<p>This study investigates the propagation and dynamics of optical waves governed by the variable-coefficient nonlinear Schrödinger equation. As a fundamental model in nonlinear wave theory, the NLSE plays a pivotal role in describing wave phenomena across diverse fields, including nonlinear optics, quantum mechanics, and fluid dynamics. Its ability to capture the interplay between dispersion and nonlinearity makes it indispensable for modeling localized wave structures, such as solitons, rogue waves, and breathers, which are critical to understanding complex wave behavior in real-world systems. Wave dynamics, including rogue waves, need to be studied to keep people safe in the ocean and to understand how they work, which helps protect marine ecosystems and make good use of ocean resources. Using the well-known enhanced modified simple equation method, we are trying to trace specific solutions that explain basic wave patterns realized in nature and the experimental phenomena, such as optical solitons, Bose–Einstein condensates, and Plasma waves. This method uses the improved modified simple equation to address the variable-coefficient nonlinear Schrödinger equation. The enhanced modified simple equation method uses a traveling wave variable called <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40042_2025_1456_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\xi\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ξ</mi> </math></EquationSource> </InlineEquation>, which is written as <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40042_2025_1456_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="120" /> </InlineMediaObject> <EquationSource Format="TEX">\(\xi =\gamma (t)x\pm \pi (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>±</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The values of γ and π change over time to show how each transition naturally happens. We can incorporate differentiable functions of time, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40042_2025_1456_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(\gamma (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40042_2025_1456_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma (t)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which renders this method highly effective for modeling more dynamic wave characteristics. Because it takes into account changes in wavelength and time, the enhanced modified simple equation method makes it easier to get more accurate answers. This work adds new features that get around methods that are static or not very flexible. The extended mapping method for this soliton equation gives us more exact solutions and helps us learn more about wave dynamics in nonlinear optics, fluid dynamics, and other areas of physics that are affected by wave modulation. The result adds to what we already know.</p>