Linear superposition solutions to nonlinear wave equations

The solutions to a linear wave equation can satisfy the principle of superposition, i.e., the linear superposition of two or more known solutions is still a solution of the linear wave equation. We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic, triangle, and exponential functions, and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics. The linear superposition solutions to the generalized KdV equation K（2,2,1）, the Oliver water wave equation, and the k（n, n） equation are given. The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed, and the reason why the solutions with the forms of hyperbolic, triangle, and exponential functions can form the linear superposition solutions is also discussed.

Linear superposition method for （2＋1）-dimensional nonlinear wave equations

New forms of different-periodic travelling wave solutions for the （2＋1）-dimensional Zakharov-Kuznetsov （ZK） equation and the Davey-Stewartson （DS） equation are obtained by the linear superposition approach of Jacobi elliptic function. A sequence of cyclic identities plays an important role in these procedures.

Superposition formulas of multi-solution to a reduced(3+1)-dimensional nonlinear evolution equation

We gave the localized solutions,the interaction solutions and the mixed solutions to a reduced(3+1)-dimensional nonlinear evolution equation.These solutions were characterized by superposition formulas of positive quadratic functions,the exponential and hyperbolic functions.According to the known lump solution in the outset,we obtained the superposition formulas of positive quadratic functions by plausible reasoning.Next,we constructed the interaction solutions between the localized solutions and the exponential function solutions with the similar theory.These two kinds of solutions contained superposition formulas of positive quadratic functions,which were turned into general ternary quadratic functions,the coefficients of which were all rational operation of vector inner product.Then we obtained linear superposition formulas of exponential and hyperbolic function solutions.Finally,for aforementioned various solutions,their dynamic properties were showed by choosing specific values for parameters.From concrete plots,we observed wave characteristics of three kinds of solutions.Especially,we could observe distinct generation and separation situations when the localized wave and the stripe wave interacted at different time points.

A millimeter wave linear superposition oscillator in 0.18 m CMOS technology

This paper presents a millimeter wave （mm-wave） oscillator that generates signal at 36.56 GHz. The ram-wave oscillator is realized in a UMC 0.18 μm CMOS process. The linear superposition （LS） technique breaks through the limit of cut-off frequency （JET）, and realizes a much higher oscillation than Jr. Measurement results show that the LS oscillator produces a calibrated 37.17 dBm output power when biased at 1.8 V; the output power of fundamental signal is -10.85 dBm after calibration. The measured phase noise at 1 MHz frequency offset is -112.54 dBc/Hz at the frequency of 9.14 GHz. This circuit can be properly applied to mm-wave communication systems with advantages of low cost and high integration density.

Different-Periodic Travelling Wave Solutions for Nonlinear Equations

Using Jacobi elliptic function linear superposition approach for the (1+1)-dimensional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) equation and the (2+1)-dimensional Nizhnik–Novikov–Veselov (NNV) equation, many new periodic travelling wave solutions with different periods and velocities are obtained based on the known periodic solutions. This procedure is crucially dependent on a sequence of cyclic identities involving Jacobi elliptic functions sn(), cn(), and dn().

Non-linearity in the dynamic world of human movement

Non-linear dynamics,fractals,periodic oscillations,bifurcations,chaos,and other terminologies have been used to describe human biological systems in the literature for a few decades.The eight manuscripts included in this special issue discussed the historical background,

A New Class of Periodic Solutions to （2＋1）-Dimensional KdV Equations

We investigate a new class of periodic solutions to (2+1)-dimensional KdV equations, by both the linear superposition approach and the mapping deformation method. These new periodic solutions are suitable combinations of the periodic solutions to the (2+1)-dimensional KdV equations obtained by means of the Jacobian elliptic function method, but they possess different periods and velocities.

Decomposition solutions and Bäcklund transformations of the B-type and C-type Kadomtsev-Petviashvili equations

This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application of this approach,we successfully ascertain decomposition solutions,Bäcklund transformations,the Lax pair,and the linear superposition solution associated with the aforementioned equation.Furthermore,we expand the utilization of this technique to the C-type Kadomtsev-Petviashvili(CKP)equation,leading to the derivation of decomposition solutions,Bäcklund transformations,and the Lax pair specific to this equation.The results obtained not only underscore the efficacy of the proposed approach,but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems.Moreover,this approach demonstrates an efficient framework for establishing interrelations between diverse systems.

Non-spherical symmetry development of underwater shock waves created by laser-induced breakdown

As a controllable alternative to cavitation collapse-induced shock waves,numerous cavitation studies on laser-induced breakdown have been carried out in hydromechanics.When the laser focusing region is not spherical,the shock waves caused by laser breakdown also exhibit non-spherical symmetry propagation.Recently,some researchers have proposed the linear superposition theory based on the far field measurement data to explain this asymmetry,assuming that it is essentially the linear superposition of multiple wave fronts caused by multiple points of laser-induced breakdown that leads to the asymmetric propagation of shock waves.In this study,measurements of shock wave propagation processes with different breakdown energies are carried out based on a nanosecond resolution photogrammetry system,and the propagation velocities of shock waves in different directions are directly measured using a double exposure technique on a single frame.In the experiment,the velocity of the shock wave at the beginning of the breakdown was measured up to nearly 4000 m/s.The early shock wave front was ellipsoidal,and the propagation velocity in the laser incident direction was generally slower than that in the perpendicular direction,decaying to the speed of sound in water within 1000 ns after the breakdown,and the wave front gradually approached to a circle.The variability of the shock wave front pressure ratio in the laser propagation direction and the vertical direction implies that the linear superposition theory applicable to the far field is not applicable to the near field.There may be more complex mechanism for the near-field shock wave propagation process.