Nth-order smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger equation

Abstract

In this paper, we investigate smooth positon and breather-positon solutions of a generalized nonlinear Schrödinger (GNLS) equation which contains higher-order nonlinear effects. With the help of generalized Darboux transformation (GDT) method, we construct Nth-order smooth positon solutions of GNLS equation. We study the effect of higher-order nonlinear terms on these solutions. Our investigations show that the positon solutions are highly compressed by higher-order nonlinear effects. The direction of positons also get changed. We also derive Nth-order breather-positon (B-P) solution with the help of GDT. We show that these B-Ps are well compressed by the effect of higher-order nonlinear terms, but the period of B-P solution is not affected as in the breather solution case.

Appendices

Appendix A: Third-order smooth positon solution of GNLS equation

Third-order smooth positon solution of GNLS equation is given by

$$\begin{aligned} q[3]= & {} \frac{A_2}{B_2}, \end{aligned}$$

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$$\begin{aligned} A_2= & {} A_{21}e^{i(x(\lambda _1-3\lambda _1^*)+2t(\lambda _1^2-3\lambda _1^{*2}-4\lambda _1^4\nu +12\lambda _1^{*4}\nu ))}+A_{22}e^{i(x(-3\lambda _1+\lambda _1^*)+2t(\lambda _1^{*2}-3\lambda _1^{2}-4\lambda _1^{*4}\nu +12\lambda _1^{4}\nu ))}\nonumber \\&+\,A_{23}e^{-i(x(\lambda _1+\lambda _1^*)+2t(\lambda _1^2+\lambda _1^{*2}-4\lambda _1^4\nu -4\lambda _1^{*4}\nu ))},\nonumber \\ A_{21}= & {} -4(\lambda _1-\lambda _1^*)\big (-3+2x^2(\lambda _1-\lambda _1^*)^2+32t^2\lambda _1^2(\lambda _1-\lambda _1^*)^2(1-8\lambda _1^2\nu )^2-4it(\lambda _1-\lambda _1^*)(-7\lambda _1+\lambda _1^*+72\lambda _1^3\nu -24\lambda _1^2\lambda _1^*\nu )\nonumber \\&-\,2x(\lambda _1-\lambda _1^*)(-3i+8t\lambda _1(\lambda _1-\lambda _1^*)(-1+8\lambda _1^2\nu ))\big )\nonumber \\ A_{22}= & {} -4(\lambda _1-\lambda _1^*)\big (-3+2x^2(\lambda _1-\lambda _1^*)^2+32t^2\lambda _1^{*2}(\lambda _1-\lambda _1^*)^2(1-8\lambda _1^{*2}\nu )^2-4it(\lambda _1-\lambda _1^*)(7\lambda _1^*-\lambda _1-72\lambda _1^{*3}\nu +24\lambda _1^{*2}\lambda _1\nu )\nonumber \\&-\,2x(\lambda _1-\lambda _1^*)(3i+8t\lambda _1^*(\lambda _1-\lambda _1^*)(-1+8\lambda _1^{*2}\nu ))\big )\nonumber \\ A_{23}= & {} -8(\lambda _1-\lambda _1^*)\bigg (-3-4x^2(\lambda _1-\lambda _1^*)^2+2x^4(\lambda _1-\lambda _1^*)^4+512t^4\lambda _1^2(\lambda _1-\lambda _1^*)^4\lambda _1^{*2}(1-8\lambda _1^2\nu )^2(1-8\lambda _1^{*2}\nu )^2-16t(\lambda _1-\lambda _1^*)^2\nonumber \\&(x^3(\lambda _1-\lambda _1^*)^2(\lambda _1+\lambda _1^*)(-1+8\lambda _1^2\nu -8\lambda _1\lambda _1^*\nu +8\lambda _1^{*2}\nu )+i(-1+9\lambda _1^2\nu +6\lambda _1\lambda _1^*\nu +9\lambda _1^{*2}\nu )-ix^2(\lambda _1-\lambda _1^*)^2(-1\nonumber \\&+\,10\lambda _1^2\nu +4\lambda _1\lambda _1^*\nu +10\lambda _1^{*2}\nu )-x(\lambda _1+\lambda _1^*)(-1+14\lambda _1^2\nu -20\lambda _1\lambda _1^*\nu +14\lambda _1^{*2}\nu ))+8t^2(\lambda _1-\lambda _1^*)^2(256x^2\lambda _1^8\nu ^2\nonumber \\&-\,256x\lambda _1^7(i+2x\lambda _1^*)\nu ^2+32x\lambda _1^5\nu (3i-24i\lambda _1^{*2}\nu +32x\lambda _1^{*3}\nu )+64\lambda _1^6\nu (-2\nu +4ix\lambda _1^*\nu +x^2(-1+4\lambda _1^{*2}\nu ))+\lambda _1^2(-1\nonumber \\&-\,48\lambda _1^{*2}\nu -192\lambda _1^{*4}\nu ^2+8x^2\lambda _1^{*2}(-3+24\lambda _1^{*2}\nu +32\lambda _1^{*4}\nu ^2)-8ix\lambda _1^*(-1+8\lambda _1^{*2}\nu +96\lambda _1^{*4}\nu ^2))-4\lambda _1^4(2\nu (-5+24\lambda _1^{*2}\nu )\nonumber \\&-\,8ix\lambda _1^*\nu (-1+24\lambda _1^{*2}\nu )+x^2(-1-48\lambda _1^{*2}\nu +512\lambda _1^{*4}\nu ^2))+\lambda _1^{*2}(-1+40\lambda _1^{*2}\nu -128\lambda _1^{*4}\nu ^2+4x^2\lambda _1^{*2}(1-8\lambda _1^{*2}\nu )^2\nonumber \\&-\,8ix(\lambda _1^*-12\lambda _1^{*3}\nu +32\lambda _1^{*5}\nu ^2))+8\lambda _1^3(2\lambda _1^*\nu (3+8\lambda _1^{*2}\nu )+ix(-1-8\lambda _1^{*2}\nu +96\lambda _1^{*4}\nu ^2)+x^2(\lambda _1^*-32\lambda _1^{*3}\nu +128\lambda _1^{*5}\nu ^2))\nonumber \\&+\,2\lambda _1\lambda _1^*(-3+24\lambda _1^{*2}\nu +4ix(\lambda _1^*-4\lambda _1^{*3}\nu +32\lambda _1^{*5}\nu ^2)+x^2(4\lambda _1^{*2}-256\lambda _1^{*6}\nu ^2)))-64t^3(\lambda _1-\lambda _1^*)^4\big (i\lambda _1^{*2}(1-8\lambda _1^{*2}\nu )^2\nonumber \\&-\,2\lambda _1\lambda _1^*(-i+2x\lambda _1^*)(1-8\lambda _1^{*2}\nu )^2-128i\lambda _1^5\lambda _1^*\nu ^2(-1+8\lambda _1^{*2}\nu )+32\lambda _1^3\lambda _1^*\nu (-1+8\lambda _1^{*2}\nu )(i-4i\lambda _1^{*2}\nu +x\lambda _1^*(-1+8\lambda _1^{*2}\nu ))\nonumber \\&+\,64\lambda _1^6\nu ^2(i-24i\lambda _1^{*2}\nu +4x\lambda _1^*(-1+8\lambda _1^{*2}\nu ))+16\lambda _1^4\nu (x(4\lambda _1^*-32\lambda _1^{*3}\nu )-i(1-32\lambda _1^{*2}\nu +64\lambda _1^{*4}\nu ^2))+\lambda _1^2(4x\lambda _1^*(-1\nonumber \\&+\,8\lambda _1^{*2}\nu )-i(-1+64\lambda _1^{*2}\nu -512\lambda _1^{*4}\nu ^2+1536\lambda _1^{*6}\nu ^3))\big )\bigg ),\nonumber \\ B_2= & {} 4e^{3i(x(-\lambda _1+\lambda _1^*)+2t(-\lambda _1^2+\lambda _1^{*2}+4\lambda _1^4\nu -4\lambda _1^{*4}\nu ))}+4e^{3i(x(\lambda _1-\lambda _1^*)+2t(\lambda _1^2-\lambda _1^{*2}-4\lambda _1^4\nu +4\lambda _1^{*4}\nu ))}\nonumber \\&+\,4B_{21}e^{-i(x(-\lambda _1+\lambda _1^*)+2t(-\lambda _1^2+\lambda _1^{*2}+4\lambda _1^4\nu -4\lambda _1^{*4}\nu ))}+4B_{22}e^{-i(x(\lambda _1-\lambda _1^*)+2t(\lambda _1^2-\lambda _1^{*2}-4\lambda _1^4\nu +4\lambda _1^{*4}\nu ))},\nonumber \\ \end{aligned}$$

