Physics

Exponential function

Meteorology

Nonlinear system

Trigonometry

Mathematics

Term (time)

Evolutionary biology

Contour line

Biology

Statistics

Epistemology

Mathematical analysis

Philosophy

Function (biology)

Trigonometric functions

Plot (graphics)

Ecology

Rogue wave

Geometry

Type (biology)

Simple (philosophy)

Quantum mechanics

Hyperbolic function

Rational function

Soliton

Heliyon

This study presents a modification form of modified simple equation method, namely new modified simple equation method. Multiple waves and interaction of soliton solutions of the Phi-4 and Klein-Gordon models are investigated via the scheme. Consequently, we derive various novels and more general interaction, and multiple wave solutions in term of exponential, hyperbolic, and trigonometric, rational function solutions combining with some free parameters. Taking special values of the free parameters, interaction of two dark bells, interaction of two bright bells, two kinks, two periodic waves, kink and soliton, kink-rogue wave solutions are obtained which is the key significance of this method. Properties of the achieved solutions have many useful descriptions of physical behavior, correlated to the solutions are attained in this work through plentiful 3D figures, density plot and 2D contour plots. The results derived may increase the prospect of performing significant experimentations and carry out probable applications.

Dynamical interaction of solitary, periodic, rogue type wave solutions and multi-soliton solutions of the nonlinear models