Two-point boundary value problems for a discrete Ermakov-Painle-ve II equation are analysed by means of topological methods. In addition, an alternative variational approach is detailed. Existence of solutions is established for appropriate choice of parameters. For more information and the latex file, see https://ejde.math.txstate.edu/Volumes/2026/39/abstr.html

Here classes of moving boundary problems of Stefan-type for both an established non-linear evolution equation of cuspon theory and novel reciprocally linked solitonic equations are shown to be solvable via Painleve' II symmetry reduction.

In 1+1-dimensions, an extension of the canonical solitonic Dym equation has previously been derived both in a geometric torsion evolution context and in the analysis of peakon solitonic phenomena in hydrodynamics. Here, a novel 2+1-dimensional S-integrable extended Dym-type equation is introduced. As Lax pair is constructed and an associated ∂̅-dressing scheme detailed. Integrable modulated versions of the 2+1-dimensional extended Dym equation are generated via application of a class of involutory transformations with genesis in classical Ermakov theory.

Here, a novel 2+1-dimensional nonlinear evolution equation with temporal modulation is introduced which admits integrable Ermakov-Painlevé II symmetry reduction. Application is made to obtain exact solution to a class of Stefan-type moving boundary problems for this 2+1-dimensional nonlinear evolution equation. Involutory transformations with origin in autonomisation of certain Ermakov-type coupled systems are extended to 2+1-dimensions and applied to derive a wide 2+1-dimensional class with temporal modulation and which inherits the property of admittance of such hybrid Ermakov-Painlevé II symmetry reduction applicable to certain moving boundary problems.

A class of moving boundary problems of Stefan-type for an extension of the 2+1- dimensional Gardner equation is shown to be amenable to exact solution by application of a Ermakov- Painlevé II symmetry reduction.

A novel 2+1-dimensional extension of the solitonic Dym equation is shown to admit a Painlevé II symmetry reduction which permits the exact solution of a class of Stefan-type moving boundary problems.

Here, a class of nonlinear moving boundary problems for a novel extension of a two-component mKdV system is shown to admit exact solution via application of a hybrid Ermakov-Ray-Reid / Painlevé II symmetry ansatz.The mKdV system has its genesis in a reduction of a coupled nonlinear NLS system incorporating deBroglie - Bohm potential terms.

A reciprocal transformation is applied to link an integrable extension of the classical solitonic mKdV to a novel nonlinear evolution equation incorporating a source term. Application of Ermakov-Painleve' II symmetry reduction is made to determine the exact solution to a class of associated moving boundary problems of Stefan-type.

Here, a procedure is presented whereby a range of modulated multi-dimensional Ermakov-type systems may be systematically reduced to their integrable canonical counterparts.

An overview is presented of quantum and resonant nonlinear Schrodinger equation links to Whitham-Broer-Kaup type systems. A novel n + 1 dimensional extension of the Whitham-Broer-Kaup hydrodynamic system is constructed with connection to an equivalent multi-dimensional resonant NLS equation. Hybrid Ermakov-Painleve II and associated Painleve XXXIV integrable similarity reductions are derived.

Classes of moving boundary problems of Stefan-type are here shown to be exactly solvable in terms of classical Airy functions both for the linearised and solitonic Korteweg–de Vries equations. In the latter case, a Miura transformation is applied to a class of Airy-type similarity solutions derived via a Painlevé II reduction of the mKdV equation. Reciprocal transformations are then applied to obtain, in turn, Airy-type solution of associated moving boundary problems for both a nonlinear evolution equation of magma theory and a novel reciprocal Korteweg–de Vries equation which incorporates a source term.

This work investigates a class of moving boundary problems related to a nonlinear evolution equation featuring an exponential source term. We establish a connection to Stefan-type problems, for different boundary conditions at the fixed face, through the application of a reciprocal transformation alongside the Cole-Hopf transformation. For specific cases, we derive explicit similarity solutions in parametric form. This innovative approach enhances our understanding of the underlying dynamics and offers valuable insights into the behavior of these systems.

