ &-\,3328 i t^8 \gamma \big (324 \alpha ^2+169 \gamma \big )-4608 t^7 \alpha \gamma \big (252 \alpha ^2+1261 \gamma \big )+15930 t \alpha \gamma ^4 \\ &\,\big (156 \alpha ^2+2231 \gamma \big )-1440 t^3 \alpha \gamma ^3 \big (2732 \alpha ^2+9741 \gamma \big )+1440 i t^4 \gamma ^3 \\ &\,\big (3108 \alpha ^2+10933 \gamma \big )+256 i t^6 \gamma ^2 \big (21924 \alpha ^2+31603 \gamma \big )+45 i \gamma ^5 \\ &\,\big (11676 \alpha ^2+748199 \gamma \big )+1536 t^9 \big (132 \alpha ^3+169 \alpha \gamma \big )\big )-6144 x^{10} \gamma ^2 \\ &\,\big (36 \alpha ^2+169 \gamma \big )^3 \big (6240 i t \alpha \gamma \big (36 \alpha ^2+169 \gamma \big )-4 t^2 \big (14256 \alpha ^4+73008 \alpha ^2 \gamma \\ &+\,28561 \gamma ^2\big )+\gamma \big (21168 \alpha ^4+382824 \alpha ^2 \gamma +305383 \gamma ^2\big )\big )+240 x^4 \gamma ^8 \\ &\,\big (15360 i t^3 \alpha \big (2700 \alpha ^2-1651 \gamma \big ) \gamma ^3-638976 i t^7 \alpha \gamma \big (108 \alpha ^2+169 \gamma \big ) \\ &+\,34560 i t \alpha \gamma ^4 \big (6940 \alpha ^2+36839 \gamma \big )+12288 i t^5 \alpha \gamma ^2 \big (21924 \alpha ^2+86021 \gamma \big ) \\ &+\,256 t^8 \big (42768 \alpha ^4+109512 \alpha ^2 \gamma +28561 \gamma ^2\big )-1792 t^6 \gamma \big (32400 \alpha ^4 \\ &+\,272376 \alpha ^2 \gamma +134017 \gamma ^2\big )+480 t^4 \gamma ^2 \big ({-}3888 \alpha ^4+1687176 \alpha ^2 \gamma \\ &+\,164437 \gamma ^2\big )-720 t^2 \gamma ^3 \big (141552 \alpha ^4+1348968 \alpha ^2 \gamma +1587599 \gamma ^2\big ) \\ &+\,45 \gamma ^4 \big (970320 \alpha ^4+19226664 \alpha ^2 \gamma +42713269 \gamma ^2\big )\big )-61440 x^9 \gamma ^3 \\ &\,\big (36 \alpha ^2+169 \gamma \big )^2 \big (52 i t^2 \gamma \big (11664 \alpha ^4+60840 \alpha ^2 \gamma +28561 \gamma ^2\big )+6 t \alpha \gamma \\ &\,\big (19440 \alpha ^4+362232 \alpha ^2 \gamma +656903 \gamma ^2\big )+i \gamma ^2 \big ({-}112752 \alpha ^4-961272 \alpha ^2 \gamma \\ &+\,5969249 \gamma ^2\big )-24 t^3 \big (4752 \alpha ^5+28392 \alpha ^3 \gamma +28561 \alpha \gamma ^2\big )\big )+1536 x^5 \gamma ^7 \\ &\,\big ({-}4160 i t^6 \gamma \big (27216 \alpha ^4+85176 \alpha ^2 \gamma +28561 \gamma ^2\big )+600 t^3 \alpha \gamma ^2 \big (14256 \alpha ^4 \\ &+\,1350216 \alpha ^2 \gamma +1161199 \gamma ^2\big )+720 i t^4 \gamma ^2 \big (438480 \alpha ^4+3089320 \alpha ^2 \gamma \\ &+\,1627977 \gamma ^2\big )-450 t \alpha \gamma ^3 \big (106704 \alpha ^4+1771944 \alpha ^2 \gamma +4105049 \gamma ^2\big ) \\ &-\,96 t^5 \alpha \gamma \big (789264 \alpha ^4+9533160 \alpha ^2 \gamma +13500565 \gamma ^2\big )-180 i t^2 \gamma ^3 \big (684720 \alpha ^4 \\ &+\,10797800 \alpha ^2 \gamma +42206567 \gamma ^2\big )+45 i \gamma ^4 \big (2545200 \alpha ^4+41227640 \alpha ^2 \gamma \\ &+\,154878191 \gamma ^2\big )+384 t^7 \big (42768 \alpha ^5+182520 \alpha ^3 \gamma +142805 \alpha \gamma ^2\big )\big )-256 x^6 \gamma ^6 \\ &\,\big (179712 i t^5 \alpha \gamma \big (27216 \alpha ^4+141960 \alpha ^2 \gamma +142805 \gamma ^2\big )-11520 i t^3 \alpha \gamma ^2 \\ &\,\big (789264 \alpha ^4+8213400 \alpha ^2 \gamma +11681449 \gamma ^2\big )+12960 i t \alpha \gamma ^3 \big (396144 \alpha ^4 \\ &+\,6663800 \alpha ^2 \gamma +29182751 \gamma ^2\big )-64 t^6 \big (10777536 \alpha ^6+68992560 \alpha ^4 \gamma \\ &+\,107960580 \alpha ^2 \gamma ^2+24134045 \gamma ^3\big )+720 t^4 \gamma \big (3592512 \alpha ^6+57198960 \alpha ^4 \gamma \\ & +\,155152140 \alpha ^2 \gamma ^2+55322657 \gamma ^3\big )-180 t^2 \gamma ^2 \big (2099520 \alpha ^6+108714960 \alpha ^4 \gamma \\ & +\,301817100 \alpha ^2 \gamma ^2+848461627 \gamma ^3\big )+135 \gamma ^3 \big (2376000 \alpha ^6+66895920 \alpha ^4 \gamma \\ & +\,480320620 \alpha ^2 \gamma ^2+2572161917 \gamma ^3\big )\big )+6144 x^7 \gamma ^5 \big ({-}15 i \gamma ^3 \big (26827200 \alpha ^6 \\ & +\,500020560 \alpha ^4 \gamma +2244182772 \alpha ^2 \gamma ^2-191958481 \gamma ^3\big )-1040 i t^4 \gamma \big (979776 \alpha ^6 \\ & +\,7665840 \alpha ^4 \gamma +15422940 \alpha ^2 \gamma ^2+4826809 \gamma ^3\big )+120 i t^2 \gamma ^2 \big (9471168 \alpha ^6 \\ & +\,128859120 \alpha ^4 \gamma +313599780 \alpha ^2 \gamma ^2+62748517 \gamma ^3\big )+270 t \alpha \gamma ^2 \big (171072 \alpha ^6 \\ & +\,7562160 \alpha ^4 \gamma +40799980 \alpha ^2 \gamma ^2+135293457 \gamma ^3\big )-80 t^3 \alpha \gamma \big (5178816 \alpha ^6 \\ & +\,102958128 \alpha ^4 \gamma +444892500 \alpha ^2 \gamma ^2+459289441 \gamma ^3\big )+96 t^5 \big (1539648 \alpha ^7 \\ & +\,13798512 \alpha ^5 \gamma +35986860 \alpha ^3 \gamma ^2+24134045 \alpha \gamma ^3\big )\big )+3840 x^8 \gamma ^4 \\ &\, \big ({-}39936 i t^3 \alpha \gamma \big (36 \alpha ^2+169 \gamma \big )^2 \big (108 \alpha ^2+169 \gamma \big )+16 t^4 \big (36 \alpha ^2+169 \gamma \big )^2 \big (42768 \alpha ^4 \\ & +\,109512 \alpha ^2 \gamma +28561 \gamma ^2\big )+768 i t \alpha \gamma ^2 \big (4059072 \alpha ^6+65926224 \alpha ^4 \gamma \\ & +\,206667396 \alpha ^2 \gamma ^2-62748517 \gamma ^3\big )+3 \gamma ^2 \big (27433728 \alpha ^8+1302448896 \alpha ^6 \gamma \\ &+\,9723935520 \alpha ^4 \gamma ^2+3364714288 \alpha ^2 \gamma ^3-245264645717 \gamma ^4\big )-72 t^2 \gamma \\ &\, \big (22954752 \alpha ^8+549110016 \alpha ^6 \gamma +3394628640 \alpha ^4 \gamma ^2+6730799504 \alpha ^2 \gamma ^3 \\ & +\,2447192163 \gamma ^4\big )\big )\big )\big )/\big (\sqrt{\gamma } \big (294912 t x^{11} \alpha \gamma \big (36 \alpha ^2+169 \gamma \big )^5+4096 x^{12} \\ &\, \big (36 \alpha ^2+169 \gamma \big )^6+96 t x \alpha \gamma ^{11} \big (3072 t^{10}-1280 t^8 \gamma +17280 t^6 \gamma ^2+73440 t^4 \gamma ^3 \\ & -\,15300 t^2 \gamma ^4+19575 \gamma ^5\big )+\gamma ^{12} \big (4096 t^{12}+6144 t^{10} \gamma +34560 t^8 \gamma ^2+149760 t^6 \gamma ^3 \\ & +\,54000 t^4 \gamma ^4+48600 t^2 \gamma ^5+2025 \gamma ^6\big )+24 x^2 \gamma ^{10} \big (21600 t^4 \big (316 \alpha ^2-561 \gamma \big ) \gamma ^3 \\ & -\,172800 t^8 \gamma \big (4 \alpha ^2+13 \gamma \big )+1024 t^{10} \big (396 \alpha ^2+169 \gamma \big )+1920 t^6 \gamma ^2 \big (948 \alpha ^2+5627 \gamma \big ) \\ & -\,2700 t^2 \gamma ^4 \big (972 \alpha ^2+10993 \gamma \big )+675 \gamma ^5 \big (1996 \alpha ^2+30607 \gamma \big )\big )+5760 t x^3 \alpha \gamma ^9 \\ &\, \big ({-}6912 t^6 \gamma \big (12 \alpha ^2+65 \gamma \big )+256 t^8 \big (132 \alpha ^2+169 \gamma \big )+240 t^2 \gamma ^3 \big (1732 \alpha ^2+3359 \gamma \big ) \\ & +\,288 t^4 \gamma ^2 \big (444 \alpha ^2+4067 \gamma \big )-45 \gamma ^4 \big (3148 \alpha ^2+59907 \gamma \big )\big )+368640 t x^9 \alpha \gamma ^3 \\ &\, \big (36 \alpha ^2+169 \gamma \big )^2 \big (4 t^2 \big (4752 \alpha ^4+28392 \alpha ^2 \gamma +28561 \gamma ^2\big )+\gamma \big ({-}14256 \alpha ^4 \\ & -\,118872 \alpha ^2 \gamma +371293 \gamma ^2\big )\big )+6144 x^{10} \gamma ^2 \big (36 \alpha ^2+169 \gamma \big )^3 \big (4 t^2 \big (14256 \alpha ^4 \\ & +\,73008 \alpha ^2 \gamma +28561 \gamma ^2\big )+\gamma \big ({-}15984 \alpha ^4-90792 \alpha ^2 \gamma +951301 \gamma ^2\big )\big ) \\ &+\,9216 t x^5 \alpha \gamma ^7 \big (20 t^2 \gamma ^2 \big (289008 \alpha ^4+3943080 \alpha ^2 \gamma -5354765 \gamma ^2\big )+64 t^6 \\ &\, \big (42768 \alpha ^4+182520 \alpha ^2 \gamma +142805 \gamma ^2\big )-16 t^4 \gamma \big (462672 \alpha ^4+4422600 \alpha ^2 \gamma \\ & +\,4932265 \gamma ^2\big )+45 \gamma ^3 \big (204048 \alpha ^4+3111560 \alpha ^2 \gamma +19527885 \gamma ^2\big )\big )+240 x^4 \gamma ^8 \\ &\, \big (1440 t^4 \gamma ^2 \big (22896 \alpha ^4+245400 \alpha ^2 \gamma -396305 \gamma ^2\big )-3328 t^6 \gamma \big (9072 \alpha ^4+68040 \alpha ^2 \gamma \\ & +\,28223 \gamma ^2\big )+256 t^8 \big (42768 \alpha ^4+109512 \alpha ^2 \gamma +28561 \gamma ^2\big )+720 t^2 \gamma ^3 \big (115920 \alpha ^4 \\ & +\,1071864 \alpha ^2 \gamma +5490797 \gamma ^2\big )+45 \gamma ^4 \big ({-}82608 \alpha ^4-316248 \alpha ^2 \gamma +35623589 \gamma ^2\big )\big ) \\ & +\,256 x^6 \gamma ^6 \big (45 \gamma ^3 \big (12094272 \alpha ^6+213073200 \alpha ^4 \gamma +1200264780 \alpha ^2 \gamma ^2-143365235 \gamma ^3\big ) \\ &+\,64 t^6 \big (10777536 \alpha ^6+68992560 \alpha ^4 \gamma +107960580 \alpha ^2 \gamma ^2+24134045 \gamma ^3\big )-240 t^4 \gamma \\ &\, \big (6858432 \alpha ^6+79606800 \alpha ^4 \gamma +156997620 \alpha ^2 \gamma ^2+30817319 \gamma ^3\big )+540 t^2 \gamma ^2 \\ &\,\big (1570752 \alpha ^6+31071600 \alpha ^4 \gamma +76350820 \alpha ^2 \gamma ^2+372920977 \gamma ^3\big )\big )+12288 t x^7 \alpha \gamma ^5 \\ &\,\big ({-}200 t^2 \gamma \big (699840 \alpha ^6+9552816 \alpha ^4 \gamma +27286740 \alpha ^2 \gamma ^2+10767497 \gamma ^3\big )+48 t^4 \\ &\,\big (1539648 \alpha ^6+13798512 \alpha ^4 \gamma +35986860 \alpha ^2 \gamma ^2+24134045 \gamma ^3\big )+15 \gamma ^2 \big (2659392 \alpha ^6 \\ &+\,79785648 \alpha ^4 \gamma +623457900 \alpha ^2 \gamma ^2+3109750241 \gamma ^3\big )\big )+3840 x^8 \gamma ^4 \\ &\,\big (16 t^4 \big (36 \alpha ^2+169 \gamma \big )^2 \big (42768 \alpha ^4+109512 \alpha ^2 \gamma +28561 \gamma ^2\big )-72 t^2 \gamma \big (16236288 \alpha ^8 \\ &+\,254741760 \alpha ^6 \gamma +926958240 \alpha ^4 \gamma ^2+243568208 \alpha ^2 \gamma ^3-815730721 \gamma ^4\big )+3 \gamma ^2 \\ &\,\big (40870656 \alpha ^8+1826489088 \alpha ^6 \gamma +23770674720 \alpha ^4 \gamma ^2+146083802800 \alpha ^2 \gamma ^3 \\ &+\,162851708851 \gamma ^4\big )\big )\big )\big ). \end{aligned}$$