Novel arbitrary-amplitude soliton solutions of the cubic-quintic complex Ginzburg-Landau equation

A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of a...

Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...

A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...

The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.

Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
A circuit representing a line section of length Δx helps in understanding the transmission line parameters. The voltage V(x) and current i(x) are measured from the...

A common physical example of wave propagation with radial symmetry is the ripple formed when a stone is dropped into a still pond. The disturbance originates at a central point and travels outward as a circular wave. As the radius of the wavefront increases, the same initial energy is distributed along a progressively larger circumference. Consequently, the amplitude, or height, of the wave decreases with distance from the center. This decay behavior cannot be captured by simple sine or cosine...