Three-dimensional simulation of the lumbar spine under lumbar spinal stenosis with different sizes of spinal canal

Abundant Closed-Form Soliton Solutions to the Fractional Stochastic Kraenkel–Manna–Merle System with Bifurcation, Chaotic, Sensitivity, and Modulation Instability Analysis

“…For deeper consideration of nonlinear occurrence and realistic challenges, it is essential to discover closed-form soliton solutions of SFPDEs. According to the quick advancements in nonlinear sciences, a variety of simple and efficient approaches have been developed to obtain closed-form soliton solutions to NLPDEs, including the Hirota method [4,5], the Bernoulli sub-equation method [6,7], the F-expansion method [8], the (G ′ /G 2 )-expansion method [9], the simple equation method [10], the modified auxiliary equation method [11,12], the two variable (G ′ /G, 1/G)-expansion method [13][14][15][16], the Lie symmetric analysis [17], the polynomial complete discriminant system [18], the tanh-coth scheme [19], the Conservation laws method [20], the generalized exponential rational function approach [21,22], the binary bell polynomials method [23], the mapping method [24], the Shehu transform scheme [25], the sine-Gordon expansion [26], the Cole-Hopf transformation method [27,28], the Fan subequation technique [29], the unified method [30], the Khater method [31], the r + mEDAM method [32], the spectral Tau method [33], the G ′ G ′ +G+A -expansion procedure [34][35][36][37][38], the sub-equation method [39], the collocation method [40], the finite element method [41], and the generalized G ′ /G-expansion method …”