Cnoidal Wave-Induced Residual Liquefaction in Loosely Deposited Seabed under Coastal Shallow Water

This study is aimed at numerically investigating the cnoidal wave-induced dynamics characteristics and the liquefaction process in a loosely deposited seabed floor in a shallow water environment. To achieve this goal, the integrated model FSSI-CAS 2D is taken as the computational platform, and the advanced soil model Pastor–Zienkiewicz Mark III is utilized to describe the complicated mechanical behavior of loose seabed soil. The computational results show that a significant lateral spreading and vertical subsidence could be observed in the loosely deposited seabed floor due to the gradual loss of soil skeleton stiffness caused by the accumulation of pore pressure. The accumulation of pore pressure in the loose seabed is not infinite but limited by the liquefaction resistance line. The seabed soil at some locations could be reached to the full liquefaction state, becoming a type of heavy fluid with great viscosity. Residual liquefaction is a progressive process that is initiated at the upper part of the seabed floor and then enlarges downward. For waves with great height in shallow water, the depth of the liquefaction zone will be greatly overestimated if the Stokes wave theory is used. This study can enhance the understanding of the characteristics of the liquefaction process in a loosely deposited seabed under coastal shallow water and provide a reference for engineering activities.

Stability analysis and novel solutions to the generalized Degasperis Procesi equation: An application to plasma physics

In this work two kinds of smooth (compactons or cnoidal waves and solitons) and nonsmooth (peakons) solutions to the general Degasperis-Procesi (gDP) equation and its family (Degasperis-Procesi (DP) equation, modified DP equation, Camassa-Holm (CH) equation, modified CH equation, Benjamin-Bona-Mahony (BBM) equation, etc.) are reported in detail using different techniques. The single and periodic peakons are investigated by studying the stability analysis of the gDP equation. The novel compacton solutions to the equations under consideration are derived in the form of Weierstrass elliptic function. Also, the periodicity of these solutions is obtained. The cnoidal wave solutions are obtained in the form of Jacobi elliptic functions. Moreover, both soliton and trigonometric solutions are covered as a special case for the cnoidal wave solutions. Finally, a new form for the peakon solution is derived in details. As an application to this study, the fluid basic equations of a collisionless unmagnetized non-Maxwellian plasma is reduced to the equation under consideration for studying several nonlinear structures in the plasma model.

Periodic and localized structures in dusty plasma with Kaniadakis distribution

The propagation of electrostatic dust-ion-acoustic nonlinear periodic waves is investigated in dusty plasma wherein electrons follow Kaniadakis distribution. The Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations are derived by employing reductive perturbation method and their cnoidal wave solutions are analysed. The effect of relevant parameters (viz., κ-deformed parameter κ and dust concentration β) on the dynamics of cnoidal structures is discussed. Further it is found that amplitude of compressive cnoidal waves increases with increasing values of β, while reverse effect is observed in case of rarefactive cnoidal structures with rising values of β. Also κ-deformed parameter κ bears no effect on cnoidal waves associated with KdV equation, whereas κ-deformed parameter κ significantly affects the cnoidal waves associated with mKdV equation.

Rigorous Asymptotics of a KdV Soliton Gas

We analytically study the long time and large space asymptotics of a new broad class of solutions of the KdV equation introduced by Dyachenko, Zakharov, and Zakharov. These solutions are characterized by a Riemann–Hilbert problem which we show arises as the limit $$N\rightarrow + \infty $$ N → + ∞ of a gas of N-solitons. We show that this gas of solitons in the limit $$N\rightarrow \infty $$ N → ∞ is slowly approaching a cnoidal wave solution for $$x \rightarrow - \infty $$ x → - ∞ up to terms of order $$\mathcal {O} (1/x)$$ O ( 1 / x ) , while approaching zero exponentially fast for $$x\rightarrow +\infty $$ x → + ∞ . We establish an asymptotic description of the gas of solitons for large times that is valid over the entire spatial domain, in terms of Jacobi elliptic functions.