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.

The Interaction and Degeneracy of Mixed Solutions for Derivative Nonlinear Schrodinger Equation

The mixed solutions of the derivative nonlinear Schrödinger equation from the trivial seed (zero solution) are derived by using the determinant representation. By adjusting the interaction and degeneracy of mixed solutions, it is possible to obtain different types of solutions: phase solutions, breather solutions, phase-breather solutions and rogue waves.

Determining osmotic suction through electrical conductivity for unsaturated low-plasticity soils

Determining osmotic suction from the electrical conductivity(EC)of soil pore water was widely reported in the literature.However,while dealing with unsaturated soils,they do not have enough soil pore water to be extracted for a reliable measurement of EC.In this paper,the chilled-mirror dew-point hygrometer and contact filter paper method were used to determine the total and matric suctions for low-plasticity soils with different salinities(0.05‰,2.1‰,and 6.76‰).A new piecewise function was proposed to calculate the osmotic suction,with the piecewise point corresponding to the first occurrence of precipitated salt in mixed salt solutions(synthetic seawater).EC,ion and salt concentrations used for osmotic suction calculation were transformed from the established relationships of mixed salt solution instead of experimental measurement.The calculated osmotic suction by the proposed equation and the equations in the literature was compared with the indirectly measured one(the difference between the measured total and matric suctions).Results showed that the calculated osmotic suction,especially the one calculated using the proposed function,was in fair agreement with the indirectly measured data(especially for specimens with higher salinity of 6.76‰),suggesting that the transformation of EC and concentrations from the established relationship is a good alternative to direct measurement for lowplasticity soil.In particular,the proposed method could be applied to unsaturated low-plasticity soils which do not have enough soil pore water for a proper EC measurement.

Dynamic behaviors of the lump solutions and mixed solutions to a(2+1)-dimensional nonlinear model

In this paper,we propose a combined form of the bilinear Kadomtsev-Petviashvili equation and the bilinear extended(2+1)-dimensional shallow water wave equation,which is linked with a novel(2+1)-dimensional nonlinear model.This model might be applied to describe the evolution of nonlinear waves in the ocean.Under the effect of a novel combination of nonlinearity and dispersion terms,two cases of lump solutions to the(2+1)-dimensional nonlinear model are derived by searching for the quadratic function solutions to the bilinear form.Moreover,the one-lump-multi-stripe solutions are constructed by the test function combining quadratic functions and multiple exponential functions.The one-lump-multi-soliton solutions are derived by the test function combining quadratic functions and multiple hyperbolic cosine functions.Dynamic behaviors of the lump solutions and mixed solutions are analyzed via numerical simulation.The result is of importance to provide efficient expressions to model nonlinear waves and explain some interaction mechanism of nonlinear waves in physics.

Dynamical rational solutions and their interaction phenomena for an extended nonlinear equation

In this paper,we analyze the extended Bogoyavlenskii-Kadomtsev-Petviashvili(eBKP)equation utilizing the condensed Hirota's approach.In accordance with a logarithmic derivative transform,we produce solutions for single,double,and triple M-lump waves.Additionally,we investigate the interaction solutions of a single M-lump with a single soliton,a single M-lump with a double soliton,and a double M-lump with a single soliton.Furthermore,we create sophisticated single,double,and triple complex soliton wave solutions.The extended Bogoyavlenskii-Kadomtsev-Petviashvili equation describes nonlinear wave phenomena in fluid mechanics,plasma,and shallow water theory.By selecting appropriate values for the related free parameters we also create three-dimensional surfaces and associated counter plots to simulate the dynamical characteristics of the solutions offered.

Exact solutions of the nonlocal Gerdjikov-Ivanov equation

The nonlocal nonlinear Gerdjikov-Ivanov(GI)equation is one of the most important integrable equations,which can be reduced from the third generic deformation of the derivative nonlinear Schr?dinger equation.The Darboux transformation is a successful method in solving many nonlocal equations with the help of symbolic computation.As applications,we obtain the bright-dark soliton,breather,rogue wave,kink,W-shaped soliton and periodic solutions of the nonlocal GI equation by constructing its 2 n-fold Darboux transformation.These solutions show rich wave structures for selections of different parameters.In all these instances we practically show that these solutions have different properties than the ones for local case.