Analytical Solutions for a New Form of the Generalized <i>q</i>-Deformed Sinh–Gordon Equation:<inline-formula><math display="inline"><semantics><mrow><mo> </mo><mfrac><mrow><msup><mo mathvariant="bold">∂</mo><mn mathvariant="bold">2</mn></msup><mi mathvariant="bold-italic">u</mi></mrow><mrow><mo mathvariant="bold">∂</mo><mi mathvariant="bold-italic">z</mi><mo mathvariant="bold">∂</mo><mi mathvariant="bold-italic">ζ</mi></mrow></mfrac><mo mathvariant="bold">=</mo><msup><mi mathvariant="bold-italic">e</mi><mrow><mi mathvariant="bold-italic">α</mi><mi mathvariant="bold-italic">u</mi></mrow></msup><msup><mrow><mo mathvariant="bold">[</mo><mi mathvariant="bold-italic">s</mi><mi mathvariant="bold-italic">i</mi><mi mathvariant="bold-italic">n</mi><msub><mi mathvariant="bold-italic">h</mi><mi mathvariant="bold-italic">q</mi></msub><mrow><mo mathvariant="bold">(</mo><msup><mi mathvariant="bold-italic">u</mi><mi mathvariant="bold-italic">γ</mi></msup><mo mathvariant="bold">)</mo></mrow><mo mathvariant="bold">]</mo></mrow><mi mathvariant="bold-italic">p</mi></msup><mo mathvariant="bold">−</mo><mi mathvariant="bold-italic">δ</mi></mrow></semantics></math></inline-formula>

Abstract

In this article, a new version of the generalized q-deformed Sinh–Gordon equation is presented, and analytical solutions are developed for specific parameter sets using those equations. There is a possibility that the new equation can be used to model physical systems that have broken symmetries and include also effects related to amplification or dissipation. In addition, we have include some illustrations that depict the varied patterns of soliton propagation.

Keywords

• soliton solutions
• a new form of <i>q</i>-deformed equation
• the generalized <i>q</i>-deformed Sinh–Gordon equation
• the Kudryashov technique