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$$\begin{aligned} B_{21}= & {} (3+4x^4(\lambda _1-\lambda _1^*)^4+1024t^4\lambda _1^2(\lambda _1-\lambda _1^*)^4\lambda _1^{*2}(1-8\lambda _1^2\nu )^2(1-8\lambda _1^{*2}\nu )^2-8x^3(\lambda _1-\lambda _1^*)^3(-i+4t(\lambda _1^2-\lambda _1^{*2})(-1\nonumber \\&+\,8\lambda _1^2\nu -8\lambda _1\lambda _1^*\nu +8\lambda _1^{*2}\nu ))+16t^2(\lambda _1-\lambda _1^*)^2(3\lambda _1^{*2}-40\lambda _1^{*4}\nu +16\lambda _1^3\lambda _1^*\nu (13-168\lambda _1^{*2}\nu )+40\lambda _1^4\nu (-1+24\lambda _1^{*2}\nu )\nonumber \\&+\,2\lambda _1\lambda _1^*(-9+104\lambda _1{^*2}\nu )+3\lambda _1^2(1-48\lambda _1^{*2}\nu +320\lambda _1^{*4}\nu ^2))+128it^3(\lambda _1-\lambda _1^*)^3(\lambda _1+\lambda _1^*)(4\lambda _1\lambda _1^*(1-8\lambda _1^{*2}\nu )^2\nonumber \\&-\,\lambda _1^{*2}(1-8\lambda _1^{*2}\nu )^2-256\lambda _1^5\lambda _1^*\nu ^2(-1+16\lambda _1^{*2}\nu )+64\lambda _1^6\nu ^2(-1+24\lambda _1^{*2}\nu )-64\lambda _1^3\lambda _1^*\nu (1-20\lambda _1^{*2}\nu +64\lambda _1^{*4}\nu ^2)\nonumber \\&+\,16\lambda _1^4\nu (1-40\lambda _1^{*2}\nu +256\lambda _1^{*4}\nu ^2)+\lambda _1^2(-1+48\lambda _1^{*2}\nu -640\lambda _1^{*4}\nu ^2+1536\lambda _1^{*6}\nu ^3))+4x^2(\lambda _1-\lambda _1^*)^2(-3\nonumber \\&-\,12it(\lambda _1^2-\lambda _1^{*2})(-1+12\lambda _1^2\nu -16\lambda _1\lambda _1^*\nu +12\lambda _1^{*2}\nu )+16t^2(\lambda _1-\lambda _1^*)^2(\lambda _1^2-16\lambda _1^4\nu +64\lambda _1^6\nu ^2+\lambda _1^{*2}(1-8\lambda _1^{*2}\nu )^2\nonumber \\&+\,32\lambda _1^3\lambda _1^*\nu (-1+8\lambda _1^{*2}\nu )+\lambda _1(4\lambda _1^*-32\lambda _1^{*3}\nu )))-16tx(\lambda _1-\lambda _1^*)^2(2048t^2\lambda _1^8\lambda _1^*\nu ^2(-1+8\lambda _1^{*2}\nu )+\lambda _1^*(3-36\lambda _1^{*2}\nu \nonumber \\&-\,32it\lambda _1^{*4}\nu +256it\lambda _1^{*6}\nu ^2)-256t\lambda _1^7\nu ^2(i+16t\lambda _1^{*2}(-1+8\lambda _1^{*2}\nu ))+\lambda _1(3+12\lambda _1^{*2}\nu -32t^2\lambda _1^{*4}(1-8\lambda _1^{*2}\nu )^2\nonumber \\&-\,8it\lambda _1^{*2}(-3+28\lambda _1^{*2}\nu +32\lambda _1^{*4}\nu ^2))+256t\lambda _1^6\lambda _1^*\nu (i\nu +t(2-24\lambda _1^{*2}\nu +64\lambda _1^{*4}\nu ^2))+32t\lambda _1^5\nu (i(1+24\lambda _1^{*2}\nu )\nonumber \\&+\,8t\lambda _1^{*2}(-3+16\lambda _1^{*2}\nu +64\lambda _1{^*4}\nu ^2))+4\lambda _1^2\lambda _1^*(3\nu -2it(3-56\lambda _1^{*2}\nu +96\lambda _1^{*4}\nu ^2)+8t^2(\lambda _1^{*2}-24\lambda _1^{*4}\nu +128\lambda _1^{*6}\nu ^2))\nonumber \\&-\,32t\lambda _1^4\lambda _1^*(i\nu (-7+104\lambda _1^{*2}\nu )+t(1-8\lambda _1^{*2}\nu -128\lambda _1^{*4}\nu ^2+1024\lambda _1^{*6}\nu ^3))+4\lambda _1^3(-9\nu +16it\lambda _1^{*2}\nu (-7+52\lambda _1^{*2}\nu )\nonumber \\&+\,8t^2(\lambda _1^{*2}+8\lambda _1^{*4}\nu -192\lambda _1^{*6}\nu ^2+512\lambda _1^{*8}\nu ^3)))),\nonumber \\ B_{22}= & {} (3+4x^4(\lambda _1-\lambda _1^*)^4+1024t^4\lambda _1^2(\lambda _1-\lambda _1^*)^4\lambda _1^{*2}(1-8\lambda _1^2\nu )^2(1-8\lambda _1^{*2}\nu )^2-8x^3(\lambda _1-\lambda _1^*)^3(i+4t(\lambda _1^2-\lambda _1^{*2})(-1\nonumber \\&+\,8\lambda _1^2\nu -8\lambda _1\lambda _1^*\nu +8\lambda _1^{*2}\nu ))+16t^2(\lambda _1-\lambda _1^*)^2(3\lambda _1^{*2}-40\lambda _1^{*4}\nu +16\lambda _1^3\lambda _1^*\nu (13-168\lambda _1^{*2}\nu )+40\lambda _1^4\nu (-1+24\lambda _1^{*2}\nu )\nonumber \\&+\,2\lambda _1\lambda _1^*(-9+104\lambda _1{^*2}\nu )+3\lambda _1^2(1-48\lambda _1^{*2}\nu +320\lambda _1^{*4}\nu ^2))-128it^3(\lambda _1-\lambda _1^*)^3(\lambda _1+\lambda _1^*)(4\lambda _1\lambda _1^*(1-8\lambda _1^{*2}\nu )^2\nonumber \\&-\,\lambda _1^{*2}(1-8\lambda _1^{*2}\nu )^2-256\lambda _1^5\lambda _1^*\nu ^2(-1+16\lambda _1^{*2}\nu )+64\lambda _1^6\nu ^2(-1+24\lambda _1^{*2}\nu )-64\lambda _1^3\lambda _1^*\nu (1-20\lambda _1^{*2}\nu +64\lambda _1^{*4}\nu ^2)\nonumber \\&+\,16\lambda _1^4\nu (1-40\lambda _1^{*2}\nu +256\lambda _1^{*4}\nu ^2)+\lambda _1^2(-1+48\lambda _1^{*2}\nu -640\lambda _1^{*4}\nu ^2+1536\lambda _1^{*6}\nu ^3))+4x^2(\lambda _1-\lambda _1^*)^2(-3\nonumber \\&12it(\lambda _1^2-\lambda _1^{*2})(-1+12\lambda _1^2\nu -16\lambda _1\lambda _1^*\nu +12\lambda _1^{*2}\nu )+16t^2(\lambda _1-\lambda _1^*)^2(\lambda _1^2-16\lambda _1^4\nu +64\lambda _1^6\nu ^2+\lambda _1^{*2}(1-8\lambda _1^{*2}\nu )^2\nonumber \\&+\,32\lambda _1^3\lambda _1^*\nu (-1+8\lambda _1^{*2}\nu )+\lambda _1(4\lambda _1^*-32\lambda _1^{*3}\nu )))-16tx(\lambda _1-\lambda _1^*)^2(2048t^2\lambda _1^8\lambda _1^*\nu ^2(-1+8\lambda _1^{*2}\nu )+\lambda _1^*(3-36\lambda _1^{*2}\nu \nonumber \\&32it\lambda _1^{*4}\nu -256it\lambda _1^{*6}\nu ^2)-256t\lambda _1^7\nu ^2(-i+16t\lambda _1^{*2}(-1+8\lambda _1^{*2}\nu ))+\lambda _1(3+12\lambda _1^{*2}\nu -32t^2\lambda _1^{*4}(1-8\lambda _1^{*2}\nu )^2\nonumber \\&8it\lambda _1^{*2}(-3+28\lambda _1^{*2}\nu +32\lambda _1^{*4}\nu ^2))+256t\lambda _1^6\lambda _1^*\nu (-i\nu +t(2-24\lambda _1^{*2}\nu +64\lambda _1^{*4}\nu ^2))+32t\lambda _1^5\nu (-i(1+24\lambda _1^{*2}\nu )\nonumber \\&+\,8t\lambda _1^{*2}(-3+16\lambda _1^{*2}\nu +64\lambda _1{^*4}\nu ^2))+4\lambda _1^2\lambda _1^*(3\nu +2it(3-56\lambda _1^{*2}\nu +96\lambda _1^{*4}\nu ^2)+8t^2(\lambda _1^{*2}-24\lambda _1^{*4}\nu +128\lambda _1^{*6}\nu ^2))\nonumber \\&-\,32t\lambda _1^4\lambda _1^*(-i\nu (-7+104\lambda _1^{*2}\nu )+t(1-8\lambda _1^{*2}\nu -128\lambda _1^{*4}\nu ^2+1024\lambda _1^{*6}\nu ^3))+4\lambda _1^3(-9\nu -16it\lambda _1^{*2}\nu (-7+52\lambda _1^{*2}\nu )\nonumber \\&+\,8t^2(\lambda _1^{*2}+8\lambda _1^{*4}\nu -192\lambda _1^{*6}\nu ^2+512\lambda _1^{*8}\nu ^3)))). \end{aligned}$$

