<p>Motivated by the ubiquitous occurrence of dispersive shock waves (DSWs) in nonlinear wave systems and the distinct non-genuine nonlinearity of the defocusing complex modified Korteweg–de Vries (cmKdV) equation, this paper investigates two typical wave-breaking scenarios: parabolic and cubic initial profiles. We derive the Whitham modulation equations that govern the slowly varying parameters of the DSWs and present the corresponding periodic wave solutions. By employing the generalized hodograph method, we obtain explicit analytical solutions for the Whitham equations as functions of the Riemann invariant <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$r_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> in each profile. Based on these solutions, we analyze the motion laws of the soliton edge and the small-amplitude edge of the DSW. Our findings reveal that for a parabolic profile, the DSW domain expands over time with its width scaling predominantly as <i>t</i>, a behavior distinct from the genuinely nonlinear KdV equation where the width grows as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>. For a cubic profile, the soliton edge moves substantially faster than the small-amplitude edge; thus, the expansion of the DSW region is dominated by the motion of the soliton edge. This behavior differs strikingly from that in the genuinely nonlinear KdV and KB systems, where the two edges propagate at comparable speeds, resulting in an overall DSW expansion that scales as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$t^{3/2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mn>3</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. These results provide new insights into the modulation dynamics of DSWs in non-genuinely nonlinear integrable systems and extend the understanding of wave-breaking phenomena beyond the classical KdV framework.</p>

Wave breaking and generation of dispersive shock waves for the defocusing complex modified KdV equation

期刊论文
发表日期：

2026年7月3日

查看全文

Ao Zhou,
Jing Chen,
Yushan Xue

摘要

Motivated by the ubiquitous occurrence of dispersive shock waves (DSWs) in nonlinear wave systems and the distinct non-genuine nonlinearity of the defocusing complex modified Korteweg–de Vries (cmKdV) equation, this paper investigates two typical wave-breaking scenarios: parabolic and cubic initial profiles. We derive the Whitham modulation equations that govern the slowly varying parameters of the DSWs and present the corresponding periodic wave solutions. By employing the generalized hodograph method, we obtain explicit analytical solutions for the Whitham equations as functions of the Riemann invariant $r_1$ $r_{1}$ in each profile. Based on these solutions, we analyze the motion laws of the soliton edge and the small-amplitude edge of the DSW. Our findings reveal that for a parabolic profile, the DSW domain expands over time with its width scaling predominantly as t, a behavior distinct from the genuinely nonlinear KdV equation where the width grows as $t^2$ $t^{2}$ . For a cubic profile, the soliton edge moves substantially faster than the small-amplitude edge; thus, the expansion of the DSW region is dominated by the motion of the soliton edge. This behavior differs strikingly from that in the genuinely nonlinear KdV and KB systems, where the two edges propagate at comparable speeds, resulting in an overall DSW expansion that scales as $t^{3/2}$ $t^{3 / 2}$ . These results provide new insights into the modulation dynamics of DSWs in non-genuinely nonlinear integrable systems and extend the understanding of wave-breaking phenomena beyond the classical KdV framework.