Similarity reductions and exact solutions of generalized Bretherton equation with time-dependent coefficients

Particle diffusion in extracellular hydrogels

“…[26][27][28] We complement the ensemble-averaged information obtained from DDM with single particle tracking, using the same time-lapse imaging data as input, in order to test for the presence of spatial and temporal heterogeneities in particle dynamics that typically arise in heterogeneous systems such as polymer gels. [29][30][31][32] In order to understand how the dynamics of hyaluronan networks are modulated by accessory extracellular molecules that introduce crosslinks, we probe singlecomponent gels with three different crosslink configurations: semidilute solutions, transiently crosslinked gels obtained by pH-triggered gelation, 33 and chemically crosslinked gels. We show that semidilute solutions and transiently crosslinked hyaluronan networks simply hinder particle transport through enhanced viscous drag, whereas permanently crosslinked gels hamper particle diffusion by size exclusion.…”

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

“…Moreover, the four models described before are also valid in the present work, but since h is assumed to be a constant, the terms associated with the coefficients B 3 and B 4 disappear [see (5i) and (5j)]. We stress that if α = 1 − √ 5 5 which belongs to the interval [0, 1], the model given by (8) and (13) can be reduced to a fourth-order system since B 2 = 0 [see (8)]. Note that if we do not consider the surface tension effect and the extra term of order O(μ 6 ) in the velocity potential expansion (see equation (20) of [32]) which appears in (13a) as the term of order O(μ 4 ) associated with B 2 as well as assuming that β 1 = β 2 = 0, then we recover the model proposed in [15].…”

“…Other nonlinear evolution equations have also been used to model these physical phenomena. Some of them are the Korteweg-de Vries equation [1][2][3]5,6], Kawahara equation [7], Bretherton equation [8], Benjamin-Bona-Mahony equation [9,10], CamassaHolm equation [11] and other types of Boussinesq equations [4,[12][13][14][15][16][17]. These nonlinear evolution equations support solitons and other types of travelling wave solutions sparking worldwide interest in the mathematical physics community.…”

Symmetries and conservation laws of a fifth-order KdV equation with time-dependent coefficients and linear damping

“…Lie's theory provides a useful, powerful tool when analysing partial differential equations. Moreover, it has numerous well-known applications, prominent among these are the obtaining of exact solutions of partial differential equations, directly or via similarity solutions [7,17], classifying invariant equations, reducing the number of independent variables or determining conservation laws [1,2,5,6,14].…”