New (3+1)-dimensional integrable fourth-order nonlinear equation: lumps and multiple soliton solutions

Purpose This paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both right- and left-going waves in a like manner to the Boussinesq equation. Design/methodology/approach This formally uses the simplified Hirota’s method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space. Findings This paper confirms the complete integrability of the newly developed (3 + 1)-dimensional model in the Painevé sense. Research limitations/implications This paper addresses the integrability features of this model via using the Painlevé analysis. Practical implications This paper presents a variety of lump solutions via using a variety of numerical values of the included parameters. Social implications This work formally furnishes useful algorithms for extending integrable equations and for the determination of lump solutions. Originality/value To the best of the author’s knowledge, this paper introduces an original work with newly developed integrable equation and shows useful findings of solitons and lump solutions.

Lump solutions to a (2+1)-dimensional fourth-order nonlinear PDE possessing a Hirota bilinear form

Through the Hirota bilinear formulation, a (2+1)-dimensional combined fourth-order nonlinear equation is proposed, which possesses lump solutions. Two classes of lump solutions are presented explicitly in terms of the coefficients in the combined nonlinear equation. A set of examples of equations is provided to show the diversity of the considered combined nonlinear equation, together with three-dimensional plots, [Formula: see text]-curves and [Formula: see text]-curves of two specific lump solutions in two cases of the combined equation.

Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation

In this paper, we consider the Cauchy problem for a second-order nonlinear equation with mixed fractional derivatives related to the fractional Khokhlov-Zabolotskaya equation. We prove the nonexistence of a classical local in time solution. The obtained instantaneous blow-up result is proved via the nonlinear capacity method.

New Travelling Wave Solutions for KdV6 Equation Using Sub Equation Method

This paper proposes obtaining the new wave solutions of time fractional sixth order nonlinear Equation (KdV6) using sub-equation method where the fractional derivatives are considered in conformable sense. Conformable derivative is an understandable and applicable type of fractional derivative that satisfies almost all the basic properties of Newtonian classical derivative such as Leibniz rule, chain rule and etc. Also conformable derivative has some superiority over other popular fractional derivatives such as Caputo and Riemann-Liouville. In this paper all the computations are carried out by computer software called Mathematica.

Computational wave solutions of generalized higher-order nonlinear Boussinesq dynamical wave equation

A Riccati equation rational expansion (RERE) method is implemented to attain solutions of nonlinear partial differential equations (NLPDEs). Subsequently, the technique is used to attain families or sets of nonlinear solutions for the third extended fifth-order nonlinear equation and the generalized (2 + 1)-dimensional Boussinesq equation. By the use of the RERE method, exciting and new forms of solution such as rational solitary wave solutions, solitary wave solutions and periodic solitary wave solutions are obtained. Some of the solutions are demonstrated graphically with specific values assigned to the parameters. It is worth noting that the application of this method is very effective and reliable.

Solving the Time-Varying Inverse Kinematics Problem for the Da Vinci Surgical Robot

A dialytic-elimination and Newton-iteration based quasi-analytic inverse kinematics approach is proposed for the 6 degree of freedom (DOF) active slave manipulator in the Da Vinci surgical robot and other similar systems. First, the transformation matrix-based inverse kinematics model is derived; then, its high-dimensional nonlinear equations are transformed to a high-order nonlinear equation with only one unknown variable by using the dialytic elimination with a unitary matrix. Finally, the quasi-analytic solution is eventually obtained by the Newton iteration method. Simulations are conducted, and the result show that the proposed quasi-analytic approach has advantages in terms of accuracy (error < 0.00004 degree (or mm)), solution speed (< 20 ms) and is barely affected by the singularity during intermediate calculations, which proves that the approach meets the real-time and high-accuracy requirements of master‒slave mapping control for the Da Vinci surgical robots and other similar systems. In addition, the proposed approach can also serve as a design reference for other types of robotic arms that do not satisfy the Pieper principle.