[Paper Review] Exact solitary wave solutions of the nonlinear Schrödinger equation with a source

Thokala Soloman Raju, C.N. Kumar+1 more|ArXiv.org|Aug 8, 2003

Nonlinear Waves and Solitons3 citations

TL;DR

This paper presents exact rational-type solitary wave solutions for the nonlinear Schrödinger equation (NLSE) with a phase-locked external source using a fractional transformation that maps solutions to elliptic functions. The method yields bright, dark, and singular solitons, as well as periodic trigonometric and hyperbolic solutions, providing nonperturbative, exact solutions for both attractive and repulsive cases, with stability confirmed numerically.

ABSTRACT

We use a fractional transformation to connect the traveling wave solutions of the nonlinear Schrödinger equation (NLSE), phase-locked with a source, to the elliptic functions satisfying, $f^{\prime\prime}\pm af\pm λf^{3}=0$. The solutions are {\it necessarily} of the rational form, containing both trigonometric and hyperbolic types as special cases. Bright, and dark solitons, respectively, for attractive and repuslive type of nonlinearities, as also singular solitons, are obtained in suitable range of parameter values.

Motivation & Objective

• To derive exact traveling wave solutions for the externally driven nonlinear Schrödinger equation (NLSE) with a phase-locked source.
• To address the lack of exact solutions for the NLSE with a source, despite extensive study via perturbation and numerical methods.
• To establish a systematic mapping between the NLSE with source and elliptic functions using a fractional transformation.
• To classify and characterize soliton types—bright, dark, singular—based on parameter regimes and modulus parameters.
• To provide nonperturbative, exact solutions that serve as improved initial conditions for studying general externally driven NLSE systems.

Proposed method

• A fractional transformation of the form $\rho(\xi) = \frac{A + B f^\delta(\xi,m)}{1 + D f^\delta(\xi,m)}$ is applied to map solutions of the NLSE with source to elliptic functions.
• The transformation reduces the NLSE with source to a nonlinear ODE $\rho'' + \rho^3 - \rho - k = 0$, which is then linked to the elliptic equation $f'' = f - f^3$.
• The method assumes a traveling wave solution $\psi(x,t) = e^{i[\chi(\xi) - \omega t]} \rho(\xi)$, with $\xi = \alpha(x - vt)$, and enforces phase-locking with the source.
• The conserved energy $E_0 = f'^2/2 + (1/4)f^4 - f^2/2$ is used to distinguish between $E_0 = 0$ and $E_0 \neq 0$ solution branches.
• Solutions are derived for different cases: trigonometric ($m=0$), hyperbolic ($m=1$), and general cnoidal ($0 < m < 1$), with constraints on parameters $A$, $B$, $D$, $\epsilon$, $k$, and $g$.
• Numerical simulations using the Crank-Nicholson finite difference method confirm the stability of the trigonometric solution, validating the analytical results.

Experimental results

Research questions

• RQ1 Can exact solitary wave solutions be derived for the nonlinear Schrödinger equation when driven by a phase-locked external source?
• RQ2 What types of solitons—bright, dark, singular—can emerge from the NLSE with source, and under what parameter conditions?
• RQ3 How does the fractional transformation map the NLSE with source to solutions of elliptic functions, and what constraints arise from this mapping?
• RQ4 What is the role of the conserved energy $E_0$ in determining the allowed solution forms, particularly in $E_0 = 0$ vs. $E_0 \neq 0$ cases?
• RQ5 Are the derived exact solutions stable under time evolution, and can they serve as robust initial conditions for numerical studies?

Key findings

• The fractional transformation maps the NLSE with source to elliptic functions, yielding exact rational-type solutions with both numerator and denominator quadratic in elliptic functions.
• For $E_0 \neq 0$, solutions with $A=0$, $B \neq 0$ are allowed, such as $\rho(\xi) = \frac{(k/4E_0)f^2}{1 + (1/8E_0)f^2}$, while $A \neq 0$, $B=0$ is forbidden.
• For $E_0 = 0$, singular solutions exist such as $\rho(\xi) = \frac{2k}{1 - f^2}$, and nonsingular solutions are possible when $B \neq 0$, $A \neq 0$, but $A=0$, $B \neq 0$ is not allowed.
• In the $m=0$ limit, trigonometric solutions emerge, such as $\rho(\xi) = \left(-\frac{2k}{\epsilon}\right)\frac{\cos^2(\xi)}{1 - \frac{2}{3}\cos^2(\xi)}$, which is numerically stable.
• For $m=1$, hyperbolic solutions are found, including the singular form $\rho(\xi) = \left(\frac{3k}{\epsilon}\right)\frac{1}{1 - \frac{3}{2}\text{sech}^2(\xi)}$, valid when $\alpha^2 = \epsilon/4$.
• For $0 < m < 1$, general cnoidal solutions are derived, such as $\rho(\xi) = \left(\frac{14k}{3\epsilon}\right)\frac{\text{cn}^2(\xi,m)}{1 + \text{cn}^2(\xi,m)}$ for $m=5/8$, with $\epsilon = 7(-gk^2/18)^{1/3}$.