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Appendix B: Second-order B-P solution of GNLS equation

The second-order B-P solution of (1) is given by

$$\begin{aligned} q[2]_{b-p}=ae^{ict}-2i\frac{A_{bp2}}{B_{bp2}}, \end{aligned}$$

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where

$$\begin{aligned} A_{bp2}= & {} \frac{8a}{\delta ^5}\eta (5a+4\eta )^2(a^2-2\eta ^2-2i\eta \delta )e^{2it(a^2+3a^4\nu )-8t\delta (\eta +2a^2\eta \nu +4\eta ^3\nu )}-\frac{8a}{\delta ^5}\eta (5a+4\eta )^2(a^2+2i\eta (i\eta +\delta ))\nonumber \\&e^{2t(3ia^4\nu +4\eta \delta (1+4\eta ^2\nu )+a^2(i+8\eta \delta \nu ))}+\frac{1}{\delta ^5}1280ia^8\eta ^2\nu te^{2ix\delta +2it(a^2+3a^4\nu )-4t\delta (\eta +2a^2\eta \nu +4\eta ^3\nu )}-\frac{1}{\delta ^2}1280ia^8\nonumber \\&\eta ^2\nu te^{2ix\delta +2it(a^2+3a^4\nu )+4t\delta (\eta +2a^2\eta \nu +4\eta ^3\nu )}-\frac{32a}{\delta ^4}\eta ^2(5a+4\eta )^2(i+8a^4\nu t+4a^2(t+8t\eta ^2\nu )-8t(\eta ^2+8\eta ^4\nu ))\nonumber \\&+\,\frac{8\eta }{\delta ^5}G_1e^{-2ix\delta +2it(a^2+3a^4\nu )-4t\delta (\eta +2a^2\eta \nu +4\eta ^3\nu )}-\frac{8\eta }{\delta ^5}G_2e^{6ia^4t\nu +2\delta (ix+2t\eta +8t\eta ^3\nu )+2a^2t(i+4\eta \delta \nu )}\nonumber \\&+\,\frac{8\eta }{\delta ^5}G_3e^{-2ix\delta +2it(a^2+3a^4\nu )+4t\delta (\eta +2a^2\eta \nu +4\eta ^3\nu )}+\frac{8\eta }{\delta ^5}G_4e^{6ia^4t\nu -2\delta (-ix+2t\eta +8t\eta ^3\nu )+a^2t(2i-8\eta \delta \nu )},\nonumber \\ G_1= & {} 160ia^8t\eta \nu -8a^7t\eta (-56i\eta +15\delta )\nu +16a^6t\eta (5i+46i\eta ^2\nu +4\eta \delta \nu )+4a^2\eta ^2(-56it\eta ^3+7i\delta +\eta ^2(-6x+16t\delta )\nonumber \\&+\,\eta (2+4ix\delta )-448it\eta ^5\nu +128t\eta ^4\delta \nu )+2a^3\eta (-336it\eta ^3+25i\delta +\eta ^2(-56x+172t\delta )+3\eta (4-5ix\delta )-2688it\eta ^5\nu \nonumber \\&+\,1376t\eta ^4\delta \nu )-32\eta ^4(-i\eta +\delta )(-i-2ix\eta +8t(\eta ^2+8\eta ^4))-16a\eta ^3(-i\eta +\delta )(-2i-7ix\eta +2t(\eta ^2+8\eta ^4\nu ))\nonumber \\&+\,a^5(-15+4it\eta (56\eta +15i\delta +336\eta ^3\nu +8i\eta ^2\delta \nu ))+4a^4(-10x\eta ^2+5i\delta +4t\eta ^2(-7i\eta +2\delta -72i\eta ^3\nu +32\eta ^2\delta \nu )),\nonumber \\ G_2= & {} 8a^7t\eta (56i\eta +15\delta )\nu +16ia^6t\eta (5+46\eta ^2\nu +4i\eta \delta \nu )-4a^2\eta ^2(56it\eta ^3+7i\delta +2\eta ^2(3x+8\delta t)+\eta (-2+4ix\delta )\nonumber \\&+\,448it\eta ^5\nu +128t\eta ^4\delta \nu )-2a^3\eta (336it\eta ^3+25i\delta +4\eta ^2(14x+43t\delta )+\eta (-12-15ix\delta )+2688it\eta ^5\nu +1376t\eta ^4\delta \nu )\nonumber \\&+\,32\eta ^4(i\eta +\delta )(-i-2ix\eta +8t(\eta ^2+8\eta ^4\nu ))+16a\eta ^3(i\eta +\delta )(-2i-7ix\eta +28t(\eta ^2+8\eta ^4\nu ))+a^5(-15\nonumber \\&+\,4t\eta (56i\eta +15\delta +336i\eta ^3\nu +8\eta ^2\delta \nu ))-4a^4(10x\eta ^2+5i\delta +4t\eta ^2(7i\eta +2\delta +72i\eta ^3\nu +32\eta ^2\delta \nu )),\nonumber \\ G_3= & {} -160ia^8t\eta \nu +8a^7t\eta (-26i\eta +15\delta )\nu +16a^6t\eta (-5i-34i\eta ^2\nu +16\eta \delta \nu )+2a^3\eta (156it\eta ^3+25i\delta -2\eta ^2(13x+4t\delta )\nonumber \\&+\,3\eta (1-5ix\delta )+1248it\eta ^5\nu -64t\eta ^4\delta \nu )-4a^2\eta ^2(16it\eta ^3-13i\delta +\eta ^2(-6x+56t\delta )+8\eta (1+2ix\delta )+128it\eta ^5\nu \nonumber \\&+\,448t\eta ^4\delta \nu )-8\eta ^4(i\eta +\delta )(-i+2ix\eta +8t(\eta ^2+8\eta ^4\nu ))-4a\eta ^3(i\eta +\delta )(-8i+13ix\eta +52t(\eta ^2+8\eta ^4\nu ))\nonumber \\&+\,4a^4(-10x\eta ^2+5i\delta +4t\eta ^2(13i\eta +8\delta +108i\eta ^3\nu +68\eta ^2\delta \nu ))+a^5(-15+4t\eta (-26i\eta +15\delta -156i\eta ^3\nu +172\eta ^2\delta \nu )),\nonumber \end{aligned}$$

