Soliton Solutions and Conservation Laws for a Self-interacting Scalar Field in \(\phi^{4}\) Theory

We calculate soliton solutions to the scalar field equation of motion that arises for the 4th-order extended Lagrangian ($\phi^{4}$ theory) in quantum field theory using the extended hyperbolic tangent and the sine-cosine methods. Using the former technique, ten complex soliton waves are obtained; we graphically represent three of these profiles using density plots. In the latter case, two real soliton solutions are obtained, of which, we demonstrate the wave profile for the positive case. Using the multiplier method, we calculate conservation laws in $(1 + 1)$-, $(2 + 1)$-, and $(3 + 1)$-dimensions producing three, six, and ten conservation laws respectively. Lastly, we reflect on the application of conservation laws in particle physics and phenomenology.