Numerical study of a multiscale expansion of KdV and Camassa-Holm equation

We study numerically solutions to the Korteweg-de Vries and Camassa-Holm equation close to the breakup of the corresponding solution to the dispersionless equation. The solutions are compared with the properly rescaled numerical solution to a fourth order ordinary differential equation, the second member of the Painlev\'e I hierarchy. It is shown that this solution gives a valid asymptotic description of the solutions close to breakup. We present a detailed analysis of the situation and compare the Korteweg-de Vries solution quantitatively with asymptotic solutions obtained via the solution of the Hopf and the Whitham equations. We give a qualitative analysis for the Camassa-Holm equationComment: 17 pages, 13 figure

Numerical study of the small dispersion limit of the Korteweg-de Vries equation and asymptotic solutions

We study numerically the small dispersion limit for the Korteweg-de Vries (KdV) equation $u_t+6uu_x+\epsilon^{2}u_{xxx}=0$ for $\epsilon\ll1$ and give a quantitative comparison of the numerical solution with various asymptotic formulae for small $\epsilon$ in the whole $(x,t)$-plane. The matching of the asymptotic solutions is studied numerically

Numerical study of the Kadomtsev\u2013Petviashvili equation and dispersive shock waves

A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev\u2013Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schr\uf6dinger equation in the semiclassical limit

On critical behaviour in generalized Kadomtsev-Petviashvili equations

An asymptotic description of the formation of dispersive shock waves in solutions to the generalized Kadomtsev\u2013Petviashvili (KP) equation is conjectured. The asymptotic description based on a multiscales expansion is given in terms of a special solution to an ordinary differential equation of the Painlev\ue9 I hierarchy. Several examples are discussed numerically to provide strong evidence for the validity of the conjecture. The numerical study of the long time behaviour of these examples indicates persistence of dispersive shock waves in solutions to the (subcritical) KP equations, while in the supercritical KP equations a blow-up occurs after the formation of the dispersive shock waves

Shock formation in the dispersionless Kadomtsev–Petviashvili equation

The dispersionless Kadomtsev-Petviashvili (dKP) equation (u(t) + uu(x))(x)= u(yy) is one of the simplest nonlinear wave equations describing two-dimensional shocks. To solve the dKP equation numerically we use a coordinate transformation inspired by the method of characteristics for the one-dimensional Hopf equation u(t) + uu(x) = 0. We show numerically that the solutions to the transformed equation stays regular for longer times than the solution of the dKP equation. This permits us to extend the dKP solution as the graph of a multivalued function beyond the critical time when the gradients blow up. This overturned solution is multivalued in a lip shape region in the (x, y) plane, where the solution of the dKP equation exists in a weak sense only, and a shock front develops. A local expansion reveals the universal scaling structure of the shock, which after a suitable change of coordinates corresponds to a generic cusp catastrophe. We provide a heuristic derivation of the shock front position near the critical point for the solution of the dKP equation, and study the solution of the dKP equation when a small amount of dissipation is added. Using multiple-scale analysis, we show that in the limit of small dissipation and near the critical point of the dKP solution, the solution of the dissipative dKP equation converges to a Pearcey integral. We test and illustrate our results by detailed comparisons with numerical simulations of both the regularized equation, the dKP equation, and the asymptotic description given in terms of the Pearcey integral

The double scaling limit method in the Toda hierarchy

Critical points of semiclassical expansions of solutions to the dispersionful Toda hierarchy are considered and a double scaling limit method of regularization is formulated. The analogues of the critical points characterized by the strong conditions in the Hermitian matrix model are analyzed and the property of doubling of equations is proved. A wide family of sets of critical points is introduced and the corresponding double scaling limit expansions are discussed.Comment: 20 page

Strength assessment of Al-Humic and Al-Kaolin aggregates by intrusive and non-intrusive methods

Made available in DSpace on 2019-10-06T15:34:15Z (GMT). No. of bitstreams: 0 Previous issue date: 2019-06-15Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Resistance to breakage is a critical property of aggregates generated in water and wastewater treatment processes. After flocculation, aggregates should ideally keep their physical characteristics (i.e. size and morphology), to result in the best performance possible by individual separation processes. The integrity of aggregates after flocculation depends upon their capacity to resist shear forces while transported through canals, passages, apertures, orifices and other hydraulic units. In this study, the strength of Al-Humic and Al-Kaolin aggregates was investigated using two macroscopic measurement techniques, based on both intrusive and non-intrusive methods, using image analysis and light scattering based equipment. Each technique generates different information which was used for obtaining three floc strength indicators, namely, strength factor (SF), local stress from the hydrodynamic disturbance (σ) and the force coefficient (γ) for two different study waters. The results showed an increasing trend for the SF of both Al-Humic and Al-Kaolin aggregates, ranging from 29.7% to 78.6% and from 33.3% to 85.2%, respectively, in response to the increase of applied shear forces during flocculation (from 20 to 120 s−1). This indicates that aggregates formed at higher shear rates are more resistant to breakage than those formed at lower rates. In these conditions, σ values were observed to range from 0.07 to 0.44 N/m2 and from 0.08 to 0.47 N/m2 for Al-Humic and Al-Kaolin, respectively. Additionally, it was found that for all studied conditions, the resistance of aggregates to shear forces was nearly the same for Al-Humic and Al-Kaolin aggregates, formed from destabilized particles using sweep coagulation. These results suggest that aggregate strength may be mainly controlled by the coagulant, emphasizing the importance of the coagulant selection in water treatment. In addition, the use of both intrusive and non-intrusive techniques helped to confirm and expand previous experiments recently reported in literature.Instituto de Geociências e Ciências Exatas Univ. Estadual Paulista (UNESP), Av. 24-A, 1515, Jardim Bela Vista, Rio ClaroPrograma de Pós-graduação em Engenharia Civil e Ambiental Univ. Estadual Paulista (UNESP), Av. 24-A, 1515, Jardim Bela Vista, Rio ClaroDepartment of Civil Engineering University of BirminghamDepartment of Civil Environmental and Geomatic Engineering University College London, Gower StInstituto de Geociências e Ciências Exatas Univ. Estadual Paulista (UNESP), Av. 24-A, 1515, Jardim Bela Vista, Rio ClaroPrograma de Pós-graduação em Engenharia Civil e Ambiental Univ. Estadual Paulista (UNESP), Av. 24-A, 1515, Jardim Bela Vista, Rio ClaroFAPESP: 2017/19195-