$$\begin{aligned} G_4= & {} 8a^7t\eta (26i\eta +15\delta )\nu +16a^6t\eta (5i+34i\eta ^2\nu +16\eta \delta \nu )+4a^2\eta ^2(16it\eta ^3+13i\delta -2\eta ^2(3x+28t\delta )+8\eta (1-2ix\delta )\nonumber \\&+\,128it\eta ^5\nu -448t\eta ^4\delta \nu )-2a^3\eta (156it\eta ^3-25i\delta +\eta ^2(-26x+8t\delta )+3\eta (1+5ix\delta )+1248it\eta ^5\nu +64t\eta ^4\delta \nu )\nonumber \\&-\,8\eta ^4(-i\eta +\delta )(-i+2ix\eta +8t(\eta ^2+8\eta ^4\nu ))-4a\eta ^3(-i\eta +\delta )(-i+2ix\eta +8t(\eta ^2+8\eta ^4\nu ))-4a\eta ^3(-i\eta +\delta )(-8i\nonumber \\&+\,13ix\eta +52t(\eta ^2+8\eta ^4\nu ))+4a^4(10x\eta ^2+5i\delta +4t\eta ^2(-13i\eta +8\delta -108i\eta ^3\nu +68\eta ^2\delta \nu ))+a^5(15\nonumber \\&+\,4t\eta (26i\eta +15\delta +156i\eta ^3\nu +172\eta ^2\delta \nu )),\nonumber \\ B_{bp2}= & {} \frac{4a^4(5a+4\eta )^2}{\delta ^6}e^{-8t\eta \delta (1+2a^2\nu +4\eta ^2\nu )}+\frac{4a^4(5a+4\eta )^2}{\delta ^6}e^{8t\eta \delta (1+2a^2\nu +4\eta ^2\nu )}+\frac{4i\eta ^4}{\delta ^7}(-24a^3+8a\eta (3\eta +5i\delta )+a^2(-30\eta \nonumber \\&+\,7i\delta )+2\eta ^2(15\eta +17i\delta ))e^{-4ix\delta }+\frac{4a\eta ^4}{\delta ^7}(-24ia^2+7a\delta +8\eta (3i\eta +5\delta ))e^{4ix\delta }-\frac{8\eta ^5}{\delta ^7}(-15ia^2+\eta (15i\eta +17\delta ))e^{4ix\delta }\nonumber \\&+\,\frac{8a\eta }{\delta ^5}H_1e^{2\delta (ix+2t\eta (1+2a^2\nu +4\eta ^2\nu ))}-\frac{8a\eta }{\delta ^5}H_2e^{-2\delta (ix+2t\eta (1+2a^2\nu +4\eta ^2\nu ))}+\frac{8a\eta }{\delta ^5}H_3e^{2\delta (-ix+2t\eta (1+2a^2\nu +4\eta ^2\nu ))}+\frac{8a\eta }{\delta ^5}H_4\nonumber \\&e^{-2\delta (-ix+2t\eta (1+2a^2\nu +4\eta ^2\nu ))}+\frac{8a}{\delta ^6}H_5,\nonumber \\ H_1= & {} 160a^6t\eta \nu +8a^5t\eta (41\eta -15i\delta )\nu +16a^4t\eta (5+50\eta ^2\nu -6i\eta \delta \nu )+4\eta ^2(-40t\eta ^3-5\delta +2i\eta ^2(5x+12t\delta )+\eta (-3i\nonumber \\&+6x\delta )-320t\eta ^5\nu +192it\eta ^4\delta \nu )+2a\eta (-164t\eta ^3-25\delta +i\eta ^2(41x+60t\delta )+15\eta (-i+x\delta )-1312t\eta ^5\nu +480it\eta ^4\delta \nu )\nonumber \\&+\,a^3(15i+4t\eta (41\eta -15i\delta )(1+8\eta ^2\nu ))+a^2(40ix\eta ^2-20\delta -16it\eta ^2(-5i\eta +3\delta )(1+8\eta ^2\nu )),\nonumber \\ H_2= & {} 160a^6t\eta \nu +8a^5t\eta \nu (41\eta +15i\delta )\nu +16a^4t\eta (5+50\eta ^2\nu +6i\eta \delta \nu )-4\eta ^2(40t\eta ^3-5\delta -2i\eta ^2(5x-12t\delta )+\eta (3i+6x\delta )\nonumber \\&+\,320t\eta ^5\nu +192it\eta ^4\delta \nu )-2a\eta (164t\eta ^3-25\delta -i\eta ^2(41x-60t\delta )+15\eta (i+x\delta )+1312t\eta ^5\nu +480it\eta ^4\delta \nu )+a^3(15i\nonumber \\&+\,4t\eta (41\eta +15i\delta )(1+8\eta ^2\nu ))+4a^2(10ix\eta ^2+5\delta +4it\eta ^2(5i\eta +3\delta )(1+8\eta ^2\nu )),\nonumber \\ H_3= & {} 160a^6t\eta \nu +8a^5t\eta (41\eta +15i\delta )\nu +16a^4t\eta (5+50\eta ^2\nu +6i\eta \delta \nu )-4\eta ^2(40t\eta ^3+5\delta +2i\eta ^2(5x+12t\delta )-3\eta (i+2x\delta )\nonumber \\&+\,320t\eta ^5\nu +192it\eta ^4\delta \nu )-2a\eta (164t\eta ^3+25\delta +i\eta ^2(41x+60t\delta )-15\eta (i+x\delta )(1+8\eta ^2\nu ))+4ia^2(-10x\eta ^2+5i\delta \nonumber \\&+\,4t\eta ^2(5i\eta +3\delta )(1+8\eta ^2\nu )),\nonumber \\ H_4= & {} -160a^6t\eta \nu +8ia^5t\eta (41i\eta +15\delta )\nu +6a^4t\eta (5+50\eta ^2\nu -6i\eta \delta \nu )+4\eta ^2(40t\eta ^3-5\delta +2i\eta ^2(5x-12t\delta )+\eta (-3i+6x\delta )\nonumber \\&+\,320t\eta ^5\nu -192it\eta ^4\delta \nu )+2a\eta (164t\eta ^3-25\delta +i\eta ^2(41x-60t\delta )+15\eta (-i+x\delta )+1312t\eta ^5\nu -480it\eta ^4\delta \nu )+4a^2\nonumber \\&(10ix\eta ^2-5\delta +4t\eta ^2(5\eta +3i\delta )(1+8\eta ^2\nu ))+ia^3(15+4t\eta (41i\eta +15\delta )(1+8\eta ^2\nu )),\nonumber \\ H_5= & {} \frac{-128\eta ^6}{(-a^2+\eta ^2)^3}(1+8x^2\eta ^2+128\eta ^4(t+8t\eta ^2\nu )^2)+3200a^{11}t^2\eta ^2\nu ^2+5120a^{10}t^2\eta ^3\nu ^2+5120a^8t^2\eta ^3\nu (1+7\eta ^2\nu )\nonumber \\&+\,128a^9t^2\eta ^2\nu (25+191\eta ^2\nu )-1280a^6t^2\eta ^3(-1-4\eta ^2\nu +32\eta ^4\nu ^2)-32a^7t^2\eta ^2(-25-164\eta ^2\nu +352\eta ^4\nu ^2)+32a^2\eta ^4\nonumber \\&(-3x+10x^2\eta +320\eta ^3(t+8t\eta ^2\nu )^2)+a\eta ^4(-43+120x\eta -72x^2\eta ^2+896\eta ^4(t+8t\eta ^2\nu )^2)-40a^4\eta (-1+32t^2\eta ^4\nonumber \\&(5+72\eta ^2\nu +256\eta ^4\nu ^2))+2a^3\eta ^2(17-60x\eta +100x^2\eta ^2+128t^2\eta ^4(15+256\eta ^2\nu +1088\eta ^4\nu ^2))-a^5(-25+32t^2\eta ^4\nonumber \\&(109+1736\eta ^2\nu +6912\eta ^4\nu ^2))-8\eta ^5(-12x\eta +40x^2\eta ^2+5(1+128\eta ^4(t+8t\eta ^2\nu )^2)).\end{aligned}$$

(29)