Here, novel geometric conservation law representations are established in two-dimensional magnetohydrodynamics whereby a triad of admitted conducting motions is derived. Application is then made of a magnetohydrodynamic superposition principle to generate extended multi-parameter classes of associated conducting motions. In addition, under a correspondence between the magnetogasdynamic system and nonlinear elastostatics an associated invariance is established for a linked canonical neo-Hookean plane strain system.

A seed class of exact solutions of the Euler system of two-dimensional relativistic gasdynamics is derived with an underlying Chaplygin–Kármán–Tsien type constitutive law. The invariance of the nonlinear relativistic system under multi-parameter reciprocal Bäcklund transformations is applied to generate a wide class of novel associated solutions with a barotropic relation. The latter reduces to the standard Chaplygin–Kármán–Tsien gas law in the non-relativistic limit. An additional invariance of the general relativistic gasdynamics system is established under a substitution principle. The latter represents an extension of a classical result in non-relativistic gasdynamics.

A class of moving boundary problems of Stefan-type is shown to be amenable to exact solution via Painlevé II similarity reduction both for the canonical solitonic Gardner equation and a reciprocally associated novel third order nonlinear evolution equation incorporating a source term.

Moving boundary problems of Stefan-type for a novel third order nonlinear evolution equation with temporal modulation are here shown to be amenable to exact Airy-type solution via a classical Ermakov equation with its admitted nonlinear superposition principle. Application of the latter together with a class of involutory transformations sets the original moving boundary problem in a wide class with temporal modulation. As an appendix, reciprocally associated exactly solvable moving boundary problems are derived.

A novel extension of the canonical solitonic mKdV equation is introduced which admits hybrid Ermakov-Painlevé II symmetry reduction. Application of the latter is made to obtain exact solution of Airy-type to a class of moving boundary problems of Stefan kind for this extended mKdV equation. A reciprocal transformation is then applied to the latter to generate an associated exactly solvable class of moving boundary problems for an extension of a base Casimir member of a compacton hierachy. The extended mKdV equation is shown to be embedded in a range of nonlinear evolution equations with temporal modulation as determined via the action of a class of involutory transformations with origin in Ermakov theory. Associated temporal modulation for the hybrid mKdV and KdV equation as embedded in the classical solitonic Gardner equation is delimited.

- A seed class of exact solutions of the Euler system of two-dimensional relativistic gasdynamics is derived with an underlying Chaplygin-K & aacute;rm & aacute;n-Tsien type constitutive law. The invariance of the nonlinear relativistic system under multi-parameter reciprocal B & auml;cklund transformations is applied to generate a wide class of novel associated solutions with a barotropic relation. The latter reduces to the standard Chaplygin-K & aacute;rm & aacute;n-Tsien gas law in the non-relativistic limit. An additional invariance of the general relativistic gasdynamics system is established under a substitution principle. The latter represents an extension of a classical result in non-relativistic gasdynamics.

A review is presented here of research to date on the application of model parameter-dependent constitutive laws for which capillarity systems admit underlying solitonic structure with their characteristic key properties such as invariance under Bu00E4cklund transformations and admittance of Painlevu00E9 reduction. The classical Korteweg capillarity system and its extensions are considered. Reductions to the canonical solitonic nonlinear Schru00F6dinger and its resonant nonlinear Schrodinger equation extension containing a de Broglie-Bohm potential are exhibited in turn for certain model constitutive relations. A capillarity analogue of the classical Ku00E1rmu00E1n-Tsien model law of gasdynamics is shown to have a key role in such canonical reductions. A novel geometric link between a Korteweg capillarity system and the classical Da Rios system of hydrodynamics is recorded. Invariance of capillarity systems under multi-parameter Bu00E4cklund transformations is detailed and applied. Gausson and q-gaussion phenomena in certain capillarity systems is described with concomitant classes of exact solutions. A Lagrangian encapsulation of a Korteweg capillarity system is presented whereby reduction is made to the canonical Boussinesq equation.